Facoltà di Scienze Matematiche, Fisiche e Naturali Corso di Laurea Specialistica in Matematica

Candidate Domenico Monaco ID number 1151621

Thesis Advisor

Co-Advisor

Prof. Gianluca Panati

Prof. Domenico Fiorenza

Academic Year 2010/2011

Thesis defended on July 19th, 2011 in front of a Board of Examiners composed by:

Domenico Monaco. A geometric approach to the decay of Wannier functions in graphene. Master thesis. Sapienza – University of Rome © 2011 VERSION : EMAIL :

July 5, 2011

[email protected]

Contents Introduction

1

1 Bloch-Floquet theory for Schrödinger operators 1.1 The Bloch-Floquet-Zak transform and its properties . . . . . . . . . . . . . 1.1.1 The Bloch-Floquet and Bloch-Floquet-Zak transforms and their periodicity properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Isometric properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Bloch-Floquet(-Zak) transform and differentiability . . . . . . . . . 1.1.4 A Paley-Wiener-type theorem . . . . . . . . . . . . . . . . . . . . . . . 1.2 Periodic Schrödinger operators: Bloch functions and Wannier functions . 1.3 Bloch bundles: the geometry behind periodic Schrödinger operators . . .

7 7 8 10 16 18 22 26

2 Decay rate of Wannier functions in graphene 2.1 What is graphene? . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Interlude: space-adiabatic perturbation theory . . . . . . . . 2.3 Conical intersections . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 The canonical Wannier functions and their decay rate 2.3.2 Invariance of the decay rate of Wannier functions . . .

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31 31 34 37 39 44

3 Bloch bundles in graphene 3.1 Avoided crossings . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Berry curvature for avoided crossings . . . . . . . . . . 3.2 Geometric identification of the composite Bloch bundle . . . 3.2.1 Detecting conical intersections: a geometric criterion 3.3 Berry curvature for conical intersections . . . . . . . . . . . .

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47 47 50 55 61 67

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71 71 72 75 79 81 86 89 92

A Fibre bundles and Chern classes A.1 Fibre bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1.1 Definition and first examples of fibre bundles . . . . . A.1.2 Principal bundles and universal bundles . . . . . . . . A.2 Chern classes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2.1 . . . of holomorphic vector bundles . . . . . . . . . . . . A.2.2 . . . of smooth vector bundles . . . . . . . . . . . . . . . A.2.3 . . . of topological vector bundles . . . . . . . . . . . . . A.2.4 Equivalence of the definitions and general properties Bibliography

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93

iii

iv

Introduction La géométrie n’est pas vraie, elle est avantageuse. H ENRI P OINCARÉ

This thesis aims to address the study of a problem arising from solid-state physics by using tools from differential geometry. The crystal structure of some solids may be described using a periodicity lattice Γ. If we think of the crystal as an infinitely-extended object, covering the whole space Rd , the lattice Γ is a subset identifiable with Zd , and it describes the position of the ionic cores of the crystal. The motion of an electron in the solid, whose modelisation can be useful, for example, in order to study conduction properties of the crystal, will be influenced by the electromagnetic field of the nuclei situated at the points of Γ and by their corresponding electronic cloud. Hence, it is physically reasonable to expect that the electron is subject to a potential V : Rd → R which is periodic with respect to translations induced by the lattice Γ, i.e. that V (x + γ) = V (x) for all x ∈ Rd and γ ∈ Γ. The Hamiltonian governing the motion of the above-mentioned electron, in a suitable system of units, will be of the form H = −∆ + VΓ

(I.1)

where ∆ is the Laplace operator in d dimensions, while VΓ is the operator multiplying wave functions times the potential V . The Hamiltonian H , defined on a suitable dense subspace DH ⊂ L 2 (Rd ), belongs to a class of operators known as periodic Schrödinger operators. Chapter 1 deals with the study of these operators. For this purpose, it is natural to try to find a basis of L 2 (Rd ) (what a physicist would call a “representation”) in which they assume a “diagonal” form. If we could concentrate only on the “discrete” part of the space Rd , namely on the crystal lattice Γ, we would know that analogous operators have a particularly simple form if one decomposes wave functions in Fourier series; as it is well known, it defines a unitary operator F : `2 (Γ) → L 2 (T∗ ), where T∗ is the torus representing the dual group of Γ. To “fill the gaps” between the points of the lattice Γ, we choose a fundamental cell W as follows: If Γ is the lattice given by integer multiples of the basis E = {e1 , . . . , ed } in Rd , we define W to be the set of points in Rd having all their coordinates with respect to the basis E between −1/2 and 1/2. This choice allows us to identify Rd , as a measure space, with the cartesian product Γ×W, simply filling all space by integer translation (i.e. the ones induced by vectors in Γ) of the fundamental cell W. 1

2

Introduction

This identification allows us, on its turn, to define isomorphisms L 2 (Rd ) ' L 2 (Γ × W) ' `2 (Γ) ⊗ L 2 (W). Thus we realize that, in order to study periodic Schrödinger operators, it may be convenient to us the unitary

B := F ⊗ 1 : L 2 (Rd ) ' `2 (Γ) ⊗ L 2 (W) → L 2 (T∗ ) ⊗ L 2 (W) ' L 2 (T∗ ; L 2 (W)) instead of the Fourier series. The operator B is called the Bloch-Floquet transform. Explicitly, if w is a function in L 2 (Rd ) which is sufficiently regular (say infinitely many times differentiable with compact support, or Schwartz class), then the transform Bw is the function X −ik·γ ˆ d , x ∈ Rd . e w(x + γ), k ∈ R (Bw) (k, x) = γ∈Γ

ˆ d and, as such, it can be thought as a It is periodic with respect to the dual lattice Γ∗ ⊂ R ˆ d /Γ∗ . function defined on the torus T∗ = R This last remark allows us to think of the space L 2 (T∗ ; L 2 (W)) of L 2 functions on the torus T∗ with values in the Hilbert space L 2 (W) as the direct integral of Hilbert spaces Z

⊕ T∗

where

d[k] Hk ,

n

o

Hk := v ∈ L 2loc (Rd ) : v(x + ω) = eik·ω v(x), for all x ∈ Rd and ω ∈ Γ . The “direct integral” notion, borrowed from analysis, can be reformulated in geometric terms as the space of locally-L 2 sections of a fibre bundle, whose base space is the torus T∗ and having the Hilbert space Hk as the fibre over the point1 [k] ∈ T∗ . Consequently, when we pass from the operator H , defined on position space Rd , to the operˆ d or better still on the torus T∗ , the latter ator B H B−1 , defined on momentum space R will be a fibred operator Z

B H B−1 =

T∗

d[k] H per ([k])

where H per ([k]) is the operator −∆ + VΓ acting on a suitable domain in the space Hk . This construction thus leads, as we set out, to decompose the original Hamiltonian into simpler operators; the main disadvantage, however, is that these act on different Hilbert spaces, depending on a parameter [k] ∈ T∗ . It would be much more convenient to be able to identify all the spaces Hk with just one fibre space Hf . This may be achieved by choosing Hf to be the fibre on the point k = 0, which is given simply by periodic function L 2 (TdW ) (where TdW is the torus Rd /Γ). One then defines the unitary operator J ([k]) : Hk → Hf ,

(J ([k])v) (x) = e−ik·x v(x), x ∈ Rd .

If J denotes the fibred operator whose fibre over [k] ∈ T∗ is the unitary J ([k]), then J goes from L 2 (T∗ ; L 2 (W)) to L 2 (B; Hf ), where B is the fundamental cell relative to the lattice Γ∗ defined by the same construction used for W. Also the space L 2 (B; Hf ) can be identified 1 The symbol [k] denotes the equivalence class of k ∈ R ˆ d modulo the subgroup Γ∗ .

Introduction

3

with a “fibred” Hilbert space: it is the space of locally-L 2 section of a fibre bundle defined on B with fibre Hf on each of its points, but satisfying the further condition φ(k − λ) = τ(λ)φ(k) where, for ψ ∈ Hf and λ ∈ Γ∗ , ¡

¢ τ(λ)ψ ([x]) := eiλ·x ψ([x]),

x ∈ Rd .

The transform which will be of interest to us will then be the Bloch-Floquet-Zak transform Z = JB, given explicitly on a sufficiently smooth function w by (Zw)(k, x) =

X γ∈Γ

e−ik·(x+γ) w(x + γ).

One checks that

ZH Z

−1

Z =

⊕ B

dk Hper (k),

where Hper (k) = (−i∇ + k)2 + VΓ

and hence Hper (k) acts on a suitable Sobolev space in Hf which is independent from k ∈ B. Spectral properties of the operators Hper (k) are well known: they have pure point spectrum accumulating at infinity. If E n (k) denotes the n-th eigenvalue (labelled in inˆ d 7→ E n (k) is called n-th Bloch band. creasing order), then the periodic function k ∈ R ˆ d 7→ φ(k) satisfying Eigenfunctions of the operator Z H Z−1 , namely those functions k ∈ R Hper (k)φ(k) = E n (k)φ(k) are called Bloch functions; their Bloch-Floquet-Zak antitransforms w = Z−1 φ ∈ L 2 (Rd ) are called Wannier functions. An interesting problem in solid-state physics is the study of localisation properties (or rather of the rate of decay at infinity) of the latter, as they represent the stationary solutions of the Schrödinger equation generated by the periodic Hamiltonian H = −∆ + VΓ . As one should expect, the Bloch-Floquet and BlochFloquet-Zak transforms share the same properties of the Fourier series, and in particular they “transform” multiplication operators times the coordinates in position space into derivatives in momentum space. Thus, a way to approach the localisation properties of Wannier functions is that of establishing the smoothness of the corresponding Bloch functions. For example, infinitely many times differentiable Bloch functions produce Wannier functions decaying at infinity faster than any polynomial, while analytic Bloch functions produce exponentially localised Wannier functions. There is a geometric reformulation of the above-mentioned problem that, from my point of view, can facilitate the search for a solution and help in giving the correct interpretation for some important phenomenological aspects, as for example the conductor or insulator nature of the solid under study, or quantum Hall effect. In fact, one can define a vector bundle P, called the Bloch bundle, on the torus T∗ , whose fibre over the point [k] ∈ T∗ is the (finite-dimensional) subspace of Hf spanned by Bloch functions associated to a certain number of relevant Bloch bands, separeted by the rest of the spectrum of the operator Hper (k) by a gap. If P (k) denotes the projection on this subspace, then its regularity with respect to the variable [k] ∈ T∗ is the same as the one of the Bloch functions; but in addition to this the triviality of the Bloch bundle P allows

4

Introduction

for example to establish the existence of a family of orthonormal bases for the subspace Ran P (k), for varying [k] ∈ T∗ (i.e. a frame for P), such that they “glue” along the boundary of the fundamental cell B and of its translations. Sufficient conditions for the existence of exponentially localised Wannier functions, for example, have been found by Gianluca Panati [20], following this approach, i.e. proving that the Bloch bundle is trivial in the category of analytic vector bundles. These hypotheses are the presence of timereversal symmetry and the low dimensionality of the base torus (d ≤ 3), independently of the dimension of the subspace Ran P (k). This result hence improves more constructive approaches like the ones of Georghe Nenciu, that apply only to the study of a single non-degenerate Bloch band (namely dim Ran P (k) = 1), or to the one-dimensional case (d = 1). In Chapter 2 we come to the heart of the thesis, with the first original contributions. The general theory of periodic Schrödinger operators and Bloch bundles is applied to a model case, that of graphene. This crystal is entirely composed of carbon atoms, and has attracted the attention of the scientific community especially since the team guided by Andre Geim and Konstantin Novoselov invented a procedure to isolate single layers, just one-atom thick; thus it can be considered a bidimensional crystal. The study of energy levels of graphene has revealed the presence of the so-called Dirac points, in which two Bloch bands cross. These intersections are called conical crossings, as in the proximity of a Dirac point k 0 the Bloch bands can be approximated as E ± (k) ≈ ±v F |k − k 0 | , and thus they describe a cone centered at k 0 . In order to elaborate a simple effective model to analyse conical intersections, we will exploit the analogy between the time evolution generated by periodic Schrödinger Hamiltonians and molecular dynamics. In the study of a molecule, it is possible to separate the dynamics of “slow” degrees of freedom (describing for example the position of nuclei) from the one of “fast” degrees of freedom (the motion of electrons), through an algorhitm known as adiabatic perturbation theory. The scale separation is measured by a parameter ε, equal to the square root of the ratio between the electron mass and the mass of a nucleus. One works in the Born-Oppenheimer approximation: roughly speaking, the nuclei move in an effective potential generated by an energy level of electrons, while the latter instantaneuosly adapt to the momentary configuration of the nuclei. The introduction of this effective potential, arising from a matrix-valued function H on the configuration space (q, p) via the substution q → x and p → iε∇ (the ε-Weyl symbol), allows to reduce drastically the number of degrees of freedom of the problem. One can thus describe the molecular system through a finite-dimensional model, whose dimension is equal to the number of relevant energy bands, if they are separated from the rest of the spectrum by a gap. The same procedure can be adapted to the study of electronic properties of crystal: rather than separate “fast” and “slow” degrees of freedom, one distinguishes between dynamics with short range (occurring in the fundamental cell W) and with long range (described by the periodicity lattice Γ). Applying this procedure to the case of graphene, we obtain a model for conical intersections given by an effective Hamiltonian represented by the 2 × 2 matrix µ

q1 Heff (q) = q2

q2 −q 1

¶

Introduction

5

where q = k − k 0 , if k 0 is the Dirac point under examination. The eigenvalues of this matrix are exactly the energy bands E ± (q) = ±|q|. The validity of this model is local, i.e. we assume that |q| is sufficiently small, say less than R. The eigenvectors of Heff (q) may be easily computed: these are µ ¶ µ ¶ cos(ϕ/2) − sin(ϕ/2) φ+ (q) = eiϕ/2 e φ− (q) = eiϕ/2 , (I.2) sin(ϕ/2) cos(ϕ/2) (ϕ denotes the angular polar coordinate of q). We will call them the canonic Bloch functions for the conical intersection. Notice that these function are singular at the Dirac point q = 0. A problem from solid-state physics which was still open required an estimate for the decay rate of Wannier functions associated to conical intersections. In this thesis, these functions (or at least the part determining their decay at infinity) are explicitly computed, starting from the definition of the Bloch-Floquet-Zak antitransform. In particular, we establish that these functions w can satisfy ¡ ¢ |w can (x)| = O |x|−3/2 . This uniform estimate has an interesting consequence in terms of the decay of the L 2 norm of the Wannier functions: in particular, the function W (x) := |x|1/2 w can (x) is not in L 2 (R2 ). Consequently, we deduce that the expectation value of the position observable |X | on the canonic Wannier functions w can , which is equal to the L 2 norm of the function W , is infinite: the conducting electrons in graphene are completely delocalised. In conclusion, the last Section of this second Chapter is devoted to show that, under mild assumption of regularity, multiplication of the Bloch function by a phase factor (or equivalently a gauge transformation) doesn’t change the rate of decay of the corresponding Wannier function. As we have already observed, the Bloch bundle associated to a conical intersection is singular at the Dirac point. This would seem to preclude the use of differentialgeometric or topological methods (that apply only to vector bundles which are at least continuous) to tackle the question of the existence of global periodic Bloch functions. In order to avoid problems arising from this singularity, we study in Chapter 3 the case of avoided crossings. Adding a “perturbative” parameter to the effective Hamiltonian Heff (q), the latter becomes µ ¶ q1 q 2 + iµ µ Heff (q) = , µ ∈ [−µ0 , µ0 ]. q 2 − iµ −q 1 p µ µ The eigenvalues of Heff (q) are E ± (q) = ± |q|2 + µ2 : energy bands do not cross anymore at the point q = 0. Consequently, we have two families of rank-1 Bloch bundles, labelled by the plus or minus sign of the energy bands and parametrised by µ ∈ [−µ0 , µ0 ]; for µ 6= 0 this are C ∞ bundles. However, it is convenient to “absorb” the parameter µ in the base space of these bundles: we thus obtain the composite Bloch bundles Pcomp,± , having the punctured cylinder C• := B R (0) × [−µ0 , µ0 ] \ {(0, 0, 0)} as their base space. As C• is homotopically equivalent to the sphere S 2 , the relevant geometric information on the composite Bloch bundle is contained in its restriction to the cylindric surface C := ∂C. We have hence found a way to avoid the singularity of the Bloch bundle corresponding to µ = 0 at the Dirac point, without losing the geometric information it carries. In particular, a simple calculation shows that the integral of the first Chern class of the bundle

6

Introduction

Pcomp,± on the cylindric surface C is equal to ±1: the bundle is thus non-trivial. As one should expect, this “non-triviality” is somewhat concentrated at the Dirac point: in fact, it is possible to show that the differential forms defining the Chern classes of the Bloch bundles for avoided crossings converge in the sense of distributions to a Dirac delta at q = 0 when the parameter µ tends to 0. One can go into the geometric interpretation of the composite Bloch bundle even further. In fact, consider the map u ± : C• → P1C ,

µ

µ

u ± (q) := Ran P ± (q)

µ

µ

where P ± (q) is the projection on the eigenspace of the effective Hamiltonian Heff (q) relµ ative to the eigenvalue E ± (q). We will show in the second Section of this third Chapter that the bundle Pcomp,± is the pullback of the tautological bundle over the complex projective line P1C via the map u ± . This allows us to come to the main original result of this thesis, namely to a geometric criterion to detect the presence of conical intersections. In fact, assume that we are given a smooth bundle Lcomp of rank 1 having as its base space the cartesian product T2 × [−µ0 , µ0 ], with the point ([k 0 ] , 0) excluded. This may be obtained from a reference bundle L0 , defined on T2 \ {[k 0 ]}, and “perturbed” by the smoothing parameter µ; this perturbation may for example arise from the discretisation of the Schrödinger problem to study its solutions numerically. Consider the map u that sends the point ([k], µ) ∈ T2 × [−µ0 , µ0 ], ([k], µ) 6= ([k 0 ] , 0), to the element of P1C given by the fibre of Lcomp in (k, µ). We can compute n=−

1 2π

Ï C

u ∗ ωP1 ∈ Z C

where C is the cylindric surface of radius R and height 2µ0 centered at ([k 0 ] , 0). Then, if n = 0 the bundle L0 can be extended to a necessarily trivial bundle on the whole T2 . If instead n = ±1, then the Bloch function can be “interpolated” with the canonic Bloch function φ± , defined in (I.2), inside the cylinder whose external boundary is C.

Chapter 1

Bloch-Floquet theory for Schrödinger operators Yes, I know you [Max Born] are fond of tedious and complicated formalism. You are only going to spoil Heisenberg’s physical ideas by your futile mathematics. W OLFGANG PAULI

This Chapter is devoted to the study of periodic Schrödinger operators. The BlochFloquet-Zak formalism used to “diagonalise” such operators is illustrated in the following section. We also introduce a geometric counterpart for the analysis of these operators which will play the main rôle in the remainder of this thesis.

1.1 The Bloch-Floquet-Zak transform and its properties We establish the basic notations needed to state the purpose of this thesis. Suppose fixed a basis E = {e1 , . . . , ed } in Rd (the position space). The dual basis E∗ = © ∗ ª ˆ d (the momentum space) is defined by the relations e1 , . . . , e∗d in R ( e∗j · ek

= 2πδ j k =

2π if j = k, 0

if j 6= k.

(1.1)

We let Γ := Ze1 ⊕ · · · ⊕ Zed ⊂ Rd be the lattice1 spanned by E, and ˆd Γ∗ := Ze∗1 ⊕ · · · ⊕ Ze∗d ⊂ R be its dual lattice. The Weigner-Seitz cell is ½ ¾ ¯ ® ¯ 1 W := x ∈ Rd : ¯ x, e j Rd ¯ ≤ for all j = 1, . . . , d 2 1 I.e., a discrete additive subgroup of Rd of maximal dimension: these are known to be isomorphic to Zd .

7

8

1. Bloch-Floquet theory for Schrödinger operators

while the (first) Brillouin zone is ½ ¾ ¯D E ¯ 1 ¯ ¯ d ∗ ˆ : ¯ k, e B := k ∈ R ¯ ≤ for all j = 1, . . . , d . j R ˆd 2 These cells will be chosen as sets of representantives for the tori TdW = Rd /Γ,

ˆ d /Γ∗ . T∗ = R

1.1.1 The Bloch-Floquet and Bloch-Floquet-Zak transforms and their periodicity properties Definition 1. Let w ∈ C 0∞ (Rd ) be a smooth function with compact support. Its BlochFloquet transform is defined to be (Bw)(k, x) :=

X γ∈Γ

ˆ d , x ∈ Rd . k ∈R

e−ik·γ w(x + γ),

(1.2)

Its Bloch-Floquet-Zak transform is defined to be (Zw)(k, x) :=

X γ∈Γ

e−ik·(x+γ) w(x + γ) = e−ik·x (Bw)(k, x),

ˆ d , x ∈ Rd . k ∈R

(1.3)

Both series in (1.2) and (1.3) are well-defined functions – they are actually finite sums because w has compact support. Notice that for all ω ∈ Γ and all λ ∈ Γ∗ (Bw)(k, x + ω) =

X γ∈Γ

e−ik·γ w(x + ω + γ) =

X γ0 =γ+ω∈Γ

0

e−ik·(γ −ω) w(x + γ0 ) =

=e (Bw)(k, x), X −i(k+λ)·γ X −iλ·γ −ik·γ (Bw)(k + λ, x) = e w(x + γ) = e e w(x + γ) = ik·ω

γ∈Γ

(1.4a)

γ∈Γ

= (Bw)(k, x)

(1.4b)

because for all γ ∈ Γ one has λ · γ ∈ 2πZ by definition of the dual lattice Γ∗ , and hence e−iλ·γ = 1. Thus we deduce that the function x 7→ (Bw)(k, x) is Γ-quasiperiodic while k 7→ (Bw)(k, x) is Γ∗ -periodic. On the other hand (Zw)(k, x + ω) =

X γ∈Γ

e−ik·(x+ω+γ) w(x + ω + γ) =

X γ0 =γ+ω∈Γ

0

e−ik·(x+γ ) w(x + γ0 ) =

= (Zw)(k, x), (1.5a) X −i(k+λ)·(x+γ) X −iλ·γ −ik·(x+γ) −iλ·x (Zw)(k + λ, x) = e w(x + γ) = e e e w(x + γ) = γ∈Γ

=e

−iλ·x

γ∈Γ

(Zw)(k, x)

(1.5b)

so that x 7→ (Zw)(k, x) is Γ-periodic while k 7→ (Bw)(k, x) is Γ∗ -quasiperiodic. For reasons that will become apparent later, it is more convenient to have periodicity in position space, and consequently to work with the Bloch-Floquet-Zak transform. The ˆ d , may thus be considered as a function defined function x 7→ (Zw)(k, x), for fixed k ∈ R on the torus TdW , i.e. as a function [x] ∈ TdW 7→ (Zw)(k, [x])

(1.6)

1.1 The Bloch-Floquet-Zak transform and its properties

9

where [x] denotes the equivalence class2 mod Γ, and x may hence be assumed to be in W. Moreover, due to Γ∗ -quasiperiodicity with respect to the variable k, the knowledge of the function (1.6) for k ∈ B is enough, because if k 0 = k + λ with k ∈ B and λ ∈ Γ∗ (and ˆ d has this form) then every k 0 ∈ R (Zw)(k 0 , [x]) = e−iλ·x (Zw)(k, [x]). Similarly, as the function k → 7 (Bw)(k, x), for fixed x ∈ Rd , is Γ∗ -periodic, it may be viewed as a function [k] ∈ T∗ 7→ (Bw)([k], x), k ∈ B, and again it suffices to know the values of this function only for x ∈ W, as (Bw)([k], x 0 ) = eik·ω (Bw)([k], x) if x 0 = x + ω, ω ∈ Γ. We will return to this considerations after we find the correct function space to set our problem. Definition 2. Let φ ∈ L 1 (T∗ ; L 2 (W)). Its Bloch-Floquet antitransform is defined by (B−1 φ)(x) =

Z T∗

d[k] eik·γ φ([k], y)

(1.7)

where x = y + γ, with y ∈ W and γ ∈ Γ, is the almost-everywhere unique decomposition of x ∈ Rd . Moreover, let φ ∈ L 1 (B; L 2 (TdW )). Its Bloch-Floquet-Zak antitransform is defined by (Z φ)(x) = −1

Z B

dk eik·x φ(k, y)

(1.8)

where x = y + γ, with y ∈ W and γ ∈ Γ, is the almost-everywhere unique decomposition of x ∈ Rd . Remark 1. 1. Here and in what follows, all L p spaces are referred to the Lebesgue ˆ d . The volume of both cells W ⊂ Rd and B ⊂ R ˆ d is then 1. measures in Rd and R d Moreover, these measures induce Lebesgue measures on the tori TW and T∗ , and these have unit volume too. The factor 2π in the definition of the dual lattice Γ∗ (see equation (1.1)), on the other hand, makes the collection of the functions εγ ([k]) := e−ik·γ ,

γ ∈ Γ, [k] ∈ T∗ ,

an orthonormal basis of L 2 (T∗ ). 2. Let H be a Hilbert space and let 1 ≤ p < ∞. The space L p (T∗ ; H) consists of those ˆ d 7→ φ(k) ∈ functions [k] ∈ T∗ 7→ φ([k]) ∈ H (or rather of Γ∗ -periodic functions k ∈ R H) such that µZ ¶ ° ° 1/p ° ° °φ° p ∗ °φ([k])°p < +∞. := d[k] L (T ;H) H T∗

2 The decomposition x = y + γ , where y ∈ W and γ ∈ Γ, is not unique for those x such that y ∈ x x x x x ∂W. However, later on we will be interested only in L 2 functions, so that a decomposition valid “almost everywhere”, like this, will be sufficient.

10

1. Bloch-Floquet theory for Schrödinger operators

If p = 2, then one can define a scalar product (and hence a Hilbert space structure) on L 2 (T∗ ; H) by Z ® ® d[k] φ1 ([k]), φ2 ([k]) H . φ1 , φ2 L 2 (T∗ ;H) = T∗

Similarly, L p (B; H) consists of those functions k ∈ B 7→ φ(k) ∈ H (or rather of Γ∗ ˆ d 7→ φ(k) ∈ H) such that quasiperiodic functions k ∈ R ° ° °φ° p L (B;H) :=

µZ B

° °p dk °φ(k)°H

¶1/p < +∞

and again L 2 (B; H) carries a natural Hilbert space structure induced by the scalar product Z ® ® φ1 , φ2 L 2 (B;H) = dk φ1 (k), φ2 (k) H . B

From these definitions we deduce that if φ is in L 1 (T∗ ; L 2 (W)) then for all γ ∈ Γ the function [k] 7→ eik·γ φ([k], ·) is still in L 1 (T∗ ; L 2 (W)), because the phase eik·γ has unit modulus. Thus the integral defining B−1 in (1.7) makes perfect sense. Clearly the same holds for the definition of Z−1 in Equation (1.8).

1.1.2 Isometric properties The goal of this section is to show that both B and Z extend to unitary operators from the space L 2 (Rd ) to some L 2 space of the kind described in the previous Remark (i.e. for an appropriate choice of the target Hilbert space H in both cases), and that B−1 and Z−1 are their respective inverses, justifying our notation. The approach that we will follow is based on the following abstract Lemma. Lemma 1. Let H1 and H2 be Hilbert spaces, and let D1 ⊂ H1 be a dense subspace. Suppose we have two linear operators U : D1 ⊂ H1 → H2

and V : H2 → H1

satisfying the following hypotheses: (a) U is isometric on D1 , namely kUxkH2 = kxkH1

for all x ∈ D1 ;

(b) V is isometric on H2 , namely kV ykH1 = kykH2

for all y ∈ H2 ;

(c) for all x ∈ D1 and y ∈ H2 one has

x,V y

®

H1

® = Ux, y H2 .

e : H1 → H2 , and Then U extends uniquely to a unitary isomorphism U e = 1H , VU 1

e = 1H . UV 2

(1.9)

1.1 The Bloch-Floquet-Zak transform and its properties

11

Proof. Let D2 ⊂ H2 be the range of U ; we want to prove that D2 is dense in H2 . In order to do so, we show that D⊥ 2 = {0}. Suppose y ∈ H2 is such that

y 0, y

®

H2

= 0 for all y 0 ∈ D2 .

As D2 = U D1 , by our hypothesis (c) this is equivalent to ® ® 0 = U x, y H2 = x,V y H1

for all x ∈ D1

⇐⇒

V y ∈ D⊥ 1.

As D1 is dense in H1 , its orthogonal space in H1 is null: V y = 0. As V is isometric by hypothesis (b), we deduce that 0 = kV ykH1 = kykH2

=⇒

y =0

as we wanted. e : D1 = By the bounded extension principle (Theorem I.7 in [22]) U extends to U H1 → D2 = H2 , which is isometric by hypothesis (a) and surjective: hence, by definie is unitary. Hypothesis (c) then implies tion, U

x,V y

®

H1

® e x, y = U H2

for all x ∈ H1 , y ∈ H2 ;

e † , the adjoint of U e . The equalities in (1.9) are just the ones definthus V coincides with U ing a unitary operator. All we have to do now is prove that the Bloch-Floquet and Bloch-Floquet-Zak transforms, defined respectively in (1.2) and (1.3), together with their antitransforms, defined respectively in (1.7) and (1.8), satisfy hypotheses (a), (b) and (c) of Lemma 13 . We also have to determine the correct Hilbert spaces H1 , H2 and the dense subset D1 ⊂ H1 on which these transforms define isometries. This is achieved in the following two Theorems. Theorem 1. The Bloch-Floquet transform, defined as in (1.2) for w ∈ C 0∞ (Rd ), extends to a unitary isomorphism B : L 2 (Rd ) → L 2 (T∗ ; L 2 (W)). Its inverse is given by the Bloch-Floquet antitransform

B−1 : L 2 (T∗ ; L 2 (W)) → L 2 (Rd ) defined as in (1.7). Remark 2. We make a few observations before proceeding with the proof of this theorem. 1. The decomposition x = y + γ, with y ∈ W and γ ∈ Γ, allows us to identify Rd with the cartesian product Γ×W as measure spaces (with respect to Lebesgue measures in Rd and W, and with the discrete “counting” measure on Γ). This amounts to the separation of fast and slow variables, corresponding respectively to W and to 3 Their extensions will again be denoted by

1.

B, Z, B−1 and Z−1 , respectively, dropping the e of Lemma

12

1. Bloch-Floquet theory for Schrödinger operators

Γ, in the study of adiabatic dynamics (compare Section 2.2). Thus, we have and induced identification L 2 (Rd ) ' L 2 (Γ × W) ' `2 (Γ) ⊗ L 2 (W). With this in mind, one can observe that the Bloch-Floquet transform, intepreted as in the statement of this Theorem as a unitary operator

B : `2 (Γ) ⊗ L 2 (W) → L 2 (T∗ ) ⊗ L 2 (W) ' L 2 (T∗ ; L 2 (W)), acts as the discrete Fourier transform F : `2 (Γ) → L 2 (T∗ ) on the first factor, and as the identity on the factor L 2 (W). This observation will make the statements and proofs of all results reported in this section and in the following more transparent. 2. In view of equation (1.4a), we can think of the space L 2 (T∗ ; L 2 (W)), appearing in the statement of this Theorem, as the direct integral of Hilbert spaces Z ⊕ HB := d[k]Hk (1.10) T∗

where n

o

Hk := v ∈ L 2loc (Rd ) : v(x + ω) = eik·ω v(x), for all x ∈ Rd , ω ∈ Γ . This point of view will be most convenient if we “visualise” HB as the space of locally square integrable sections of an infinite-dimensional vector bundle over T∗ , whose fibre over the point [k] ∈ T∗ is the space Hk . Proof of Theorem 1. With the notations of Lemma 1, we have set

H1 = L 2 (Rd ), H2 = L 2 (T∗ ; L 2 (W)) and D1 = C 0∞ (Rd ). First we prove that B preserves L 2 -norms, i.e. is isometric (hypothesis (a) in Lemma 1). Let w ∈ C 0∞ (Rd ); for γ ∈ Γ, we denote by Tγ w the function ¡ ¢ Tγ w (x) := w(x + γ), x ∈ Rd . We have kBwk2L 2 (T∗ ;L 2 (W))

Z =

T∗

Z =

T∗

d[k] k(Bw)([k], ·)k2L 2 (W) = + * X −ik·γ X −ik·ω d[k] e Tγ w, e Tω w γ∈Γ

Z = = = =

d[k]

X Z γ,ω∈Γ W

X Z γ,ω∈Γ W

W

dy

!Ã X γ∈Γ

e

ik·γ

w(y + γ)

dy w(y + γ)w(y + ω)

Z T∗

! X

e

−ik·ω

ω∈Γ

γ∈Γ W−γ

Z dy w(y)w(y) =

= kwk2L 2 (Rd ) .

Rd

w(y + ω) =

d[k] eik·(γ−ω) =

dy w(y + γ)w(y + ω)δγ,ω =

XZ

=

L 2 (W)

Ã

Z T∗

ω∈Γ

dx |w(x)|2 =

1.1 The Bloch-Floquet-Zak transform and its properties

13

We used the fact that, as w is smooth and has compact support, the series are all finite sums, and we may perform all integrals in the order we prefer; moreover, as the trans© ª lated cells W − γ = y − γ, y ∈ W are “almost-everywhere disjoint” (meaning that their intersections are of Lebesgue measure zero) and they cover all Rd , integration over all position space may be performed by integrating at first on all the cells and then adding up all these contributions. Next we show that also B−1 defines an isometry (hypothesis (b) in Lemma 1): this will also guarantee that B−1 maps4 L 2 (T∗ ; L 2 (W)) onto L 2 (Rd ). Let φ ∈ L 2 (T∗ ; L 2 (W)): then Z XZ ° −1 °2 ¯ −1 ¯ ¯ ¯2 °B φ° 2 d = ¯(B φ)(x)¯2 = dx dy ¯(B−1 φ)(y + γ)¯ = L (R ) Rd

γ∈Γ W

¯2 ¯ = d[k] e φ([k], y)¯¯ = ∗ T γ∈Γ W Z ¯2 Z X ¯¯ ® ¯ = dy dy kφ(·, y)k2L 2 (T∗ ) = ¯ εγ , φ(·, y) L 2 (T∗ ) ¯ = XZ

W

=

Z

ZW T

ik·γ

W

γ∈Γ

Z =

¯Z ¯ dy ¯¯

dy

T∗

d[k] |φ([k], y)|2 =

d[k] kφ([k], ·)k2L 2 (W) ∗

Z

Z T∗

d[k]

W

dy |φ([k], y)|2 =

= kφk2L 2 (T∗ ;L 2 (W)) .

(1.11)

To exchange the sum and the integral on the second and third line, we argue as follows. The sum over Γ may be thought of as an integration over Γ, when the latter is endowed with the discrete measure. Hence this exchange is a change in the order of integration, and may be performed if the following hypothesis of Fubini theorem is satisfied: The ¯ ¯2 ® ¯ ¯ integrand ¯ εγ , φ(·, y) L 2 (T∗ ) ¯ (which is a positive function) is a measurable function of ¡ ¢ γ, y ∈ Γ × W ' Rd (see the first item in the above Remark 2). In order to check this, it suffices to show that the integrand can be estimated from the above with a function in L 1 (Rd ) ' L 1 (Γ × W). Now, Cauchy-Schwartz inequality gives ¯ ¯2 ° ° ° °2 ° °2 ® ¯ ¯ 2 ¯ εγ , φ(·, y) L 2 (T∗ ) ¯ ≤ °εγ °L 2 (T∗ ) °φ(·, y)°L 2 (T∗ ) = °φ(·, y)°L 2 (T∗ ) because the εγ ’s are normalised functions (actually an orthonormal basis) in L 2 (T∗ ). To see that the right-hand side of the above inequality defines a function in L 1 (W), we notice that Z Z Z ° °2 ° ° dy φ(·, y) L 2 (T∗ ) = dy d[k] |φ([k], y)|2 = W W T∗ Z Z = d[k] dy |φ([k], y)|2 = kφk2L 2 (T∗ ;L 2 (W)) T∗

W

where we changed the order of integration like we did in the second-to-last line in (1.11). This operation is again legit by Fubini theorem because the function ([k], y) ∈ T∗ × W 7→ ¯ ¯ ¯φ([k], y)¯2 is positive and measurable. All that remains to prove is hypothesis (c) of Lemma 1, namely that B−1 is the formal adjoint of B, at least on the dense subspace C 0∞ (Rd ) ⊂ L 2 (Rd ). Thus, let w ∈ C 0∞ (Rd ) and 4 Remark that L 2 (T∗ ; L 2 (W)) is a subspace of L 1 (T∗ ; L 2 (W)), where

T∗ has finite measure (in particular equal to 1).

B−1 was originally defined, because

14

1. Bloch-Floquet theory for Schrödinger operators

φ ∈ L 2 (T∗ ; L 2 (W)). Then

® w, B φ L 2 (Rd ) = −1

Z

¡ ¢ dxw(x) B−1 φ (x) = Z XZ d[k]eik·γ φ([k], y) = = dy w(y + γ) Rd

T∗

γ∈Γ W

=

XZ γ∈Γ W

Z dy

Z =

d[k]e−ik·γ w(y + γ)φ([k], y) =

Z T∗

Z

T∗

d[k]

W

dy

X γ∈Γ

e−ik·γ w(y + γ)φ([k], y) =

d[k] (Bw) ([k]), φ([k]) T∗ ® = Bw, φ L 2 (T∗ ;L 2 (W)) . =

®

L 2 (W)

=

Again, we may bring the series over Γ inside the integral because it is actually a finite sum, as w has compact support. Moreover, we may use Fubini theorem to change the order of the integrals between the third line and the fourth line because Ã ! ¯ ¯ ¯ ¯ ¯ ¯ −ik·γ w(y + γ)φ([k], y)¯ ≤ sup |w(x)| ¯φ([k], y)¯ ¯e x∈W+γ

and the right-hand side of the above estimate defines a measurable function of ([k], y) ∈ T∗ × W (the supremum is finite because w ∈ C 0∞ (Rd ) is continuous over the compact set W + γ, for all γ ∈ Γ). An analogous result holds for the Bloch-Floquet-Zak transform. Theorem 2. The Bloch-Floquet-Zak transform, defined as in (1.3) for w ∈ C 0∞ (Rd ), extends to a unitary isomorphism

Z : L 2 (Rd ) → L 2 (B; L 2 (TdW )). Its inverse is given by the Bloch-Floquet-Zak antitransform

Z−1 : L 2 (B; L 2 (TdW )) → L 2 (Rd ) defined as in (1.8). Remark 3. As for the previous Theorem, we give an alternative description of the space L 2 (B; L 2 (TdW )), which is suggested by Equation (1.5b). Let Λ be the group of lattice translations on Γ∗ (which can be naturally identified with Γ∗ itself ) and define, for λ ∈ Λ and ψ ∈ L 2 (TdW ) ¡ ¢ τ(λ)ψ ([x]) := eiλ·x ψ([x]), x ∈ Rd . (1.12) Then τ is a unitary representation of the group Λ on the space L 2 (TdW ). Define n

o

Hτ := φ ∈ L 2loc (Rˆ d ; L 2 (TdW )) : φ(k − λ) = τ(λ)φ(k), for all λ ∈ Λ .

(1.13)

There is a natural isomorphism between Hτ and L 2 (B; L 2 (TdW )): in one sense it is restricˆ d to B, in the other sense it consists in τ-equivariant continuation – see the tion from R

1.1 The Bloch-Floquet-Zak transform and its properties

15

above discussion on the (quasi)periodicity properties of the Bloch-Floquet-Zak transform. We want to compare the two identifications Z ⊕ 2 ∗ 2 L (T ; L (W)) ' HB = d[k]Hk and L 2 (B; L 2 (TdW )) ' Hτ T∗

from a geometric point of view. As was already pointed out in the second part of Remark 2, the direct integral appearing above can be viewed as the space of locally L 2 sections of a vector bundle over the torus T∗ : the fibre over the point [k] ∈ T∗ is the infinitedimensional Hilbert space Hk . Thus the fibres of this vector bundle depend explicitly on the point on which they “sit”: we would like to identify all these different space with just one space Hf , or, as a geometer would say, we would like to trivialise the bundle over the largest open set possible5 . This can indeed be done if we choose Hf := H0 = L 2 (TdW ): the fibre Hk is identified with Hf via the unitary isomorphism J ([k]) : Hk → Hf ,

(J ([k])v) (x) = e−ik·x v(x) for all x ∈ Rd .

From the definition of Hk , one sees immediately that for all ω ∈ Γ (J ([k])v) (x + ω) = e−ik·(x+ω) v(x + ω) = e−ik·x e−ik·ω eik·ω v(x) = = e−ik·x v(x) = (J ([k])v) (x) i.e. J ([k])v is a Γ-periodic function, defining an element of Hf . The above definition of J ([k]) makes sense for all k ∈ B◦ , the interior of the Brillouin zone B; thus, over the image on the torus T∗ of these points via the mod Γ projection map, the bundle is trivialised, being isomorphic to B◦ × Hf . The (a priori) non-triviality of the bundle is now all “concentrated” on the boundary ∂B; we have to specify how these fibres are treated when we identify the sides of the Brillouin zone to obtain the torus T∗ . These boundary conditions consist exactly in the τ-covariance expressed in the definition of the Hilbert space Hτ ; so we see that the latter gives an equivalent representation of the bundle. The convenience to have periodicity in position space, i.e. to work with the Bloch-Floquet-Zak representation, lies essentially in the fact that this bundle “looks” trivial on “most” of the torus, having simple (τ-covariant) boundary conditions. Proof of Theorem 2. The second equality in (1.3) stresses the fact that ¡ ¢ Z = JB, where Jφ (k, y) := e−ik·y φ(k, y). The operator J : L 2 (T∗ ; L 2 (W)) → L 2 (B; L 2 (TdW )) is clearly unitary, because it essentially consist in multiplying by a phase factor – its adjoint is given by multiplication by the conjugate phase eik·y . As we already know that B is unitary, we deduce the first statement of the theorem. As for the assertion on the inverse of Z, we compute Z ¡ ¢ −1 −1 −1 (Z φ)(x) = (B J φ)(x) = dk eik·γ J−1 φ (k, y) = B Z Z ik·γ ik·y = dk e e φ(k, y) = dk eik·(y+γ) φ(k, y) = B ZB ik·x = dk e φ(k, y). B

5 This bundle is indeed trivial, because the “infinite unitary group” U (∞), which is the structure group of the bundle, is contractible: however, as for now, we are just trying to give a “mental image” of these abstract objects, so we will not need this powerful result.

16

1. Bloch-Floquet theory for Schrödinger operators

We remark that the identification of L 2 (T∗ ; L 2 (W)) with the direct integral of Hilbert spaces defined in (1.10) allows us to interpret J as a fibred operator Z J= d[k]J ([k]) T∗

where J ([k]) : Hk → Hf is the unitary operator defined in the preceding Remark.

1.1.3 Bloch-Floquet(-Zak) transform and differentiability As was already pointed out in Remark 2, there is a very close analogy between the BlochFloquet(-Zak) transform and the Fourier series expansion. Accordingly, the former will be our main tool for studying Schrödinger operators with periodic coefficients: in the Bloch-Floquet-Zak representation, these will be “diagonal”. As we will soon be dealing with differential operators, we study how the unitaries B and Z behave with respect to derivatives. Let X j be the operator on L 2 (Rd ) defined as ¡

¢ X j w (x) := x j w(x)

for suitable w ∈ L 2 (Rd ). Theorem 3. Let φ ∈ C 1 (B; Hf ) and let w := Z−1 φ ∈ L 2 (Rd ). Then X j w is still in L 2 (Rd ) for all j = 1, . . . , d and ¡ ¢ Z −iX j w = ∂k j φ, i.e. Z−1 ∂k j Z = −iX j . Proof. By (1.8) ³

´

Z−1 ∂k j φ (x) =

Z B

dkeik·x ∂k j φ(k, y)

where y = [x] denotes the equivalence class of x mod Γ. Integration by parts yields ³

´

Z−1 ∂k j φ (x) = −

Z

Z ³ ´ dk∂k j eik·x φ(k, y) = −ix j dkeik·x φ(k, y) = B B ¡ ¢ = −ix j Z−1 φ (x) = −ix j w(x),

where we have not taken into account the boundary terms because these vanish. In fact, the boundary ∂B of the Brillouin zone can be decomposed as ∂B =

d [ j =1

F j+ ∪ F j− ,

¾ ½ 1 where F j± := k = (k 1 , . . . , k d ) ∈ B : k j = ± . 2

The two faces F j+ and F j− are clearly isomorphic to a Brillouin zone B0 of dimension d −1; ¡ ¢ we will denote by k 0 the point k 0 = k 1 , . . . , k j −1 , k j +1 , . . . , k d ∈ B0 . The above decomposition thus gives Z ∂B

dke

ik·x

=

φ(k, y) =

d Z X 0 j =1 B

d Z X

dke

+ j =1 F j ³ ´ 0 1 ∗ 0 i k + 2 e j ·x

dk e

ik·x

φ(k, y) −

Z F j−

dkeik·x φ(k, y) =

³ ´ ¶ Z µ ¶ 0 1 ∗ 1 ∗ 1 ∗ 0 i k − 2 e j ·x 0 φ k + e j , y − dk e φ k − e j , y . (1.14) 2 2 B0

µ

0

1.1 The Bloch-Floquet-Zak transform and its properties

17

The minus sign in front of the integral over F j− is due to the way we run along ∂B – there is a change of orientation from F j+ to F j− . By (1.13) we deduce that µ ¶ µ ¶ ¶ ³ ´ µ 1 ∗ 1 ∗ 1 ∗ 0 0 ∗ ∗ 0 φ k − ej , y = φ k + ej − ej , y = τ ej φ k + ej , y = 2 2 2 ¶ ¶ µ µ ∗ 1 1 ie ·x ie∗j ·y ∗ ∗ 0 0 j φ k + ej , y =e φ k + ej , y = e 2 2 where we also used the fact that γ := x − y ∈ Γ has integer components, so that by (1.1) e

ie∗j ·x

= ei2πx j = ei2π(y j +γ j ) = ei2πy j ei2πγ j = ei2πy j = e

ie∗j ·y

.

In conclusion we have ´ ³ ´ ³ µ ¶ Z ¶ µ Z 0 1 ∗ 0 1 ∗ 1 ∗ 1 ∗ 0 0 i k − 2 e j ·x ie∗j ·x 0 0 i k − 2 e j ·x φ k − ej , y = dk e e φ k + ej , y = dk e 2 2 B0 B0 ³ ´ ¶ µ Z 0 1 ∗ ∗ 1 0 i k − 2 e j +e j ·x ∗ 0 = dk e φ k + ej , y = 2 B0 ³ ´ µ ¶ Z 0 1 ∗ 1 0 i k + 2 e j ·x 0 ∗ dk e = φ k + ej , y 2 B0 so that the two integrals in (1.14) cancel out. Iteration of this Theorem immediately yields the following. Theorem 4. Let J = ( j 1 , . . . , j n ), where each j ` is in {1, . . . , d }, and M = (m 1 , . . . , m n ), where ¡ ¢M ¡ ¢m ¡ ¢m each m ` is in N, be multiindices, and let X J denote the operator X j 1 1 · · · X j n n . ¡ ¢M Let φ ∈ C m (B; Hf ), m ≥ 1, and define w := Z−1 φ ∈ L 2 (Rd ). Then X J w is in L 2 (Rd ) for all multiindices J and M such that m 1 + · · · + m n = m, and

Z −iX J ¡

¢M

w = ∂kMJ φ,

¡ ¢M i.e. Z−1 ∂kMJ Z = −iX J ,

m

m

where ∂kM stands for ∂k 1 · · · ∂k n . J

jn

j1

¡ ¢M In particular, if φ ∈ C (B; Hf ) then X J w is in L 2 (Rd ) for all multiindices J and M . ∞

We now want to give the analogous staments for the Bloch-Floquet transform B. These have a slightly different formulation. Theorem 5. Let φ ∈ C 1 (T∗ ; L 2 (W)) and define w := B−1 φ ∈ L 2 (Rd ). Then ¯ ¯ sup ¯γ¯ kwkL 2 (W−γ) < +∞. γ∈Γ

If φ ∈ C m (T∗ ; L 2 (W)), m ≥ 1, then ¯ ¯m sup ¯γ¯ kwkL 2 (W−γ) < +∞. γ∈Γ

¯ ¯ © ª If φ ∈ C ∞ (T∗ ; L 2 (W)) then kwkL 2 (W−γ) γ∈Γ decays faster than any power of ¯γ¯.

18

1. Bloch-Floquet theory for Schrödinger operators

Proof. Obviously it suffices to prove only the first statement, because the other two follow from its iteration. Let γ ∈ Γ be arbitrary. Clearly we have ¯ ¯ ¯γ¯ kwk

L 2 (W−γ)

¯ ¯ = ¯γ¯

µZ W−γ

¯ ¯2 dy ¯w(y)¯

¶1/2

µZ =

W

¯ ¯2 ¯ ¯2 dy ¯γ¯ ¯w(y + γ)¯ Ã =

d Z X j =1 W

¶1/2 = !1/2

¯ ¯2 ¯ ¯2 dy ¯γ j ¯ ¯w(y + γ)¯

.

Notice that by (1.7) w(y + γ) =

Z T∗

d[k]eik·γ φ([k], y)

so that integration by parts yields γ j w(y + γ) =

Z T∗

d[k]γ j e

ik·γ

Z

³ ´ φ([k], y) = −i d[k]∂k j eik·γ φ([k], y) = T∗ Z ³ ´ = −i d[k]eik·γ ∂k j φ([k], y) = −i B−1 ∂k j φ (y + γ). T∗

The absence of boundary terms is just due to the Γ∗ -periodicity of the function k 7→ eik·γ φ([k], y). We deduce that Ã ¯ ¯ ¯γ¯ kwk

L 2 (W−γ)

=

d Z X j =1 W

¯³ ¯2 ´ ¯ ¯ dy ¯ B−1 ∂k j φ (y + γ)¯

!1/2

Ã

°2 d ° X ° −1 ° = °B ∂k j φ° 2 j =1

L (W−γ)

!1/2 .

As B−1 is unitary, we have ° °2 ° −1 ° °B ∂k j φ° 2

L (W−γ)

≤

°2 X° ° −1 ° °B ∂k j φ° 2

γ0 ∈Γ

L (W−γ0 )

° °2 ° ° = °B−1 ∂k j φ° 2

L (Rd )

° °2 ° ° = °∂k j φ° 2

L (T∗ ;L 2 (W))

.

(1.15) ¯ ¯ We deduce that ¯γ¯ kwkL 2 (W−γ) can be estimated with the Euclidean norm of the vector ° ° ° ° in Rd whose j -th component is °∂k j φ° 2 ∗ 2 , and hence is finite. The thesis follows L (T ;L (W))

from the arbitrarity of γ ∈ Γ and the fact that the right-hand side in (1.15) is independent of γ.

1.1.4 A Paley-Wiener-type theorem We now come to a Paley-Wiener-type theorem: it establishes a link between exponential decay of a function in L 2 (Rd ) and analyticity of its Bloch-Floquet-Zak transform. It will be convenient to look at the latter as an element of the space Hτ , defined in (1.13) and isomorphic to L 2 (B; Hf ). ˆ d the complexification of the momentum space, namely C ˆd ≡ R ˆd ⊕ We denote by C ˆ d . Clearly a function φ ∈ Hτ is real-analytic if and only if it is the restriction to R ˆ d of iR d ˆ ˆd. an analytic function defined in some domain in C which contains the “real axis” R d ˆ ; this is not the case of interest to us. However, this domain may get arbitrarily close to R The domain of analiticity of our function will be a strip ¾ ½ ¯ ¯ a d ¯ ¯ ˆ Ωa := z = (z 1 , . . . , z d ) ∈ C : ℑz j < p , for all j = 1, . . . , d 2π d

1.1 The Bloch-Floquet-Zak transform and its properties

19

for some a > 0. For the same a > 0, let E a be the operator acting on functions L 2 (Rd ) as follows: (E a w) (x) := ea|x| w(x), for suitable w ∈ L 2 (Rd ). Theorem 6. Let n

o

HτC := Φ ∈ L 2loc (Cˆ d ; L 2 (TdW )) : Φ(z − λ) = τ(λ)Φ(z), for all λ ∈ Λ

where Λ is the group of lattice translations on Γ∗ and τ(λ) is defined as in (1.12). Let ˆ d of a function Φ ∈ HτC analytic in the strip Ωa , a > 0. Assume φ ∈ Hτ be the restriction to R moreover that Z dk kΦ(k + ih, ·)k2Hf ≤ C for all z = k + ih ∈ Ωa (1.16) B

where C is a constant, and this estimate is thus uniform in Ωa . Define w := Z−1 φ ∈ L 2 (Rd ). Then E a w is still in L 2 (Rd ). The following proof is mimicked from its analogue for the Fourier transform given in [13, Chap. VI, Sec. 7]. Proof. Define, for z = k + ih ∈ Ωa , φh (k) := Φ(k + ih) ∈ Hτ . Suppose we prove that ¡ ¢ w h (x) := Z−1 φh (x) = w(x)eh·x .

(1.17)

Then, as for all z = k + ih ∈ Ωa one has6 Ã |h · x| ≤ 2π|h||x| = 2π

d X j =1

!1/2 h 2j

Ã |x| < 2π

d X j =1

µ

a p 2π d

¶2 !1/2 |x| = a|x|,

we could deduce that kE a wk2L 2 (Rd ) =

Z Rd

¯ ¯2 dx ¯ea|x| w(x)¯ = sup

Z

z∈Ωa Rd z=k+ih

° °2 ° ° = sup °eh·x w(x)° 2

L (Rd )

z∈Ωa z=k+ih

¯ ¯2 ¯ ¯ dx ¯eh·x w(x)¯ =

= sup kw h k2L 2 (Rd ) = z∈Ωa z=k+ih

°2 ° °2 ° = sup °Z−1 φh °L 2 (Rd ) = sup °φh °L 2 (B;Hf ) z∈Ωa z=k+ih

z∈Ωa z=k+ih

Z = sup

z∈Ωa B z=k+ih

dk kΦ(k + ih, ·)k2Hf ≤ C

by unitarity of Z−1 and our hypothesis (1.16). 6 To justify the presence of a factor 2π in the duality dot product h · x, remember our definition (1.1) of

the dual lattice.

20

1. Bloch-Floquet theory for Schrödinger operators

Thus to prove that E a w is in L 2 (Rd ) we just have to prove (1.17), i.e. explicitly Z B

dkeik·x Φ(k + ih, [x]) = eh·x

Z B

dkeik·x φ(k, [x]) =

Z

dkei(k−ih)·x φ(k, [x]).

B

(1.18)

We consider h fixed once and for all. Without loss of generality, we may assume that the function Φ(z) can be factorised as Φ(z 1 , . . . , z d ) = Φ1 (z 1 ) · · · Φd (z d ) and thus also φ(k 1 , . . . , k d ) = φ1 (k 1 ) · · · φd (k d ). The general case will follow from a density argument7 . Let n o ˆ d : ℜz ∈ B and − h j ≤ ℑz j ≤ 0, for all j = 1, . . . , d . D := z = (z 1 , . . . , z d ) ∈ C The function z 7→ eiz·x Φ(z + ih, [x]) is analytic in D with values in Hf and hence by the residue theorem Z ∂D

dzeiz·x Φ(z + ih, [x]) =

d Z Y j =1 ∂D j

dz j e

iz j e∗j ·x

Φj

³¡

´ ¢ z j + ih j e∗j , [x] = 0

© ª ˆ denotes the projection of D on the complex line z j 0 = 0 for all j 0 6= j – see where D j ⊂ C Figure 1.1, where the boundary ∂D j is drawn with thicker lines. ˆ iR 1/2

−1/2

ˆ R

Dj −h j p −a/2π d Figure 1.1. The region D j 7 The function z → 1 7 Φ(z 1 , z 2 , . . . , z d ) is analytic in the strip

½

ˆ : |ℑz 1 | < z1 ∈ C

a p 2π d

¾

which is the projection of Ωa onto the complex line {z 2 = · · · = z d = 0}. Thus it has a Taylor series expansion Φ(z 1 , z 2 , . . . , z d ) =

∞ X n=0

Φ(n) (z 2 , . . . , z d )z 1n = lim

N X

N →∞ n=0

Φ(n) (z 2 , . . . , z d )Φ(n) 1 (z 1 ),

n where Φ(n) 1 (z 1 ) := z 1 .

Consequently, Φ(z 1 , z 2 , . . . , z d ) is the limit of a sequence of functions, each of which is a sum of other functions that have the variable z 1 factorised. A simple induction on the dimension d leads to the above mentioned factorisation.

1.1 The Bloch-Floquet-Zak transform and its properties

21

Consequently, all factors in the above product vanish. After noticing that e∗j · x = 2πx j , we fix j ∈ {1, . . . , d } and drop the subscript j , as if we were in dimension d = 1. We deduce from the above equation that ¶ µ Z 0 Z 1/2 ¡1 ¢ 0 1 dh 0 ei2π 2 +ih x Φ + ih + ih 0 , [x] − dkei2πkx Φ (k + ih, [x]) − 2 −h −1/2 ¶ µ Z 1/2 Z 0 ¡ ¢ 1 i2π(k−ih)x 0 i2π − 12 +ih 0 x 0 − dke Φ (k + ih − ih, [x]) + dh e Φ − + ih + ih , [x] = 0. 2 −1/2 −h (1.19) We focus our attention on the second and fourth summand. By definition of the space HτC we have, for all h 00 := h + h 0 ∈ [0, h], µ ¶ µ ¶ µ ¶ 1 1 1 Φ − + ih 00 , [x] = Φ + ih 00 − 1, [x] = τ(1)Φ + ih 00 , [x] = 2 2 2 µ µ ¶ ¶ 1 1 = ei2π[x] Φ + ih 00 , [x] = ei2πx Φ j + ih 00 , [x] 2 2 because, as γ := x − [x] is an integer, ei2πx = ei2π([x]+γ) = ei2π[x] ei2πγ = ei2π[x] . Consequently µ µ ¶ Z h ¶ Z 0 ¡ 1 ¢ ¡ 1 ¢ 0 0 1 1 dh 0 ei2π − 2 +ih x Φ − + ih + ih 0 , [x] = dh 0 ei2π − 2 +ih x ei2πx Φ + ih + ih 0 , [x] = 2 2 −h −h µ ¶ Z 0 ¡ 1 ¢ 1 0 i2π − 2 +ih 0 +1 x 0 dh e Φ + ih + ih , [x] = = 2 −h ¶ µ Z 0 ¡1 ¢ 0 1 0 i2π 2 +ih x 0 dh e = Φ + ih + ih , [x] , 2 −h so that the second summand in (1.19) cancels out the fourth. After this cancellation, (1.19) reduces to Z 1/2 Z 1/2 i2πkx dkei2π(k−ih)x Φ (k + ih − ih, [x]) = Φ (k + ih, [x]) = dke −1/2 −1/2 Z 1/2 = dkei2π(k−ih)x φ (k, [x]) −1/2

which is exactly (1.18). To state a similar result which is valid for the Bloch-Floquet transform, first we have ˆ d as to introduce some notation. We embed the torus T∗ in C o n ¯ ¯ ˆ d : ¯z j ¯ = 1 for all j = 1, . . . , d . T∗ := z = (z 1 , . . . , z d ) ∈ C We will need a tubular neighbourhood of this torus, namely o n ¯ ¯ Da := z = (z 1 , . . . , z d ) ∈ Cˆ d : e−a < ¯z j ¯ < ea for all j = 1, . . . , d ,

with a > 0.

We will call Da a complex annulus. Theorem 7. Let φ ∈ L 2 (T∗ ; L 2 (W)) be the restriction to T∗ of a function Φ analytic in the complex annulus Da , a > 0. Define w := B−1 φ ∈ L 2 (Rd ). Then sup eb |γ| kwk 2 < +∞ for all 0 ≤ b < a. γ∈Γ

L (W−γ)

For the proof of this Theorem, we refer the reader to [15, Theorem 2.2.2].

22

1. Bloch-Floquet theory for Schrödinger operators

1.2 Periodic Schrödinger operators: Bloch functions and Wannier functions In this section, we finally state the aim of this thesis. We will be primarly interested in the study of periodic Schrödinger equations, governing the motion of electrons in a crystal. We consider the crystal as an infinitely-extended solid, so that it can be identified with the whole space Rd . The crystal periodic structure is best described via the periodicity lattice Γ ⊂ Rd , so that the Wigner-Seitz cell W represents the typical cell of the crystal. We assume that the potential V = V (x) to which the free electron gas is subjected – for example the Coulomb potential generated by the nuclei and the electronic cloud – is periodic with respect to Γ: V (x + γ) = V (x) for all x ∈ Rd , γ ∈ Γ. We choose units so that all the relevant physical constants (the Planck constant, twice the electron mass and the Fermi velocity, to be defined in Section 2.1) are equal to 1. The Hamiltonian of the Schrödinger equation we mentioned above is thus H = −∆ + VΓ

(1.20)

where ∆ = ∂2x1 + · · · + ∂2xd is the Laplacian in d dimensions and VΓ is the operator defined by (VΓ w) (x) := V (x)w(x). The operator H acts on a suitable subspace DH of L 2 (Rd ), to be determined in a moment, containing C 0∞ (Rd ). This kind of operators are thouroughly studied in [21, Sec. XIII.16] and [15, Sec. 4.5]; we will refer to these books for the proof of most results quoted in this section. In particular, from now on we assume that the potential operator VΓ is ε-bounded with respect to −∆ for all ε > 0, meaning that for fixed ε > 0 there exists κ > 0 such that kVΓ wkL 2 (Rd ) ≤ ε k−∆wkL 2 (Rd ) + κ kwkL 2 (Rd ) , for all w ∈ DH . (1.21) This allows to use Kato-Rellich and Weyl theorems to study the spectrum of the Hamiltonian operator, and to control its self-adjoint extensions. We will now show that the Bloch-Floquet-Zak transform “diagonalises” the operator H = −∆ +VΓ : more precisely, one finds that Z H Z−1 , acting on a subspace of L 2 (B; Hf ), is a fibred operator of the form

ZH Z−1 =

Z

⊕ B

dk Hper (k),

where Hper (k) = (−i∇ + k)2 + VΓ .

(1.22)

The operator ∇ appearing in the above equation has τ-covariant boundary conditions, see Equation (1.13). Under the above assumption (1.21) on VΓ , one gets that the domain of definition of Hper (k) is the subspace W 2,2 (TdW ) ⊂ Hf , the Sobolev space of L 2 functions on TdW whose second-order distributional derivatives are still represented by functions in L 2 (TdW ). This domain of definition is thus independent of k ∈ B; this would not be the case if we used the Bloch-Floquet representation, which is the main reason for modifying it into the Bloch-Floquet-Zak transform. Accordingly, we may take

DH = Z−1

Z

⊕ B

dkW 2,2 (TdW ).

1.2 Periodic Schrödinger operators: Bloch functions and Wannier functions

23

Moreover, each Hper (k) has compact resolvent: its spectrum coincides with the discrete set of its eigenvalues, accumulating at infinity. We label them in increasing order: E 0 (k) ≤ E 1 (k) ≤ · · · and call the function k 7→ E n (k) the n-th Bloch band. σ(Hper O (k)) E 4 (k) .. E 3 (k)

..

..

...

...

. ...

. .. .

. . .......

... ....

....

..

. . .. . . . . . . . ......

....... . . .

....

E 2 (k)

E 1 (k) . .. ....... .. . . . . . . .

/ k ∈B

.....

E 0 (k)

.....

......

. . . . . . . . . . . . . . . . . .....

.... .... . . .

.....

...

Figure 1.2. Bloch bands

We come to the proof of (1.22). Let φ ∈ L 2 (B; Hf ) be such that φ(k) ∈ W 2,2 (TdW ) for all k ∈ B. Clearly we have Z Z ¡ ¡ ¢¢ ¡ ¢ VΓ Z−1 φ (x) = V (x) dkeik·x φ(k, [x]) = dkeik·x V ([x])φ(k, [x]) = Z−1 VΓ φ(k) (x) B

B

because V is Γ-periodic, so that V (x) = V ([x] + γ) = V ([x]). On the other hand Z Z h i ¡ ¢ −∆ Z−1 φ (x) = −∆ dkeik·x φ(k, [x]) = − dk∆ eik·x φ(k, [x]) . B

B

Notice that for all j = 1, . . . , d , h i ∂x j eik·x φ(k, [x]) = ik j eik·x φ(k, [x]) + eik·x ∂x j φ(k, [x]), h i ∂2x j eik·x φ(k, [x]) = −k 2j eik·x φ(k, [x]) + 2ik j eik·x ∂x j φ(k, [x]) + eik·x ∂2x j φ(k, [x]),

so that h i h i d X £ ¤ −∆ eik·x φ(k, [x]) = − ∂2x j eik·x φ(k, [x]) = eik·x |k|2 − 2i (k · ∇) − ∆ φ(k, [x]). j =1

Consequently ¡ ¢ −∆ Z−1 φ (x) =

Z B

¡ ¢ eik·x (−i∇ + k)2 φ(k, [x]) = Z−1 (−i∇ + k)2 φ(k) (x).

24

1. Bloch-Floquet theory for Schrödinger operators

Thus we deduce that ¡

ZH Z−1 φ (k) = Z (−∆ + VΓ ) Z−1 φ (k) = (−i∇ + k)2 + VΓ φ(k) = Hper (k)φ(k) ¢

¢

¡

£

¤

as was to prove. Remark 4. Using the same type of calculations, one can see that

B H B−1 =

Z T∗

d[k] H per ([k]),

where H per ([k]) = −∆ + VΓ .

Here the Laplace operator ∆ appearing in the definition of H per ([k]) acts on a suitable domain in Hk (compare Remark 2). This essentially follows from the fact that, when ¡ ¢ computing ∆B−1 φ (x +γ), the Laplacian acts only on the function φ(·, x) and not on the phase eik·γ which is present in the definition (1.7) of the Bloch-Floquet antitransform. Hence the fibre operators of the decomposition of B H B−1 , as was already noticed, have a simpler expression, but they act on different (i.e. k-dependent) Hilbert spaces. ˆ d ; thus we get fibre The expression we found for Hper (k) makes sense for all k ∈ R Hamiltonian operators on the whole momentum space. However, the knowledge of these operators for k ∈ B will suffice, because they enjoy a covariance property with respect to the unitaries τ(λ): namely Hper (k + λ) = τ(λ)−1 Hper (k) τ(λ).

(1.23)

In fact, using the isomorphism between L 2 (B; Hf ) and the Hilbert space Hτ defined in ˆ d then (1.13), we see that if the map k 7→ φ(k) is such that φ(k) ∈ W 2,2 (TdW ) for all k ∈ R £ ¤ τ(λ)Hper (k + λ)φ(k + λ) = τ(λ) −∆ − 2i(k + λ) · ∇ + |k + λ|2 φ(k + λ) = £ ¤ = −∆ − 2i(k + λ − λ) · ∇ + |k + λ − λ|2 φ(k + λ − λ) = £ ¤ = −∆ − 2ik · ∇ + |k|2 τ(λ)φ(k + λ) = = Hper (λ)τ(λ)φ(k + λ). Equation (1.23) also implies that the spectrum of Hper (k + λ) coincides with the one of Hper (k), because the former is obtained from the latter by the adjoint action of the unitary τ(λ)−1 : ˆ d , λ ∈ Γ∗ and for all n ∈ N. E n (k + λ) = E n (k) for all k ∈ R ˆ d 7→ E n (k) are thus Γ∗ -periodic real-valued functions. The Bloch bands k ∈ R As is usual in operator theory and quantum mechanics, one is interested in the study of eigenvalue problems for the Hamiltonian operator H . We assume that we know how to solve the equations Hper (k)φ(k) = E n (k)φ(k) (1.24) for all k ∈ B and n ∈ N. The functions k 7→ φ(k) are called Bloch functions: thus these are the eigenfunctions of the Hamiltonian, or equivalently the stationary solutions of the associated Schrödinger equation relative to an energy equal to the n-Bloch band, written in the Bloch-Floquet-Zak representation.

1.2 Periodic Schrödinger operators: Bloch functions and Wannier functions

25

As the fibre Hamiltonian operators Hper (k) satisfy the τ-covariance property (1.23), we may glue the φ’s together by τ-equivariant prolongation and obtain Γ∗ -quasiperiodic ˆ d . In fact we have Bloch functions whole R E n (k)φ(k + λ) = E n (k + λ)φ(k + λ) = Hper (k + λ)φ(k + λ) = = τ(λ)−1 Hper (k)τ(λ)φ(k + λ) = τ(λ)−1 Hper (k)φ(k + λ − λ) = = τ(λ)−1 Hper (k)φ(k) = τ(λ)−1 E n (k)φ(k) which implies φ(k + λ) = τ(λ)−1 φ(k),

or equivalently φ(k − λ) = τ(λ)φ(k).

One is naturally led to address the usual questions: How many Bloch functions are there for a fixed energy band; or equivalently, is the Bloch band degenerate? How regular are these Bloch functions? If we restrict our attention to normalised eigenfunctions of the operator Hper (k) (i.e. we require the L 2 -norm of φ(k) to be equal to 1), then the Bloch functions are defined only up to multiplication by a phase factor, that is, a complex number of unit absolute value. This phase, in general, is k-dependent. We may exploit this “gauge” in order to obtain locally smooth Bloch functions8 , i.e. such that k 7→ φ(k) depends smoothly on k, at least away from points in which the Bloch bands cross (compare Figure 1.2); this is because k ∈ B 7→ Hper (k) is smooth in the norm-resolvent sense. The non-trivial question is now: Can we piece these functions together in order to obtain globally smooth Bloch functions? The answer to this question, unfortunately, is in general false: the 2-dimensional model that we will study in Chapter 2, for example, falls into this class. In the next section we will see that this question, however physical in nature, has a geometric reformulation that allows us to see clearly where the “obstruction” for this gluing procedure to succed comes from. ˆ d is the energyBefore getting there, we have to give another definition. The space R momentum space; we would like to “translate” the Bloch-function language to position space. The Bloch-Floquet-Zak transform allows us to do this: thus we may define ¡ ¢ w(x) := Z−1 φ (x), x ∈ Rd . The function w is called the Wannier function associated to the Bloch function φ. Theorems 4 and 6 tell us that smoothness (respectively analiticity) of Bloch functions may be interpreted as localisation properties in L 2 (Rd ) of the corresponding Wannier functions. Remark 5. Bloch functions are such that φ(k) is in the domain W 2,2 (TdW ) of Hper (k) = (−i∇ + k)2 + VΓ for all k ∈ B. If d ≤ 3, this implies that the associated Wannier functions w = Z−1 φ are well-defined and continuous at x = 0. In fact, by the Sobolev embedding theorem (see [1, Thm. 2.10]), the space W 2,2 (TdW ) is contained in C 0 (TdW ) whenever d < 4. By definition Z w(x) =

B

dk eik·x φ(k, [x])

and as integration is performed on a compact set, the function x 7→ w(x) is continuous at least for those x in the interior of W – the function x 7→ [x] has jump discontinuities on ∂W. 8 This may be proved using Nagy’s formula, as will be done in next section.

26

1. Bloch-Floquet theory for Schrödinger operators

1.3 Bloch bundles: the geometry behind periodic Schrödinger operators In order to give a geometric interpretation of Bloch functions, we need to make a simple but crucial observation: they came up when solving the eigenvalue problem (1.24). As basic linear algebra tells us, the eigenvectors of a certain operator relative to a fixed eigenvalue (the energy Bloch band, in the present case) form a subspace of the space where the operator acts, which in our case is the infinite-dimensional Hilbert space Hf = L 2 (TdW ). (We will assume, for the sake of semplicity, that the former subspace is finitedimensional: we don’t want to deal with infinite degeneracy of the nenergy levels.) Thus, o

(n) a choice of (normalised, smooth, quasiperiodic) Bloch functions φ(n) 1 (k), . . . , φm (k) , where m is the dimension of this subspace and the label n reminds us that we are considering the n-th Bloch band E n , is nothing but a choice of a (possibly orthormal) basis in this eigenspace, and thus it is highly non-canonical. We want to find a way to keep track of the vector-space structure of this eigenspace, without having to deal explicitly with one basis. One way to do this is to consider the eigenprojector P n (k) on the eigenspace: in Dirac’s notation, this would be defined by

P n (k) =

¯ ED m ¯ X ¯ (n) ¯ (k) ¯φ j (k) φ(n) ¯. j

j =1

This is a projection in the algebra B(Hf ) of bounded linear operators acting on Hf , meaning that P n (k)† P n (k) = P n (k),

or equivalently P n (k)2 = P n (k) = P n (k)† .

As it is well known, projections in B(Hf ) are in one-to-one correspondence with closed subspaces in Hf : the subspace associated to the projection P is just its range – the space on which it projects! Hence P n (k) encodesn all the relevant geometric features of the o

eigenspace spanned by the Bloch functions φ1(n) (k), . . . , φ(n) m (k) . For the regularity and covariance properties of the Hamiltonian operator H to fall to the projections P n (k), we have to require a gap condition, namely, that the set σ0 (k) of the N physically relevant Bloch bands {E n (k), . . . , E n+N −1 (k)} under study is separated by ¡ ¢ the rest of the spectrum σ Hper (k) of the fibre Hamiltonian operator, in the sense that ¡ ¡ ¢ ¢ dist σ0 (k), σ Hper (k) \ σ0 (k) ≥ g > 0, for all k ∈ B. (1.25) For example, in Figure 1.2 we have σ0 (k) = {E 1 (k), E 2 (k), E 3 (k)}, and the dotted lines show that the gap condition is satisfied. ˆ d 7→ P (k) is a smooth (and, in some If we let P (k) := P n (k)+· · ·+P n+N −1 (k), then k ∈ R cases, also analytic) function, and it is τ-covariant, in the sense that (compare (1.23)) P (k + λ) = τ(λ)−1 P (k)τ(λ),

ˆ d , λ ∈ Γ∗ . for all k ∈ R

(1.26)

ˆ d a vector space Ran P (k) in a smooth In this way, we can associate to each point k ∈ R way; formally, the geometric tool to deal with similar situations is provided by vector bundles (see Appendix A, Section A.2 for their definition and main properties). This is ˆ d × Hf : defined as follows. Consider the following equivalence relation on the set R ¡ ¢ ¡ ¢ k, φ ∼τ k 0 , φ0 if and only if there exists λ ∈ Γ∗ such that k 0 = k + λ and φ0 = τ(λ)−1 φ.

1.3 Bloch bundles: the geometry behind periodic Schrödinger operators

27

¡ ¢ ˆ d × Hf with respect to this equivalence relaThe equivalence class of a couple k, φ ∈ R £ ¤ tion will be denoted by k, φ τ . The vector bundle P associated to the family of projec³ ´ π tions {P (k)}k∈Rˆ d , called the Bloch bundle, is then given by P = E − → T∗ where E :=

n£ ´ o ¤ ³ d ˆ × Hf / ∼τ : φ ∈ Ran P (k) k, φ τ ∈ R

(1.27)

¡£ ¤ ¢ and π k, φ τ = [k] ∈ T∗ is the class of k mod Γ∗ . These definitions of the total space E and of the map π do not depend on the choice of the representatives in the class of ¡ ¢ ˆ d × Hf . In fact, by definition k and k 0 = k + λ define the same class modΓ∗ ; k, φ ∈ R ¡ ¢ ¡ ¢ moreover, if φ ∈ Ran P (k), i.e. if P (k)φ = φ, and if k 0 , φ0 = k + λ, τ(λ)−1 φ , then P (k 0 )φ0 = P (k + λ)τ(λ)−1 φ = τ(λ)P (k)φ = τ(λ)φ = φ0 , by the covariance property (1.26), so that φ0 ∈ Ran P (k 0 ). The typical fibre of this vector bundle P is Cm , where m is the sum of all degeneracies of the Bloch bands E n , . . . , E n+N −1 ; one says that P is a vector bundle of rank m. To check that it is locally trivial, we argue as follows. As by hypothesis k 7→ P (k) is smooth, then it ˆ d there exists an open neighbourhood U ⊂ R ˆd is also continuous, so that for fixed k 0 ∈ R such that kP (k) − P (k 0 )kB(Hf ) < 1 whenever k ∈ U . From this follows at once that ° ° 2 °[P (k) − P (k 0 )]2 ° B(Hf ) ≤ kP (k) − P (k 0 )kB(Hf ) < 1 and hence the operator A(k; k 0 ) := 1 − [P (k) − P (k 0 )]2 is invertible. As it is self-adjoint (because both P (k) and P (k 0 ) are projections), the above inequality assures us that it is also positive9 . Hence we may define (this is Nagy’s formula) W (k; k 0 ) := A(k; k 0 )−1/2 {P (k)P (k 0 ) − [1 − P (k)] [1 − P (k 0 )]} . Notice that P (k)P (k 0 ) − [1 − P (k)] [1 − P (k 0 )] = P (k)P (k 0 ) − 1 + P (k) + P (k 0 ) − P (k)P (k 0 ) = = P (k) + P (k 0 ) − 1 and hence P (k)P (k 0 ) − [1 − P (k)] [1 − P (k 0 )] is a self-adjoint operator. Moreover ¯ ¯2 n o2 ¯ ¯ ¯P (k)P (k 0 ) − [1 − P (k)] [1 − P (k 0 )] ¯ = P (k)P (k 0 ) − [1 − P (k)] [1 − P (k 0 )] = = (P (k) + P (k 0 ) − 1)2 = = P (k)2 + P (k 0 )2 + 1 − 2P (k) − 2P (k 0 ) + P (k)P (k 0 ) + P (k 0 )P (k) = = 1 − P (k) − P (k 0 ) + P (k)P (k 0 ) + P (k 0 )P (k). On the other hand we have that [P (k) − P (k 0 )]2 = P (k)2 − P (k)P (k 0 ) − P (k 0 )P (k) + P (k 0 )2 = = P (k) − P (k)P (k 0 ) − P (k 0 )P (k) + P (k 0 ) 9 The norm of a self-adjoint operator equals its spectral radius. As the norm of the positive operator [P (k) − P (k 0 )]2 is less than 1, its spectrum is contained in the interval [0, 1], so that also

³ ´ ³ ´ σ 1 − [P (k) − P (k 0 )]2 = 1 − σ [P (k) − P (k 0 )]2 is contained in [0, 1]. (Actually it does not contain 0, because 1 − [P (k) − P (k 0 )]2 is invertible.)

28

1. Bloch-Floquet theory for Schrödinger operators

so that

¯ ¯2 ¯ ¯ ¯P (k)P (k 0 ) − [1 − P (k)] [1 − P (k 0 )] ¯ = A(k; k 0 ).

Thus the presence of A(k; k 0 )−1/2 in the definition of W (k; k 0 ) makes it unitary. Now we will show that P (k 0 ) commutes with [P (k) − P (k 0 )]2 . In fact we have P (k 0 ) [P (k) − P (k 0 )]2 = P (k 0 )P (k) − P (k 0 )P (k)P (k 0 ) − P (k 0 )2 P (k) + P (k 0 )2 = = −P (k 0 )P (k)P (k 0 ) + P (k 0 )2 = = P (k)P (k 0 ) − P (k)P (k 0 )2 − P (k 0 )P (k)P (k 0 ) + P (k 0 )2 = = [P (k) − P (k 0 )]2 P (k 0 ). As A(k; k 0 )−1/2 is a functional calculus of [P (k) − P (k 0 )]2 , we also have P (k 0 )A(k; k 0 )−1/2 = A(k; k 0 )−1/2 P (k 0 ). We may now deduce that W (k; k 0 ) is a unitary intertwiner between P (k 0 ) and P (k), namely that W (k; k 0 )P (k)W (k; k 0 )−1 = P (k 0 ). (1.28) In fact, on the one hand we have W (k; k 0 )P (k) = A(k; k 0 )−1/2 [P (k) + P (k 0 ) − 1] P (k) = £ ¤ = A(k; k 0 )−1/2 P (k)2 + P (k 0 )P (k) − P (k) = A(k; k 0 )−1/2 P (k 0 )P (k) while on the other hand P (k 0 )W (k; k 0 ) = P (k 0 )A(k; k 0 )−1/2 [P (k) + P (k 0 ) − 1] P (k) = = A(k; k 0 )−1/2 P (k 0 ) [P (k) + P (k 0 ) − 1] = £ ¤ = A(k; k 0 )−1/2 P (k 0 )P (k) + P (k 0 )2 − P (k 0 ) = = A(k; k 0 )−1/2 P (k 0 )P (k). Identifying the generic fibre Cm of the Bloch bundle P with Ran P (k 0 ), we see from © ª (1.28) that if we choose an arbitrary orthonormal basis φ j 1≤ j ≤m spanning Ran P (k 0 ), © ª then W (k; k 0 )φ j 1≤ j ≤m is an orthonormal basis for Ran P (k). Hence we may define a local trivialisation of the bundle P via the map Ã m

−1

ΦU : U × C → π U ,

k,

m X j =1

!

"

c j φ j 7→ k,

m X j =1

# c j W (k; k 0 )φ j τ

¢¢ and the condition π ΦU k, φ = φ for all k ∈ U and φ ∈ Cm ≡ Ran P (k 0 ) is satisfied10 ; moreover ΦU is smooth because W (k; k 0 ) depends smoothly on k ∈ U . ¡

¡

10 The structure group of every rank-m complex vector bundle, which is a subgroup of the linear group

GL(m, C), may always be reduced to the group of unitary matrices U (m). In this case this reduction has ˆ d is contained in an open subset V ⊂ R ˆ d trivialising P and if k ∈ U ∩ V , already occurred, because if k 1 ∈ R then the transition function gUV (k) = W (k; k 0 )−1 W (k; k 1 ) is given by a unitary matrix.

1.3 Bloch bundles: the geometry behind periodic Schrödinger operators

29

We have thus proved that P is indeed a vector bundle. As each fibre is a subspace of the Hilbert space Hf , there is an induced Hermitian structure on P, defined by ® 〈v 1 , v 2 〉π−1 ([k]) := φ1 , φ2 Hf ,

£ ¤ £ ¤ if v 1 = k, φ1 τ , v 2 = k, φ2 τ .

(1.29)

£ ¤ In the following, when considering Bloch bundles, we will write φ(k) for k, φ τ . The scalar product just defined will be denoted by 〈·, ·〉P ; the same symbol will denote all scalar products which are induced by the latter on tensor bundles associated to P. Now that we have introduced the Bloch bundle P, we may give a geometrically transparent formulation of the question posed in the previous section, namely that of existence of smooth (respectively analytic) global Bloch functions, spanning the range of P (k). In fact, these function form nothing but a frame of global sections of the bundle P, viewed as a smooth (respectively analytic) vector bundle; this will exist if and only if P is isomorphic to the trivial bundle. Thus one may pursue the search for a collection of smooth eigenfunctions of the periodic Schrödinger Hamiltonian, in the Bloch-FloquetZak representation, by studying the geometric properties of the associated Bloch bundle P. An important geometric tool in the study of Bloch bundles was brought to the attention of theoretical physicists in 1983 by Michael V. Berry (see [4] and [5]). He noticed that the study of the adiabatic time evolution of the Hamiltonian eigenfunctions along a closed path in configuration space produced a “non-integrable” phase (i.e. a phase depending on the geometry of the path), which was henceforth called Berry’s phase in the physics community. Barry Simon [24] interpreted it as an (an)holonomy term, which could be explained intepreting the adiabatic evolution as a “parallel transport” along the chosen path. Berry’s phase can be expressed as the integral along the path of a 1-form (more precisely, of a matrix in u(m), the Lie algebra of Hermitian matrices, whose coef© ª ficients are 1-forms) called the Berry connection: if φ1 (k), . . . , φm (k) is an orthonormal set of eigenvectors relative to one of the eigenvalues of the effective Hamiltonian governing the motion, and the configuration space is a d -dimensional Brillouin zone B, then the latter is given by11

A(k)ab :=

E d D X ® i φa (k), ∂k j φb (k) dk j = i φa (k), dφb (k) P ,

j =1

P

a, b = 1, . . . , m.

One easily checks how A(k) changes after a gauge transformation on the eigenvec© ª © ª e1 (k), . . . , φ em (k) are obtained from φ1 (k), . . . , φm (k) by the gauge transformators: if φ tion G = G(k) ∈ U (m) then e(k) = G(k)−1 A(k)G(k) + iG(k)−1 dG(k). A This is exactly how connections relative to a (smooth) vector bundle transform after a change of local coordinates (compare Equation (A.9)), thus justifying the terminology 11 This definition is intrinsic, i.e. it does not depend on the choice of coordinates on the base space. This

is because the differentials of the coordinates dk j , spanning the cotangent space at each point, are “dual” objects to the partial derivatives ∂k j , spanning the tangent space at each point. Consequently, on the one hand a “change of base” in tangent space is performed by the Jacobian matrix of a change of coordinates in configuration space, while on the other hand the same “change of base” in cotangent space is performed by the transpose of the same matrix. Thus the composition of the two yields the identity matrix. Remark, on the contrary, that the gradient of a function is not an intrinsic object, but depends on the choice of a metric on configuration space.

30

1. Bloch-Floquet theory for Schrödinger operators

“Berry connection”. From a differential-geometric point of view, the Berry connection is the one induced on P by the immersion in the trivial bundle T∗ × Hf . Recall that, whenever a connection A is given on a vector bundle V of rank m, we have the associated curvature ωA , which is the u(m) matrix of 2-forms defined by ωA = dA − iA ∧ A,

i.e. explicitly (ωA )ab = dAab − i

m X

Aac ∧ Acb , a, b = 1, . . . , m.

c=1

The (real) first Chern class of the vector bundle is then defined by12 Ch1 (V) = −

¸ ·m 1 1 X (ωA )aa [tr ωA ]dR = − 2π 2π a=1 dR

where [·]dR denotes a de Rham cohomology class. This cohomology class Ch1 does not depend on the connection A, but just on the geometry of the vector bundle. Moreover, the number Ï ch1 (γ, V) = Ch1 (V), (1.30) γ

where γ is an arbitrary integer 2-cycle in the configuration space, is always an integer, and it is called a Chern number. For details on connections and Chern classes, the reader is addressed to Appendix A, Section A.2.2. One can thus approach the problem of finding global Bloch functions by calculating the Chern numbers of the Bloch bundle P; in fact, Chern classes are characteristic classes of a complex vector bundle, meaning that they contain a lot of information on the bundle itself – sometimes they even determine uniquely its isomorphism class. This is what has been done by Gianluca Panati [20]. He proved that the Bloch bundle is trivial in dimension d ≤ 3 (which are of physical interest) whenever time-reversal symmetry is present. In fact, bundles on low-dimensional base spaces are completely determined, up to isomorphism, by their first Chern class, meaning that bundles having the same Chern class are isomorphic. In particular, the bundle is trivial if and only if its first Chern class vanishes (the sufficiency of this condition is peculiar to the low-dimensionality of the base space). Time-reversal symmetry, on the other hand, implies that the trace of the Berry curvature is odd with respect to the parameter [k] ∈ T∗ , and hence its integral over a certain set of 2-cycles generating the homology group of T∗ vanish. By de Rham’s duality theorem, this in turn implies that Ch1 (P) is zero. Thus we see that the classification theory of smooth vector bundles is of great interest also for physical reason. In next chapter, instead, we will illustrate the case of a Bloch bundle on the 2-torus which is not smooth – actually, not even continuous – and has a singularity at the so-called Dirac point. However, the geometric approach will nonetheless allow us to determine and study its principal features.

12 In mathematical literature, a further factor i is added. We consider it “absorbed” in the connection, in

order to make it real-valued: compare Appendix A, Section A.2.2.

Chapter 2

Decay rate of Wannier functions in graphene Although graphene [. . . ] has been presumably produced every time someone writes with a pencil, it was only isolated 440 years after its invention. A NTONIO H. C ASTO N ETO

In this Chapter, we will deepen our understanding of the Bloch-Floquet-Zak theory of periodic Schrödinger operators and of Bloch bundles by studying a concrete example. We will consider a crystal called graphene, which in the last years captured the interest of a large part of the physics community; studying it has earned Andre Geim and Konstantin Novoselov the Nobel Prize in Physics 2010.

2.1 What is graphene? Graphene is a crystal made entirely of carbon atoms. Recently [6], Geim and Novoselov succeded in isolating single layers of graphene from small pieces of graphite, so that this crystal may be considered 2-dimensional. The carbon atoms are arranged in a honeycomb structure made out of hexagons, as shown in the next Figure 2.1. The basis E that generates the lattice Γ is given by p ³ p ³ ´ 3 p ´ 3 p e1 = a 3, 1 , e2 = a 3, −1 2 2 where a ≈ 1.42 Å is the distance between two subsequent carbon atoms. The dual basis E∗ is then given by p ´ 2π ³ 2π ³ p ´ e∗1 = 1, 3 , e∗2 = 1, − 3 . 3a 3a Both position space and momentum space are depicted in Figure 2.2: the Wigner-Seitz cell W and the Brillouin zone B are the areas contained between the dotted lines in both images. To construct the Hamiltonian for this system, we will work in the tight-binding approximation: this means that an electron orbiting near the carbon nucleus situated, say, 31

32

2. Decay rate of Wannier functions in graphene

Figure 2.1. Carbon atoms in graphene

ky e∗1 k0 W

e1 e2

B k 00

kx

e∗2

Figure 2.2. Wigner-Seitz cell (left) and Brillouin zone (right) in graphene

at the origin, can only “hop on” the nearest atoms (the white spots on the inner circle in the left image of Figure 2.2) or to the next-nearest atoms (the black spots on the outer circle in the same image). Experimental values of the nearest atom hopping energy give t ≈ 2.8 eV, while the next-nearest neighbour hopping energy can be estimated to be t 0 ≈ 0.1 eV. There is a general agreement between physicists (altough as yet there is no rigorous mathematical proof of this fact) that the energies involved in “further jumps” of the electrons tend to zero sufficiently fast, so we may neglect third-nearest hoppings and so on. In momentum space, the tight-binding approximation means that we take into account only the first two Fourier harmonics of the energy bands. The first calculations with this approximations were made in [28] (see also [3] for the expression of the tight-binding Hamiltonian both in first- and second-quantised language, and [23] for computations involving also third-nearest neighbour jumps), where

2.1 What is graphene?

33

it is shown that the energy bands are given by E ± (k) = ±t where ¡

e(k) = e k x , k y

¢

p

3 + e(k) − t 0 e(k),

(2.1)

Ãp ! µ ¶ ´ ³p 3 3 := 2 cos 3ak y + 4 cos ak y cos ak x 2 2

or equivalently e(k) = e (k 1 , k 2 ) := 2 cos [2π (k 1 − k 2 )] + 4 cos [π (k 1 − k 2 )] cos [π (k 1 + k 2 )] . o n ¡ ¢ ˆ 2 with respect to the canonical basis e∗x , e∗y , Here k x , k y are the coordinates of k ∈ R while (k 1 , k 2 ) are the coordinates of the same point with respect to E∗ . The energy bands (2.1) are illustrated in the next Figure 2.3: the plus sign applies to the upper band, whereas the minus sign applies to the lower band.

Figure 2.3. Energy bands in graphene: a conical intersection is zoomed on the right

From the picture, we see that the energy band E + (k) and E − (k) cross each other when k = k 0 (in the picture, this corresponds to the circled area, which is zoomed in the right-hand image) or k = k 00 (the one just on its left) where µ ¶ 2π 1 1, p , k0 = 3a 3

k 00

µ ¶ 2π 1 = 1, − p . 3a 3

Due to periodicity, crossing points occur in all other vertices of the hexagon illustrated on the right in Figure 2.2. Moreover at all these points the energy E ± (k) is zero. Expanding E ± (k) around k 0 or k 00 , i.e. setting k = k 0 + q for sufficiently small |q|, one gets [28] ¡ ¢ E ± (k) = ±v F |q| + o |q|/|k 0 | (2.2) where v F is the so-called Fermi velocity: 3 v F = at ≈ 106 m/s. 2

34

2. Decay rate of Wannier functions in graphene

The physical interpretation of the Fermi velocity is as follows. Let {Λn }n∈N be the increasing sequence of boxes Λn := [−2n −1, 2n +1] ⊂ R covering all R. Define the trace per unit volume1 of a self-adjoint operator acting on an Hilbert space H as ¡ ¢ 1 Tr AχΛn (1H ) n→∞ |Λn |

e Tr(A) := lim

where |Λn | = 4n + 2 denotes the Lebesgue measure of Λn , and χΛn denotes the characteristic function of the set Λn . Let χλ be the characteristic function of the interval (−∞, λ) ⊂ R. Then the function ¡ ¢ e χλ (H ) λ 7→ N (λ) := Tr is an increasing function of λ, measuring the number of states of the physical system having an energy less than λ. The unique value E F ∈ R such that N (E F ) equals the number of electrons of the system under consideration is called the Fermi energy (or chemical potential) of the system. The Fermi energy determines the so-called Fermi surface S F := {k ∈ B : E n (k) = E F } . If k F ∈ S F , then the modulus of the gradient of the n-th Bloch band E n , evaluated at k F , is called the Fermi velocity (it can be shown that this value is independent of the point we choose on the Fermi surface). In next sections, we introduce a finite-dimensional effective model to study the dynamics around one of these point k 0 and k 00 .

2.2 Interlude: space-adiabatic perturbation theory We make a momentary digression to illustrate the analogy between the problem at hand, namely the periodic Schrödinger equation, and molecular dynamics. If we consider the physical system given by a complex molecule, there is a clear separation of scales, called adiabatic decoupling. In fact, as the electron mass m el is at least three orders of magnitude smaller than the mass m n of the nuclei, we expect the former to move much faster than the latter: thus the electrons are usually called the fast degrees of freedom, while nuclei are the slow degrees of freedom. To be more precise, energy equipartition of the molecule system implies that the kynetic energies of the nuclei and of the electrons are comparable. Thus one can see that the velocities scale as |v n | ≈ ε |v el | ,

where ε :=

r

m el . mn

As the dimensionless adiabatic parameter ε is very small, one can use it as a perturbation parameter, and try to elaborate an approximated description of the dynamics up to the first orders in ε. This was first realised by Max Born and Robert Oppenheimer in the late Twenties, and was henceforth called the dynamical Born-Oppenheimer approximation (see [26] for an extensive account of this theory). The strategy is as follows. If X denotes the configuration space for nuclei and Y the one for electrons – i.e. X = R3K 1 For physical systems which are bounded, this coincides with the usual trace of a trace-class operator.

2.2 Interlude: space-adiabatic perturbation theory

35

and Y = R3N if the molecule is composed of K nuclei and N electrons –, the complete “molecular” Hamiltonian, in an appropriate system of units, reads ε Hmol = Hnε + Hel

where Hel = Hel (x) = −∆ y + Vel + Vn−el (x),

x ∈ X,

(2.3)

is the Hamiltonian operator responsible for the motion of the electrons, acting on the Hilbert space Hel which is the projection of L 2 (Y ) onto skew-symmetric wave functions, while Hnε = −ε2 ∆x + Vn is the “nuclear” Hamiltonian, regulating the motion of the nuclei and hence acting on ε Hn = L 2 (X ) (possibly with some statistics encoded). The Hilbert space on which Hmol is thus H = Hn ⊗ Hel ' L 2 (X ; Hel ). (2.4) As one can see, the nuclear Hamiltonian Hnε depends explicitly on the adiabatic parameter ε; hence it can be considered as a small perturbation of the operator Hel (x). Operators of the form (2.3) have spectral properties which are very similar to the ones of periodic Schrödinger operators: the main difference is that, in addition to bound states, labelled in increasing order E n (x) ≤ E n+1 (x), they also posses continuous spectrum. Thus the eigenvalues of “electronic” Hamiltonian Hel present a band structure. We assume the solution to the electronic structure problem Hel (x)χn (x) = E n (x)χn (x),

χn (x) ∈ Hel ,

to be known, for every nuclei configuration x ∈ X . We may thus consider states of the molecule in which the “electronic part” of the wave function lies in the subspace generated by eigenfunctions relative to a portion σ0 (k) = {E n (k), . . . , E n+N −1 (k)} of N energy bands, separated from the rest of the spectrum by a gap (compare Equation (1.25)); one says that these states are concentrated on σ0 . These will be of the form Ψ(x, ·) =

n+N X−1

ψ j (x)φ j (x, ·), ψ j ∈ Hn .

(2.5)

j =n

If we denote by Π ∈ B(H) the projection operator on states of the form (2.5), one can prove that, under the gap assumption, states which evolve from a Ψ0 ∈ Ran Π will stay localised in the same subspace up to terms of order O(ε): formally, one shows that ° ° ε ° ° (2.6) °(1H − Π) e−iHmol t /ε ΠΨ0 ° = O(ε). H

For this reason the subspace Ran Π is called the almost-invariant adiabatic subspace. The leading order in the dynamical Born-Oppeheimer approximation thus consists in restricting the attention to this adiabatic subspace, and replacing the original Hamiltoε nian ΠHmol Π with n+N X−1 HBO = −ε2 ∆ + Vn + Ej j =n ε acting on the space Hn ; the above formula (2.6) shows that ΠHmol Π = HBO + O(ε).

36

2. Decay rate of Wannier functions in graphene

Going on to higher orders in ε, one can show that there exists a projection Πε ∈ B(H) such that Πε = Π+O(ε) and that Ran Πε is almost-invariant up to errors smaller than any power of ε: this error scales linearly with respect to time and to a kinetic energy cut-off. Moreover, one can choose a unitary operator U ε : Ran Πε → Hn such that the effective Hamiltonian ¡ ¢ b ε := U ε Πε H ε Πε U ε † H eff mol is the ε-Weyl quantisation2 of a function h ε (q, p) = h 0 (q, p) + O(ε),

where h 0 (q, p) := p 2 1CN + Heff (q).

(Actually also the terms of order ε and ε2 in the above formula can be computed, and may be intepreted highlighting both their physical and geometric relevance.) The function Heff (q) is matrix-valued, and the order of this matrix is exactly the number N of relevant Bloch bands (or equivalently the dimension of the almost-invariant subspace b ε acts on the reference space Ran Πε ). Consequently, the effective Hamiltonian H eff

Href = Hn ⊗ CN ' L 2 (X ; CN ). To make the analogy of the problem of the periodic Schrödinger equation with its analogue in molecular dynamics more evident, we consider the case where the crystal interacts with an external electromagnetic field created in the laboratory. We denote by φ the electrostatic potential and by A the magnetic vector potential, as is customary. Typically, the scale of variation L of these fields is much greater than the distance a between two consecutive nuclei; thus, rather then separating slow and fast degrees of freedom, we have a clear distinction between short-range and long-range interactions. This leads to the so-called space-adiabatic perturbation theory (see [26]), and the adiabatic parameter is replaced by a ε := . L The Schrödinger Hamiltonian, to be compared with (1.20), is thus3 £ ¤2 H ε = −i∇x − A ε + VΓ + φε where A ε and φε are the operators on L 2 (Rd ) defined by ¡ ε ¢ ¡ ¢ A w (x) := A(εx)w(x), φε w (x) := φ(εx)w(x). The operator H ε acts on the Hilbert space Hτ ' L 2 (B; Hf ); this is to be compared with (2.4). This Schrödinger operator H ε is not periodic, unless φ = 0 and A = 0, i.e. there no external interacting fields, or ε = 0. Nonetheless, one may write H ε in the BlochFloquet-Zak representation, and obtain, with the same type of calculations carried out in Section 1.2, that £ ¤2 ZH ε Z−1 = −i∇ y + k − A (iε∇k ) + VΓ + φ (iε∇k ) 2 The ε-Weyl quantisation of a smooth function defined on the phase space (q, p) is a possibly unbounded operator acting on Hn ; functions f (q) are mapped to the multiplication operator times f (x), while functions g (p) are mapped to g (iε∇x ). Details may be found in [26, Apps. A and B]. 3 Notice that the associated Schrödinger equation reads

iε∂t ψ(x, t ) = H ε ψ(x, t ), i.e. is referred to the macroscopic time scale t = εs; the time scale s regulates, on the other hand, short-range interactions.

2.3 Conical intersections

37

(here y = [x] denotes the coordinate on TdW ). One can deduce from this expression that the spectrum of Z H ε Z−1 is obtained from the one of Hper (k) (i.e. from the case ε = 0, in which conjugation by the unitary Z leads indeed to a diagonalisation of the periodic Hamiltonian) by a translation determined by A; thus also Z H ε Z−1 has a band structure for its eigenvalues. One can carry the analogy with molecular dynamics further, and show that if σ0 (k) is a set of N relevant Bloch bands isolated from the rest of the spectrum of Hper (k) and if P ε (k) ∈ B(Hτ ) denotes the projection operator on states which are concentrated on σ0 (k), then the dynamics of a state which is initially concentrated on Ran P ε (k) stays localised in the same spectral region up to any order of ε, b ε , acting on the reference space and may be described by an effective Hamiltonian H eff Href := L 2 (B; CN ), which is the ε-Weyl quantisation of a matrix-valued function on the “configuration space” (k, r ).

2.3 Conical intersections We apply the space-adiabatic perturbation theory, introduced in the previous section, to the case of graphene. We consider the case ε = 0, where no interaction is present. £ ¤ Following George Hagedorn [10], we say that [k 0 ] = k 01 , k 02 ∈ T∗ is a Dirac point if the following holds. We suppose the spectral projection P (k) has a 2-dimensional range: equivalently, we assume that there are two relevant Bloch bands which are separated from the rest of the spectrum of the fibre Hamiltonian operators by a gap. The associated Bloch bundle P is thus trivial (see [20]). Consequently, we can identify the range of the spectral projection P (k) with the one of P (k 0 ), for k in some neighbourhood U of the Dirac point k 0 : this corresponds to choosing a local trivialisation of the Bloch bundle around [k 0 ]. For the sake of simplicity, we let U := ΦB R (0) where ˆ 2 → B, Φ : B R (0) ⊂ R

q 7→ k 0 + q

(2.7)

and R > 0 is so small that B R (k 0 ) is all contained in B. In particular, k 0 = Φ(0). In the following, we will use the q-coordinates, and denote every function of k ∈ U and its corresponding function of q ∈ B R (0) by the same symbol: for example, P (q) will mean P (Φ(q)). We fix an orthonormal basis {ψ1 , ψ2 } ⊂ Hf of the range of the spectral projection P (0): then the above-mentioned identification between P (q) and P (0) allows us to write every element of Ran P (q), q ∈ B R (0), as µ ¶ φ1 (q) φ(q) = , φ2 (q)

meaning φ(q, [x]) = φ1 (q)ψ1 ([x]) + φ2 (q)ψ2 ([x]),

x ∈ R2 .

The functions φ1 and φ2 are thus complex-valued. In addition to this, as by hypothesis ® ψ j , ψ j 0 H = δ j j 0 , the Hf -norm of the function φ(q) is just the Euclidean norm in C2 of f the vector (φ1 (q), φ2 (q)): kφ(q)k2Hf = |φ1 (q)|2 + |φ2 (q)|2 .

(2.8)

Of course, the same holds for scalar products. Another consequence of our assumption on the dimension of Ran P (q) is that we may study the effective dynamics through a (2 × 2)-matrix-valued Hamiltonian Heff (q).

38

2. Decay rate of Wannier functions in graphene

We assume that4 µ

q1 Heff (q) = q2

¶ µ ¶ cos ϕ sin ϕ q2 = |q| , sin ϕ − cos ϕ −q 1

¶ cos ϕ if q = |q| , |q| < R. sin ϕ µ

We will also call q = 0 a conical intersection point (or a codimension-2 intersection point), due to the fact that the eigenvalues of Heff (q) (i.e. the energy levels under consideration) are given by E ± (q) := ±|q| and hence describe a “cone” (see the picture on the right of Figure 2.3). Comparing this energy bands with (2.2), we see that in this model we have chosen units such that the Fermi velocity v F equals 1. We want to find eigenvectors for Heff (q) (normalised to 1) for q 6= 0. First of all, we observe that Heff (q) is a real symmetric matrix, so we may assume without loss of generality that these eigenvectors have real components. Now, in order to compute them, we notice that if (v 1 , v 2 ) = C (cos θ, sin θ) is such that ( ( cos θ cos ϕ + sin θ sin ϕ = cos(ϕ − θ) = ± cos θ, v 1 cos ϕ + v 2 sin ϕ = ±v 1 , ⇐⇒ cos θ sin ϕ − sin θ cos ϕ = sin(ϕ − θ) = ± sin θ, v 1 sin ϕ − v 2 cos ϕ = ±v 2 , then we must have

( ϕ−θ =

θ

if plus sign holds,

θ − π if minus sign holds.

This means that we have the two eigenvectors ¶ µ cos(ϕ/2) , ψ+,C (q) = C sin(ϕ/2)

¶ ¶ µ µ − sin(ϕ/2) cos(ϕ/2 + π/2) =C ψ−,C (q) = C cos(ϕ/2) sin(ϕ/2 + π/2)

where C = C (q) is a scalar (complex-valued) arbitrary function. If we want that kψ±,C (q)kHf = 1, then we must have |C | = 1

=⇒

C (q) = eiΘ(q) .

One easily checks that ¡ ¢ ∂|q| ψ±,C = i ∂|q| Θ ψ±,C ,

¡ ¢ 1 ∂ϕ ψ±,C = i ∂ϕ Θ ψ±,C ± ψ∓,C . 2

A particularly “symmetric” choice is then to take Θ(q) = Θ0 (q) satisfying ∂|q| Θ0 = 0,

∂ϕ Θ0 =

1 2

=⇒

Θ0 (q) =

ϕ 2

so that, if we put φ± (q) := ψ±,eiϕ/2 (q), i.e. explicitly φ+ (q) = e

iϕ/2

µ ¶ cos(ϕ/2) , sin(ϕ/2)

φ− (q) = e

iϕ/2

µ

¶ − sin(ϕ/2) , cos(ϕ/2)

4 In the following, we switch freely between cartesian and polar coordinates, both in R2 and in R ˆ 2.

(2.9)

2.3 Conical intersections

39

we then have

¢ 1¡ ±φ∓ + iφ± . (2.10) 2 This choice of the phase factor also makes φ+ (q) and φ− (q) single-valued when we identify ϕ = 0 with ϕ = 2π in the polar coordinates. We call φ± the canonical eigenvectors associated to the Dirac point under examination. This means that if {ψ1 , ψ2 } ⊂ Hf is the chosen orthonormal basis for the range of P (0) then h ³ϕ´ ³ϕ´ i φ+ (q, [x]) = eiϕ/2 cos ψ1 ([x]) + sin ψ2 ([x]) , (2.11a) 2³ ´ 2³ ´ h i ϕ ϕ φ− (q, [x]) = eiϕ/2 − sin ψ1 ([x]) + cos ψ2 ([x]) (2.11b) 2 2 ∂|q| φ± = 0,

∂ϕ φ± =

are the canonical Bloch functions. Remark 6. The main rôle in the geometric riformulation of the study of Bloch (and Wannier) functions is played by the eigenprojections of the Hamiltonian, i.e. the projection matrices on its eigenfunctions (compare Section 1.3). We compute them for the canonical Bloch functions of a conical intersection. One has, for the eigenvector φ+ (q), ¯ ¯ À ¿ ¯ ¯ ¯ cos(ϕ/2) −iϕ/2 iϕ/2 cos(ϕ/2) ¯ ® ¯= P + (q) = ¯φ+ (q) φ+ (q)¯ = ¯¯ e e sin(ϕ/2) sin(ϕ/2) ¯ ³ ´ ³ϕ´ ³ ϕ ´ 1 + cos ϕ sin ϕ 2 ϕ cos sin cos 2 2 2 2 2 = = = ³ϕ´ ³ϕ´ ³ϕ´ sin ϕ 1 − cos ϕ 2 sin sin cos 2 2 2 2 2 ¶ µ 1 cos ϕ + 1 sin ϕ = . sin ϕ − cos ϕ + 1 2 Similarly one finds for φ− (q) ¶ µ ¯ ¯ ® 1 cos ϕ − 1 sin ϕ ¯ ¯ P − (q) = φ− (q) φ− (q) = − . sin ϕ − cos ϕ − 1 2 In cartesian coordinates, these read µ ¶ 1 q 1 ± |q| q2 P ± (q) = ± . q2 −q 1 ± |q| 2|q|

(2.12)

The above-mentioned spectral projection P (q) thus decomposes as P (q) = P + (q) + P − (q).

2.3.1 The canonical Wannier functions and their decay rate We now want to compute the Bloch-Floquet-Zak antitransform of the Bloch functions defined in (2.11a) and (2.11b), in order to get the canonical Wannier functions of a conical intersection. As we will be interested in finding the rate of decay of these Wannier functions, we first make the following observation. Let χR (k) be a smoothed characteristic function for the set B R (k 0 ) ⊂ B. Then the canonical Bloch functions k 7→ φcan (k) may be written as ¡ ¢ φcan (k) = χR (k)φcan (k) + 1 − χR (k) φcan (k).

40

2. Decay rate of Wannier functions in graphene

The work done in the previous section allows us to identify the first summand with φ+ (k −k 0 ) (respectively φ− (k −k 0 )) for k ∈ B R (k 0 ); thus, χR (k)φcan (k) is a C ∞ function on B \ {k 0 }. On the other hand, as we are conducting a local analysis, we may assume that ¡ ¢ e φ(k) := 1 − χU (k) φ(k) is a C ∞ function on the whole Brillouin zone (all singularities are concentrated at the Dirac point [k 0 ]). This means that the Wannier function w can corresponding to φcan (k) may be decomposed as e e w can (x) = w + (x) + w(x) (respectively w can (x) = w − (x) + w(x)),

(2.13)

e Theorem 4 implies that w(x) e := Z−1 φ. e where w ± := Z−1 φ± and w decreases faster than any polinomial when |x| → ∞; thus we get that the decay rate of the canonical Wannier function w can is the same as that of w ± . We now proceed with the calculation of the canonical Wannier function of a conical intersection. First we notice that the change of coordinates Φ defined in (2.7) satisfies |det DΦ| = 1 and hence Ï Ï U

dk φ(k) =

B R (0)

dq φ(Φ(q)).

By definition Ï w ± (x) =

U

dk eik·x φ± (k, [x]) = eik0 ·x

Ï B R (0)

dq eiq·x φ± (q, [x]).

First we notice that in polar coordinates dq = |q| d|q| dϕ,

¶ ¶ µ µ cos θ cos ϕ . , x = |x| q · x = 2π|q| |x| cos(ϕ − θ) if q = |q| sin θ sin ϕ

We choose a frame in R2 such that θ = π/2, so that ³ π´ cos(ϕ − θ) = cos ϕ − = − sin ϕ. 2 We set R

Z w c (x) :=

2π

Z

dϕ e−i2π|q| |x| sin ϕ eiϕ/2 cos

³ϕ´

, 2 Z R Z 2π ³ϕ´ d|q| |q| dϕ e−i2π|q| |x| sin ϕ eiϕ/2 sin w s (x) := , 2 0 0 0

d|q| |q|

0

so that £ ¤ w + (x) = eik0 ·x w c (x)ψ1 ([x]) + w s (x)ψ2 ([x]) , £ ¤ w − (x) = eik0 ·x −w s (x)ψ1 ([x]) + w c (x)ψ2 ([x]) . Notice that ¢ eiϕ/2 + e−iϕ/2 1 ¡ iϕ = e +1 , 2 2 2 ³ϕ´ iϕ/2 −iϕ/2 ¢ e − e 1 ¡ iϕ = eiϕ/2 eiϕ/2 sin = e −1 , 2 2i 2i

eiϕ/2 cos

³ϕ´

= eiϕ/2

2.3 Conical intersections

41

so that Z Z 2π Z Z 2π 1 R 1 R i(ϕ−2π|q| |x| sin ϕ) w c (x) = d|q| |q| dϕ e + d|q| |q| dϕ e−i2π|q| |x| sin ϕ , 2 0 2 0 0 0 Z 2π Z 2π Z Z 1 R 1 R i(ϕ−2π|q| |x| sin ϕ) dϕ e − dϕ e−i2π|q| |x| sin ϕ . d|q| |q| d|q| |q| w s (x) = 2i 0 2i 0 0 0 Thus, we see that in both functions integrals of the form 2π

Z 0

dϕ ei(nϕ−z sin ϕ) =

2π

Z 0

¡ ¢ dϕ cos nϕ − z sin ϕ + i

2π

Z 0

¡ ¢ dϕ sin nϕ − z sin ϕ , n = 0, 1, z = 2π|q| |x| (2.14)

emerge. First we focus on the second summand in (2.14). We integrate with respect to the variable ϕ0 = ϕ − π and obtain 2π

Z 0

¡ ¢ dϕ sin nϕ − z sin ϕ =

Z

π

¡ ¢ dϕ0 sin nϕ0 + nπ − z sin(ϕ0 + π) = −π Z π ¢ ¡ dϕ0 sin nϕ0 + z sin(ϕ0 ) . = (−1)n −π

Define f (ϕ0 ) := sin nϕ0 + z sin(ϕ0 ) . Then ¡

¢

¡ ¢ ¡ ¡ ¢¢ f (−ϕ0 ) = sin −nϕ0 + z sin(−ϕ0 ) = sin − nϕ0 + z sin(ϕ0 ) = ¡ ¢ = − sin nϕ0 + z sin(ϕ0 ) = − f (ϕ0 ) i.e. f is odd, so that the second summand on the right-hand side of (2.14) is zero. On the other hand (see [17, Sec. 1.4.5, Eqn. (1)])5 , 2π

Z 0

¡ ¢ dϕ cos nϕ − z sin ϕ =: 2πJ n (z)

where J n is the Bessel function of order n. One has (see [17, Sec. 2.3, Eqn. (1) and Sec. 2.4, Eqn. (1)])6 µ ¶ ν−µ+1 Z s ∞ 2s µ X 2 ¶t J ν+2t +1 (s) = (ν + 2t + 1) µ (2.15a) dz z µ J ν (z) = ν+µ+3 µ + ν + 1 t =0 0 2 t µ ¶ (µ + ν + 1)/2 s2 s µ+ν+1 ;− = ν 1 F2 2 (µ + ν + 1)Γ(ν + 1) ν + 1, (µ + ν + 3)/2 4

(2.15b)

5 This was taken by Bessel as the definition of J (z) in his orginal article published in 1826 in Berliner n

Abh., page 34. 6 As this formula is valid for all µ and ν satisfying ℜ(µ + ν) > −1, all the calculations that follow may by adapted in higher dimensional problems (say, in dimension d ) in order to calculate the Wannier functions associated to Bloch functions of the form ³ ´ const × ein1 ϕ1 + · · · + eind −1 ϕd −1 ˆ d ) provided none of the n j ’s is less than −d . (where (|q|, ϕ1 , . . . , ϕd −1 ) are polar coordinates in R

42

2. Decay rate of Wannier functions in graphene

where ℜ(µ + ν) > −1, Γ(·) is the Euler Gamma function, (a)t denotes the Pochhammer symbol Γ(a + t ) (a)0 = 1, (a)t = = a(a + 1) · · · (a + t − 1), Γ(a) and 1 F 2 is a generalised hypergeometric function µ 1F 2

¶ ∞ X a (a)t yt ;y = . b1 , b2 t =0 (b 1 )t (b 2 )t t !

In the present case, we have to evaluate, assuming |x| > 0, R

Z 0

1 d|q| |q| J n (2π|q| |x|) = (2π|x|)2

2πR|x|

Z

dz z J n (z)

0

(as before, n = 0, 1 and z = 2π|q| |x|). Using both expressions above, we get R

R J 1 (2πR|x|) 2π|x| 0 µ ¶ Z R πR 3 3/2 (2πR|x|)2 |x| 1F 2 d|q| |q| J 1 (2π|q| |x|) = ;− 3 4 2, 5/2 0 Z

d|q| |q| J 0 (2π|q| |x|) =

by (2.15a), by (2.15b).

Thus we deduce that (|x| > 0) ¶ µ π2 R 3 R 3/2 (2πR|x|)2 w c (x) = |x| 1F 2 + J 1 (2πR|x|), ;− 3 4 2|x| 2, 5/2 ¶ µ π2 R 3 R 3/2 (2πR|x|)2 w s (x) = |x| 1F 2 − J 1 (2πR|x|). ;− 3i 4 2i|x| 2, 5/2 We are interested, as was already mentioned, in the decay rate of these Wannier functions. It is known (see [17, Sec. 1.3.3 and Sec. 1.4.6, Eqn. (3)]) that if s ∈ R, s → +∞, then µ ¶1/2 · µ ¶∞ ¶∞ µ ¸ 2 3 X 3 X −1/2 t −2t t −(2t +1) J 1 (s) ∼ s cos s − π (−1) a 2t s − sin s − π (−1) a 2t +1 s , π 4 t =0 4 t =0 µ ¶ µ 2¶ · µµ ¶ ¶ µµ ¶ ¶¸ s Γ(b 1 )Γ(b 2 ) 1 iπ/2 2 1 −iπ/2 2 a s2 ∼L + K se +K se , ;− 1F 2 b1 , b2 4 4 Γ(a) 2 2 where µµ K

1 u 2

eu u ρ

∞ X

2ρ+1 π1/2

t =0

¶2 ¶ =

L(u) = u

∞ X −a

c t u −t ,

ρ=

1 + a − b1 − b2 , 2

(−1)t d t u −t .

t =0

The latter are to be understood as formal series; the a t ’s, c t ’s and d t ’s are suitable coefficients – in particular a 0 = c 0 = d 0 = 1. Both the functions that appear as summands in the expressions we found for w c and w s can thus be expanded, for large arguments, in powers of s −1 , where s = 2πR|x| ∝ |x|. As we are interested in the decay rate, we may neglect all higher orders and keep track only of the lowest (corresponding to t = 0 in every

2.3 Conical intersections

43

series): thus as |x| → ∞ J 1 (s) ∼

¸ µ ¶1/2 · µ ¶ ¡ ¢ 2 3 s −1/2 cos s − π + O s −1 π 4

∼ const × |x|−1/2 , µ ¶ µ 2 ¶−3/2 £ ¡ ¢¤ 3/2 s2 s ;− ∼ 1 + O s −2 + 1F 2 2, 5/2 4 4 # " ¡ ¡ ¢−5/2 ¢−5/2 ¡ −7/2 ¢ + e−is e−iπ/2 s 3 eis eiπ/2 s +O s + 2 2−5/2 π1/2 ∼ const × |x|−3 + const × |x|−5/2 ∼ const × |x|−5/2 . In conclusion, we obtain µ ¶ 3/2 2 2 2 ; −π R |x| ∼ |x|−3/2 , |x| 1F 2 2, 5/2

|x|−1 J 1 (2πR|x|) ∼ |x|−3/2 ,

so that the rate of decay at infinity of both w c (x) and w s (x) (and consequently that of w ± (x)) is that of |x|−3/2 . We summarise the consequences of the above calculations in the following results. Theorem 8. Let w can ∈ L 2 (R2 ) be the canonical Wannier function of a conical intersection defined in (2.13), and let w ± = Z−1 φ± where φ± are as in (2.9). Then the rate of decay at infinity of w can is the same as the one of w ± , and thus satisfies ¡ ¢ |w can (x)| = O |x|−3/2 , for x → ∞. (2.16) An immediate corollary of this result carries a profound physical meaning. Corollary 1. Let w can ∈ L 2 (R2 ) be a canonical Wannier function of a conical intersection. Let α be a positive real number, and denote by |X |α the operator acting on a suitable domain in L 2 (R2 ) by ¡ α ¢ |X | w (x) = |x|α w(x). Then |X |α w can ∈ L 2 (R2 )

⇐⇒

1 0<α< . 2

(2.17)

In particular ° 1/2 ° °|X | w can ° 2 2 = 〈w can , |X |w can 〉 2 2 = ∞. L (R ) L (R )

(2.18)

Proof. Write ° α ° °|X | w can °2 2 2 = L (R )

Ï R2

dx|x|2α |w can (x)|2 = Ï Ï = dx|x|2α |w can (x)|2 + B 1/4 (0)

R2 \B 1/4 (0)

dx|x|2α |w can (x)|2 .

The first summand in the above sum is finite for all α > 0, because the function x 7→ |x|2α |w can (x)|2 is continuous on the bounded set B 1/4 (0) ⊂ W: this was proved in Remark 5 (we are using the fact that we are dealing with a 2-dimensional problem). As for the second summand, we use the explicit rate of decay of the function w can provided by Equation (2.16); thus we have Z Z ∞ 2α 2 dx|x| |w can (x)| ∝ d|x| |x| |x|2α |x|−3 R2 \B 1/4 (0)

1/4

and this last integral is convergent if and only if 3 − 1 − 2α > 1, i.e. α < 1/2.

44

2. Decay rate of Wannier functions in graphene

The property of the canonical Wannier functions of a conical intersection stated in Equation (2.18) has a clear physical interpretation: the expectation value of the “position” operator |X | is infinite. Equivalently, one can say that electrons in graphene are completely delocalised.

2.3.2 Invariance of the decay rate of Wannier functions Now we study how a different choice of the phase of the Bloch function affects the decay rate of the Wannier function. In other words, we consider the function where F (q) := eiΘ(q) .

φ(q) := F (q)φ± (q),

We consider F : U ⊂ B → S 1 ⊂ C as an L 2 function; consequently we have a Fourier decomposition F=

X γ∈Γ

c γ εγ ,

® where εγ (k) := e−ik·γ , c γ := εγ , F L 2 (U ) ,

© ª where the sequence c γ γ∈Γ is in `2 (Γ). Define w to be the Bloch-Floquet-Zak antitransform of φ. A straightforward computation yields Ï w(x) =

dk e U

ik·x

φ(k, [x]) =

X γ∈Γ

Ï cγ

U

dk eik·(x−γ) φ(k, [x − γ]) =

X γ∈Γ

c γ w ± (x − γ).

We used the fact that [x] = [x − γ] as these are classes modulo Γ. To see if w(x) has the same rate of decay in the L 2 -sense of w can , i.e. if also w satisfies (2.17), we fix 0 < α < 1/2. We have by the triangle inequality ° ° ° °X ° α ° ° °|x| w(x)° 2 2 = ° c γ |x|α w ± (x − γ)° ° L (R ) °γ∈Γ °

L 2 (R2 )

≤

X ¯ ¯° α ° ¯c γ ¯ °|x| w ± (x − γ)°

L 2 (R2 ) .

γ∈Γ

(2.19)

Now notice that ° α ° °|x| w ± (x − γ)°2 2 2 = L (R )

Ï

¯2 w ± (x − γ)¯ =

Ï

dx |x + γ|2α |w ± (x)|2 ≤ Ï Ï ¡ ¢2α ¡ ¢ 2 ≤ dx |x| + |γ| |w ± (x)| ≤ dx |x|2α + |γ|2α |w ± (x)|2 = 2 2 R R °2 ° α 2α 2 ° ° kw = |x| w ± (x) L 2 (R2 ) + |γ| ± (x)kL 2 (R2 ) . R2

dx |x|

2α ¯

¯

R2

In the second-to-last step we used the fact that, for all a, b ≥ 0 and 0 < p < 1, (a + b)p ≤ a p + b p

(2.20)

2.3 Conical intersections

45

(set a = |x|, b = |γ| and p = 2α)7 . Consequently h° i1/2 ° α ° ° °|x| w ± (x − γ)° 2 2 ≤ °|x|α w ± (x)°2 2 2 + |γ|2α kw ± (x)k2 2 2 ≤ L (R ) L (R ) L (R ) ° α ° ≤ °|x| w ± (x)°L 2 (R2 ) + |γ|α kw ± (x)kL 2 (R2 ) again by (2.20). Plugging this in into (2.19), we get ´ X ¯ ¯ ³° α ° α ° ° °|x| w(x)° 2 2 ≤ ¯c γ ¯ °|x| w ± (x)° 2 2 + |γ|α kw ± (x)k 2 2 = L (R ) L (R ) L (R ) γ∈Γ

Ã

! Ã ! X¯ ¯ ° α X ¯ ¯α ¯ ¯ ° ¯c γ ¯ °|x| w ± (x)° 2 2 + ¯γ¯ ¯c γ ¯ kw ± (x)k 2 2 = = L (R ) L (R ) γ∈Γ

γ∈Γ

° ° = kc γ k`1 (Γ) °|x|α w ± (x)°L 2 (R2 ) + k|γ|α c γ k`1 (Γ) kw ± (x)kL 2 (R2 ) . © ª This last expression is finite provided the Fourier coefficients sequence c γ γ∈Γ ∈ `2 (Γ) © ª is also in `1 (Γ) and is such that |γ|α c γ γ∈Γ is in `1 (Γ), for all 0 < α < 1/2. By standard

Fourier theory (see [13, Chap. I, Sec. 4]) both this conditions are satisfied if F (q) = eiΘ(q) (or equivalently Θ(q)) is in C 1,α (B), the space of all differentiable functions whose partial derivatives are Hölder functions of exponent α, for all α ∈ (0, 1/2).

7 We prove inequality (2.20). It is trivially verified if b = 0, so that we may suppose b > 0. Dividing both sides by b p > 0 we get ³a ´p ³ a ´p

+1 ≤ + 1. b b Let t := a/b > 0, and consider the function g (t ) := (t + 1)p − t p − 1: our inequality is then equivalent to g (t ) ≤ 0. Notice that g (0) = 0, so it only remains to check that g is decreasing. The derivative of g is h i g 0 (t ) = p (t + 1)p−1 − t p−1

and as p − 1 < 0 we have t +1 > t which concludes the proof.

=⇒

(t + 1)p−1 < t p−1

=⇒

g 0 (t ) < 0

46

2. Decay rate of Wannier functions in graphene

Chapter 3

Bloch bundles in graphene There are perhaps more fibre bundles in physics than a local, gauge-theory shaped eye allows to see. H ELMUTH K. U RBANTKE

In this section, we focus on the behaviour of the Bloch bundle around the Dirac point q = 0. To this purpose, we “perturb” the effective Hamiltonian Heff (q), in order to avoid the crossing of the Bloch bands – this is done in the following section. Next, we introduce the composite Bloch bundle, in order to be able to use differential-geometric methods to study the singular Bloch bundle; the perturbation introduced in the effective Hamiltonian allows us to stay away from the singularity. After that, a clear geometric interpretation of this composite Bloch bundle is provided. Lastly, we extend the notion of Berry curvature to the singular Bloch bundle associated with a conical intersection by using differential forms with distributional coefficients.

3.1 Avoided crossings £ ¤ We add a small parameter µ ∈ −µ0 , µ0 , for fixed µ0 > 0, to our theory, and define for q ∈ B R (0) (compare [11]) µ Heff (q) :=

µ

q1 q 2 − iµ

¶ ¶ µ q 2 + iµ cos ϕ sin ϕ + iη , = |q| −q 1 sin ϕ − iη − cos ϕ

where η = ηµ (q) :=

µ . |q|

In this case, the Bloch bands are q q µ E ± (q) := ± |q|2 + µ2 = ±|q| 1 + η2 . Clearly for µ = 0 we obtain the analogous quantities studied in the preceeding section; however, for µ 6= 0 these functions are smooth, also for q = 0. µ This time, finding eigenvectors for Heff (q) (with µ 6= 0) requires a little more than the µ simple trigonometry used in the case of the conical intersection. In fact, as Heff (q) is Hermitian but not real, we cannot expect to find real eigenvectors. Hence we study the system µ ¶µ ¶ µ ¶ q cos ϕ sin ϕ + iη v 1 + iu 1 v 1 + iu 1 2 . = ± 1+η v 2 + iu 2 sin ϕ − iη − cos ϕ v 2 + iu 2 47

48

3. Bloch bundles in graphene

Let v = (v 1 , v 2 ) and u = (u 1 , u 2 ). After separating real and imaginary parts and using the notations µ ¶ µ ¶ q q⊥ cos ϕ − sin ϕ e|q| = = , eϕ = = , sin ϕ cos ϕ |q| |q| we obtain the following system: p v |q| := e|q| · v = ± 1 + η2 v 1 + ηu 2 , v := e · v = ∓p1 + η2 v + ηu , ϕ ϕ 2 1 p (3.1a) u |q| := e|q| · u = ± 1 + η2 u 1 − ηv 2 , p u ϕ := eϕ · u = ∓ 1 + η2 u 2 − ηv 1 . © ª ˆ 2 which is obtained from the On the other hand, e|q| , eϕ is the orthonormal basis for R © ª ∗ ∗ ∗ fixed one E = e1 , e2 , in which the coordinates of q, x and y are originally written, by a counterclock-wise rotation of an angle ϕ. Consequently, the vectors (v |q| , v ϕ ) and (u |q| , u ϕ ) are obtained respectively from v and u by the same rotation. Thus we deduce that v |q| = cos ϕ · v 1 + sin ϕ · v 2 , v = − sin ϕ · v + cos ϕ · v , ϕ 1 2 (3.1b) u |q| = cos ϕ · u 1 + sin ϕ · u 2 , u ϕ = − sin ϕ · u 1 + cos ϕ · u 2 . We may now view systems (3.1a) and (3.1b) as a whole and solve this linear system with respect to the eight unknowns v 1 , v 2 , u 1 , u 2 , v |q| , v ϕ , u |q| , u ϕ . As we obtained it from an eigenvalue problem, we know it to have a (complex) 1-dimensional space of solutions, and hence we can fix two of these (real) parameters to find a unique solution (up to a phase). As we look at the avoided crossing as a perturbation of the conical crossing, µ we require the following to hold: When q 6= 0, the eigenvectors of Heff (q) must tend to ψ±,1 (q) as µ approaches 0. Now, the two vectors ¶ ¶ µ µ − sin(ϕ/2) cos(ϕ/2) and ψ−,1 (q) = ψ+,1 (q) = cos(ϕ/2) sin(ϕ/2) µ

have real components; we want the imaginary part of the eigenvectors of Heff (q) to vanish as µ → 0. Thus, we can fix the two parameters v = (v 1 , v 2 ) in order to have v = ψ±,1 , i.e. v 1 = cos(ϕ/2), v 2 = sin(ϕ/2) (respectively v 1 = − sin(ϕ/2), v 2 = cos(ϕ/2)). Plugging this in (3.1a) and (3.1b), a straightforward calculation yields p p 1 − 1 + η2 |q| − |q|2 + µ2 µ u = αψ∓,1 , where α = α (q) = = . η µ As wanted, αµ (q) tends to 0 as µ vanishes. We still have a choice on the phase of these eigenvectors – this means that we can replace ψ±,1 with ψ±,eiΘ in the preceeding calculations. Our preferred choice will again be the phase eiϕ/2 , which gives the canonical eigenvectors for an avoided crossing £ ¤ 1 µ φ± (q) := p φ± (q) + iαφ∓ (q) . 1 + α2 µ

With the factor (1 + α2 )−1/2 , φ± (q) are normalised eigenvectors.

(3.2)

3.1 Avoided crossings

49 µ

µ

Remark 7. It is interesting to compute the eigenprojections P ± (q) associated to φ± (q): these satisfy the relation µ

µ

µ

Heff (q) = E + (q)P + (q) + E −µ (q)P −µ (q). µ

We start from P + (q). By definition ¯ µ ® µ ¯ µ P + (q) = ¯φ+ (q) φ+ (q)¯ = ¯ ¯ À ¿ 1 ¯¯ cos(ϕ/2) − iα sin(ϕ/2) −iϕ/2 iϕ/2 cos(ϕ/2) − iα sin(ϕ/2) ¯¯ e e = = 1 + α2 ¯ sin(ϕ/2) + iα cos(ϕ/2) sin(ϕ/2) + iα cos(ϕ/2) ¯

ϕ ϕ + α2 sin2 2 2

cos2

1 = 1 + α2 cos ϕ sin ϕ (1 − α2 )+ 2 2 h ϕ ϕi +iα cos2 + sin2 2 2

sin

ϕ ϕ cos (1 − α2 )− 2 2 h i ϕ ϕ −iα sin2 + cos2 2 2 = ϕ ϕ sin2 + α2 cos2 2 2

1 + cos ϕ sin ϕ 2 1 − cos ϕ 2 + α (1 − α ) − iα 1 2 2 2 = = 2 1+α sin ϕ 1 − cos ϕ 1 + cos ϕ (1 − α2 ) + iα + α2 2 2 2 1 + α2 2α sin ϕ − i 2 cos ϕ + 1 1−α 1 − α2 1 − α2 = . 2 1 + α2 1 + α2 2α − cos ϕ + sin ϕ + i 1 − α2 1 − α2

From the definition of α we deduce that ³ ´ p p 2 1 − 1 + η2 1 + η −2 p 1 + α2 = 1 − 1 + η2 η2 ´ ³ p =⇒ α= η −2 1 − 1 + η2 2 . 1 − α = η2

and (3.3)

Thus 1 + α2 = 1 − α2

q

1 + η2 ,

2α = −η 1 − α2

and we may write ! p cos ϕ + 1 + η2 sin ϕ + iη p = p sin ϕ − iη − cos ϕ + 1 + η2 2 1 + η2 Ã ! p 1 q 2 + iµ q 1 + |q|2 + µ2 p = p . q 2 − iµ −q 1 + |q|2 + µ2 2 |q|2 + µ2

µ P + (q) =

1

Ã

(3.4a) (3.4b)

50

3. Bloch bundles in graphene

A similar computation shows that ! p cos ϕ − 1 + η2 sin ϕ + iη p = p sin ϕ − iη − cos ϕ − 1 + η2 2 1 + η2 Ã ! p 1 q 1 − |q|2 + µ2 q 2 + iµ p =− p . q 2 − iµ −q 1 − |q|2 + µ2 2 |q|2 + µ2

P −µ (q) = −

Ã

1

(3.4c) (3.4d)

We now observe that in q = 0 µ P ± (0) = ±

µ ¶ µ ¶ 1 1 ±i 1 ±µ iµ = 2µ −iµ ±µ 2 ∓i 1

which is independent of µ. Thus we may define P ± (0) ≡ P ±0 (0) :=

µ ¶ 1 1 ±i 2 ∓i 1

and get a coherent – yet not continuous – prolongation of the Bloch bundle in the Dirac point also in the case of a conical intersection.

3.1.1 Berry curvature for avoided crossings We come to the differential-geometric study of the Bloch bundle in graphene. In the present case of avoided crossings, we consider the bundles associated with (the projecµ tion on) the eigenvector φ+ (q) or φµ− (q), defined over the ball B R (0); hence these bundles are 1-dimensional (m = 1 in the notations of Section 1.3). However, rather than considering a family of Bloch bundles parametrised by µ ∈ [−µ0 , µ0 ] – many bundles on one fixed base –, one should think of all these bundles as separated objects, each one lying on a different copy of the same base space. Thus we consider the cartesian product £ ¤ C := B R (0)× −µ0 , µ0 as the base space for these composite Bloch bundles Pcomp,± ; the fibre of the latter over the point (q 1 , q 2 , µ) ∈ C is the 1-dimensional vector space consisting µ of the range of the eigenprojection P + (q) (respectively P −µ (q)), where q = (q 1 , q 2 ). However, we know that for µ = 0 the Bloch bundle is not defined in q = 0, while for all µ 6= 0 the eigenprojections are smooth functions of q ∈ B R (0); thus we have to restrict the base space of these bundles to the punctured cylinder C• := C \ {(0, 0, 0)}. From a topological point of view, the latter is homotopically equivalent to the 2-sphere S 2 – just blow inside the hole to smooth up the edges! As S 2 has non-trivial cohomology groups (in particular 2 HdR (S 2 ) ' R), a topological obstruction may rise in the extension of this family of vector bundles at the point (0, 0, 0). µ

O

Ctop

µ0 −

q1

−µ0 −

q2

/

C

... ... .. .. .. .. .. .. .. .. .. ... ... .... .................... .. .. .. .. • (0,0,0) .. .. .. .. .. .. ... ... .... .................... ... ... .. .. .. .. .. .. .. .. .. .. .. .. .. ...

... .... .................... ... ... .. .. .. .. .. .. .. Cbottom

Figure 3.1. The cylinder C

Clateral

3.1 Avoided crossings

51

Before computing the Berry curvature corresponding to the Berry connection, we d X make one more general remark. For any u(1)-connection A(q) = A j (q) dq j , the A j ’s j =1

are just scalar functions, so that1

A∧A =

X

A j A j 0 dq j ∧ dq j 0 + dq j 0 ∧ dq j = 0. ¢

¡

j

® Consequently, when A j (q) = i φ(q), ∂q j φ(q) P is the local expression for the Berry connection over a d -dimensional configuration space B, the associated Berry curvature is just ® ωA (q) = dA(q) = i d φ(q), dφ(q) P = ® ® = i dφ(q), dφ(q) P + i φ(q), d2 φ(q) P = ® = i dφ(q), dφ(q) P . This is a compact notation, that explicitly stands for ωA (q) =

d X j , j 0 =1

D E i ∂q j φ(q), ∂q j 0 φ(q) dq j ∧ dq j 0 = P

E D E ´ X ³D = i ∂q j φ(q), ∂q j 0 φ(q) − ∂q j 0 φ(q), ∂q j φ(q) dq j ∧ dq j 0 = P

j

P

µ D E ¶ E X D = dq j ∧ dq j 0 = i ∂q j φ(q), ∂q j 0 φ(q) − ∂q j φ(q), ∂q j 0 φ(q) =

X j

P

P

j

D

−2ℑ ∂q j φ(q), ∂q j 0 φ(q)

E P

dq j ∧ dq j 0 .

In the case under examination, the base space C• of the composite Bloch bundle Pcomp,± is 3-dimensional. The canonical eigenvectors for avoided crossings are defined via equation (3.2) through their counterparts for conical intersections; accordingly, it is µ very easy to compute the partial derivatives of φ± (q) in cylindric coordinates (|q|, ϕ, µ), using equation (2.10). The Berry curvature thus reads ³ ® µ µ µ ω± (q) := −2 ℑ ∂|q| φ± (q), ∂ϕ φ± (q) P d|q| ∧ dϕ+ comp,± ® µ µ + ℑ ∂|q| φ± (q), ∂µ φ± (q) P d|q| ∧ dµ+ comp,± ® µ µ + ℑ ∂ϕ φ± (q), ∂µ φ± (q) P

comp,±

´ dϕ ∧ dµ . (3.5)

In order to compute all derivatives appearing in the above equation, first we notice µ that φ± (q), defined as in (3.2), depends on |q| and µ only through α = αµ (q), while its 1 On higher-dimensional vector bundles, the

A j ’s would be Hermitian matrices, so that the above sum

would read

A∧A =

X h j

h

i

i

A j , A j 0 dq j ∧ dq j 0 .

The commutators A j , A j 0 may very well be non-zero.

52

3. Bloch bundles in graphene

dependence on ϕ lies only in φ± (q). Consequently, we have ¶ µ α 1 µ φ± + i p φ∓ = ∂|q| φ± (q) = ∂|q| p 1 + α2 1 + α2 p α 1 + α2 − α p µ ¶ 1 α 1 + α2 φ∓ ∂|q| α = = φ± + i − 1 + α2 p 2 1+α 1 + α2 · = − =

α (1 + α2 )3/2 i

¡

(1 + α2 )3/2

φ± + i

1 + α2 − α2 (1 + α2 )3/2

¢ φ∓ + iαφ± ∂|q| α =

¸

φ∓ ∂|q| α =

1

¡

(1 + α2 )3/2

¢ −αφ± + iφ∓ ∂|q| α =

i µ φ∓ ∂|q| α. 2 1+α

The partial derivative with respect to µ has a very similiar expression, because it involves only the corresponding derivative of α: µ

∂µ φ± (q) =

i µ φ ∂µ α. 1 + α2 ∓ µ

On the other hand, to compute the partial derivative of φ± (q) with respect to ϕ we invoke equation (2.10), which yields £¡ ¢ ¡ ¢¤ 1 µ ∂ϕ φ± (q) = p ∂ϕ φ± (q) + iα ∂ϕ φ∓ (q) = 1 + α2 · ¸ ¢ ¢ 1¡ 1¡ 1 =p ±φ∓ + iφ± + iα ∓φ± + iφ∓ = 2 1 + α2 2 ¡ ¢ 1 1 ±φ∓ + iφ± ∓ iαφ± − αφ∓ = = p 2 1 + α2 £ ¤ 1 1∓α ¡ ¢ 1 1 = p ±φ∓ + iφ± . (±1 − α) φ∓ + i (1 ∓ α) φ± = p 2 1 + α2 2 1 + α2 Using the relations

φ± (q), φ± (q)

®

Hf

= 1 and

φ± (q), φ∓ (q)

®

Hf

= 0,

we can easily calculate the Pcomp,± -scalar products of couples of partial derivatives of µ φ± (q). In fact, from the above equalities it follows that

µ

φ∓ (q), ±φ∓ (q) + iφ± (q)

®

Hf

® 1 =p φ∓ (q) + iαφ± (q), ±φ∓ (q) + iφ± (q) Hf = 1 + α2 1 1±α =p (±1 + α) = ± p . 2 1+α 1 + α2

This yields

µ

µ

∂|q| φ± (q), ∂ϕ φ± (q)

®

Pcomp,±

® i 1∓α µ φ∓ (q), ±φ∓ (q) + iφ± (q) H ∂|q| α = 2 3/2 f 2 (1 + α ) 2 i (1 ∓ α)(1 ± α) i 1−α =± ∂|q| α = ± ∂|q| α 2 2 2 (1 + α ) 2 (1 + α2 )2 ³ α ´ i = ± ∂|q| 2 1 + α2 =

3.1 Avoided crossings

53

and analogously

® µ µ ∂µ φ± (q), ∂ϕ φ± (q) P

i ³ α ´ . = ± ∂µ comp,± 2 1 + α2

(Notice that the latter is not the scalar product appearing in (3.5), but rather its comµ plex conjugate: the partial derivatives of φ± (q) are in a different order.) The last scalar product

µ

µ

∂|q| φ± (q), ∂µ φ∓ (q)

®

=−

Pcomp,±

µ ® 1 µ φ∓ (q), φ∓ (q) H ∂|q| α ∂µ α 2 3 f (1 + α )

is of no interest to us, because it is a real number: its imaginary part, which appears in µ the expression for ω± (q) we found before, vanishes. By (3.3) we know that p 1 + η2 2 1 + α = −2α η and consequently µ |q|

α 1 η 1 =− p =− s 2 1+α 2 1 + η2 2

1+

µ2 |q|2

1 µ =− p . 2 |q|2 + µ2

Thus we may write ! Ã 1 µ ℑ = ∓ ∂|q| p , 4 |q|2 + µ2 Ã ! ® 1 µ µ µ = ± ∂µ p ℑ ∂ϕ φ± (q), ∂µ φ± (q) P . comp,± 4 |q|2 + µ2

® µ µ ∂|q| φ± (q), ∂ϕ φ± (q) P comp,±

(3.6a) (3.6b)

In conclusion, putting together Equations (3.5), (3.6a) and (3.6b), we deduce that 1 µ ω± (q) = −

"

Ã

µ

!

Ã

µ

!

#

d|q| ∧ dϕ ± ∂µ p dϕ ∧ dµ = ∓∂|q| p |q|2 + µ2 |q|2 + µ2 ! Ã 1 µ =± d p ∧ dϕ. 2 |q|2 + µ2 2

(3.7a) (3.7b)

We are ready to compute the Chern number (1.30) of the composite Bloch bundle on

C• . As we already remarked, this space is homotopically equivalent to S 2 , which means that we can continuously retract it on its boundary C := ∂C. Hence, the only relevant Chern number is the one associated with (the homology class of ) C: µ ch±

1 := − 2π

Ï

µ

C

ω± (q).

We divide the boundary of C into three parts (see Figure 3.1),

C = Ctop ∪ Clateral ∪ Cbottom ,

54

3. Bloch bundles in graphene

where Ctop denotes the top of the cylinder, Clateral is its side surface and Cbottom its bottom: n o Ctop := (q1 , q2 , µ) ∈ C : µ = µ0 , n o Clateral := (q1 , q2 , µ) ∈ C : |q| = R , n o Cbottom := (q1 , q2 , µ) ∈ C : µ = −µ0 . µ

Accordingly, the integral defining ch± divides into three summands: we compute them separately. On Ctop , µ is constant, and hence dµ = 0. Thus the second summand on the righthand side of (3.7a) vanishes; on the other hand, the first summand must be evaluated at µ = µ0 . Consequently Ï Ï Z 2π Z R dϕ µ0 1 1 1 µ µ − ω± (q) = − ω±0 (q) = ∓ ∂|q| q d|q| = 2π Ctop 2π B R (0) 2 0 2π 0 |q|2 + µ20 |q|=R 1 µ0 = ∓ q 2 |q|2 + µ2 0

|q|=0

1 µ0 = ∓ q − 1 = 2 R 2 + µ2 0

µ0 1 = ± 1 − q . 2 2 2 R +µ 0

The integral of the Berry curvature on Cbottom is formally the same: we just have to evaluate this 2-form at µ = −µ0 . However, Cbottom has a different orientation than Ctop . One can visualise this in Figure 3.1 by thinking at the normal vectors to these two surfaces; a vector that is normal to Ctop points upwards (i.e. towards increasing µ’s) while a vector that is normal to Cbottom points downwards (i.e. towards decreasing µ’s). A change of orientation amounts to a change of sign when calculating an integral; as also the integrand changes sign when we evaluate it at µ = −µ0 , these two changes cancel with each other. Thus we have exactly the same result: Ï Ï 1 1 1 µ0 µ −µ ω± (q) = − ω± 0 (q) = ± 1 − q − . 2π Cbottom 2π B R (0) 2 2 2 R + µ0 Finally we compute the integral on the cylinder side Clateral . This time |q| is constant, so that d|q| = 0; the only contribution to this integral comes from the second summand on the right-hand side of (3.7a). We have Ã ! Ï Ï Z Z 1 1 1 2π dϕ µ0 µ µ µ − ω (q) = − ω (R, ϕ) = ± ∂µ p dµ = 2π Clateral ± 2π B R (0) ± 2 0 2π −µ0 R 2 + µ2 #µ=µ0 " 1 µ 1 µ0 −µ0 =± = ± q −q p = 2 2 2 2 R +µ R 2 + µ2 R 2 + µ2 µ=−µ0

µ0

= ±q . 2 2 R + µ0

0

0

3.2 Geometric identification of the composite Bloch bundle

55

We are finally able to state that the Chern number of the composite Bloch bundle equals µ0 µ0 1 µ0 1 µ ch± = ± 1 − q = ±1. ± 1 − q ± q 2 2 R 2 + µ2 R 2 + µ2 R 2 + µ2 0

0

(3.8)

0

These vector bundles cannot be extended at the point (0, 0, 0); this would require the above number to vanish (see Appendix A, Section A.2.3).

3.2 Geometric identification of the composite Bloch bundle The physical information about the system under study is encoded in the geometric framework by the map ¡ ¢ µ ˆ 2 → G 1; C2 , u ± : B R (0) ⊂ R

µ

q 7→ Ran P ± (q).

(3.9)

¡ ¢ Here G 1; C2 denotes the complex Grassmannian of complex 1-dimensional vector sub¡ ¢ spaces (i.e. lines) in C2 . Clearly this manifold coincides with P C2 ' P1C , the complex projective line. A line bundle L over P1C is completely determined by its first Chern class, or equivalently by the integer Chern number ch1 (P1C , L) ∈ Z – see Example 6 in Appendix A. In particular, all line bundles can be obtained by taking tensor powers of the following two: 1. the tautological bundle: the fibre over the point in P1C represented by the line Cψ, 0 6= ψ ∈ C2 , is just the line itself; its Chern number is −1, and thus is denoted by O(−1); 2. the hyperplane bundle: it is the dual of the tautological bundle, or equivalently the one whose fibre over the line Cψ is the ortogonal complement of the line itself with respect the usual Hermitian structure in C2 ; its Chern number is minus the Chern number of the tautological bundle (being its dual), and hence it is equal to +1, leading to the notation O(1). Clearly these constructions can be generalised and lead to the definition of tautological and anti-tautological bundles over all Grassmannians G(k; Cn ) of k-dimensional vector subspaces of Cn . More details on O(1) and O(−1) may be found in Appendix A. Basic differential geometry tells us that P1C may be thought of as the one-point compactification of C ' R2 , and hence is diffeomorphic to the sphere S 2 via the stereographic µ projection map p. The map u ± is smooth on the whole ball B R (0) (as we will explicitly check in a moment) whenever µ 6= 0; thus we can pull-back the Fubini-Study 2form ωP1 , defined as the curvature of the unique connection on the tautological bundle C

O(−1) which is compatible with the complex structure and the Hermitian metric on P1C

(see footnote on page 86). However, it is more convenient to exploit the identification S 2 ' P1C , where the isomorphism is given by the map p. It is known (see Appendix A, Example 3) that the pullback bundle p ∗ O(−1) on S 2 is related to the so-called Hopf bundle, and we will prove that p ∗ ωP1 is just 1/2 times the usual area 2-form on S 2 , to be denoted C

µ

µ

by ωS 2 . Thus, we will consider the map v ± (q) := p −1 ◦ u ± (q) and compute ¡

µ ¢∗

v±

¡ µ ¢∗ ¡ ¢∗ ¡ µ ¢∗ ωS 2 = u ± p −1 ωS 2 = 2 u ± ωP1 . C

(3.10)

56

3. Bloch bundles in graphene

We will then compare this pull-back with the Berry curvature, and this will make the geometric interpretation of the latter more clear. We start from recalling the isomorphism between P1C and S 2 . We view S 2 as the set © ª S 2 = (x, y, z) ∈ R3 : x 2 + y 2 + z 2 = 1 and map it via stereographic projection on the plane {z = 0}, with coordinates (x, y). We will identify this plane with C with coordinate ζ = x + iy, and look at it as the affine line ª © C = [ζ0 ; ζ1 ] ∈ P1C : ζ1 6= 0 ⊂ P1C ,

so that ζ = ζ0 /ζ1 .

(3.11)

z N

y =0

Q

X

P x

O

Figure 3.2. Stereographic projection in R2

To establish the diffeomorphism between S 2 and P1C , we first start from a lower dimensional case, namely S 1 ' P1R . Consider the plane {y = 0}; then we identify S 1 with the set {x 2 + z 2 = 1}. In this case the formula can be deduced from Figure 3.2; by definition, the stereographic projection through the “North pole” N = (0, 1) of the point X = (cos θ1 , sin θ1 ) ∈ S 1 is the point of intersection of the line N X with the line {z = 0}. We denote by P = (x, 0) this intersection point. From the similitude between the triangles NQ X and NOP , we deduce that x 1 = cos θ1 1 − sin θ1

=⇒

x=

cos θ1 . 1 − sin θ1

We know realise the sphere S 2 rotating the hemicircle © ª (x, 0, z) ∈ R3 : x 2 + z 2 = 1, x > 0 around the z axis. Thus we get the parametrisation2

S2 :

x = cos θ1 cos θ2 , y = cos θ1 sin θ2 ,

z = sin θ , 1

³ π π´ θ1 ∈ − , , θ2 ∈ [0, 2π) 2 2

(3.12)

2 Notice that in most of the mathematical literature polar coordinates on the 2-sphere are denoted by θ = π/2 − θ1 ∈ (0, π) (called the inclination) and ϕ = θ2 ∈ [0, 2π) (called azimuth or longitude). Our angle θ1 is sometimes called the elevation or latitude. Thus sin θ1 = cos θ while cos θ1 = sin θ.

3.2 Geometric identification of the composite Bloch bundle

57

(the “North pole” N = (0, 0, 1) and the “South pole” S = (0, 0, −1) would correspond to θ1 = π/2 and θ1 = −π/2, respectively). The stereographic projection through N of X ∈ S 2 having these coordinates is thus obtained by rotating the whole space of an angle θ2 after applying the formula we found above to the point (cos θ1 , 0, sin θ1 ): thus we get that µ ¶ ³ ´ cos θ1 cos θ1 x y P= cos θ2 , sin θ2 , 0 = , ,0 . 1 − sin θ1 1 − sin θ1 1−z 1−z In this geometric construction, points approaching N on the sphere are mapped to points approaching infinity in C ' R2 : thus we may extend this definition to the complex projective line P1C and set p: S

2

→ P1C ,

£ ¤ (x, y, z) such that z 6= 1 7→ x + iy; 1 − z , (0, 0, 1) 7→ [1; 0].

(3.13)

Figure 3.3. Stereographic projection in R3

Now we want to find an expression for the inverse of this map. Given a point P = (x, y, 0) ∈ R3 , we want to find the intersection of the line N P with S 2 . The former is given in parametric form by X = N + t (P − N ), t ∈ R. Thus we have to solve with respect to the parameter t the system (

X = N + t (P − N ), |X |2 = 1.

A substitution of the first equation in the second leads to ¯ ¡ ¢¯ ¯(0, 0, 1) + t x, y, −1 ¯2 = 1 ⇐⇒

⇐⇒ t 2 x 2 + t 2 y 2 + (1 − t )2 = 1 ⇐⇒ ¡ ¢ £ ¡ ¢ ¤ t 2 x 2 + y 2 + 1 − 2t = 0 ⇐⇒ t t x 2 + y 2 + 1 − 2 = 0

58

3. Bloch bundles in graphene

which descarding the trivial solution t = 0 (that corresponds to the North pole) yields t=

2 . x2 + y 2 + 1

In conclusion, we have that the inverse stereographic projection is given by µ ¶ 2 2ℜζ 2ℑζ |ζ|2 − 1 (ℜζ, ℑζ, −1) = , , , [ζ; 1] 7→ (0, 0, 1) + 2 |ζ| + 1 |ζ|2 + 1 |ζ|2 + 1 |ζ|2 + 1 p −1 : P1C → S 2 , [1; 0] 7→ (0, 0, 1). (3.14) Notice that if ξ is a non-zero real number then ¯ ¯ ζ ζ ¯¯ ζ ¯¯2 µ· ¸¶ 2ℜ 2ℑ −1 ζ ξ ξ ¯ξ¯ p −1 ([ζ; ξ]) = p −1 ; 1 = ¯ ¯2 , ¯ ¯2 , ¯ ¯2 = ¯ζ¯ ¯ζ¯ ¯ζ¯ ξ ¯ ¯ +1 ¯ ¯ +1 ¯ ¯ +1 ¯ξ¯ ¯ξ¯ ¯ξ¯ ¶ µ 2ξℑζ |ζ|2 − ξ2 2ξℜζ , , . (3.15) = |ζ|2 + ξ2 |ζ|2 + ξ2 |ζ|2 + ξ2 Finally, recall that the parametrisation (3.12) of the sphere S 2 allows us to compute the area 2-form, that we will later compare with the Berry curvature (3.7b). This is given by ® ® ¶ µ q ∂θ 1 , ∂θ 1 R3 ∂θ 1 , ∂θ 2 R3 ® ® ωS 2 := det g dθ1 ∧ dθ2 , where g = . ∂θ 2 , ∂θ 1 R3 ∂θ 2 , ∂θ 2 R3 From (3.12) we immediately deduce that ∂θ1 = (− sin θ1 cos θ2 , − sin θ1 sin θ2 , cos θ1 ) , so that

∂θ2 = (− cos θ1 sin θ2 , cos θ1 cos θ2 , 0)

¯ ¯2 ¯ ¯ 2 ®2 det g = ¯∂θ1 ¯ ¯∂θ2 ¯ − ∂θ1 , ∂θ2 R3 = 1 · cos2 θ1 − 0 = cos2 θ1 .

In conclusion the area 2-form on S 2 is given by3 ωS 2 = cos θ1 dθ1 ∧ dθ2 . We will need an expression for this 2-form using the cartesian coordinates of R3 . We immediately notice that cos θ1 dθ1 = dz while θ2 = arctan In conclusion ωS 2 = −

³y´ x

=⇒

dθ2 = −

y x dx + 2 dy. x2 + y 2 x + y2

¯ x y ¯ dz ∧ dx + dz ∧ dy ¯ 2 2 2 . 2 2 2 2 x +y +z =1 x +y x +y

(3.16)

Consider now the Fubini-Study form (compare Equation (A.7); the change of sign is due to the conventional orientation in P1C ) ωP1 = −i C

dζ ∧ dζ da ∧ db =2 , 2 2 (1 + |ζ| ) (1 + a 2 + b 2 )2

3 When θ is in (−π/2, π/2), cos θ is positive, so that 1 1

if ζ = a + ib.

p cos2 θ1 = | cos θ1 | = cos θ1 .

3.2 Geometric identification of the composite Bloch bundle

59

This is the curvature form for the Levi-Civita connection on the bundle O(−1) defined on P1C (see footnote on page 86); the above expression holds on the affine line (3.11). We want to pullback this 2-form on the sphere S 2 via the stereographic projection map p, and compare it with the volume form ωS 2 . In order to do so, we just notice that (3.13) gives x y x + iy p ∗ζ = =⇒ p ∗ a = , p ∗b = . 1−z 1−z 1−z From these equalities it follows that ¡ ¢ 1−z 2 x 2 + y 2 x 2 + y 2 + 1 + z 2 − 2z = =2 = p ∗ 1 + |ζ|2 = 1 + 2 2 2 (1 − z) (1 − z) (1 − z) 1−z where we used the fact that on S 2 we have x 2 + y 2 + z 2 = 1. Moreover p ∗ da =

(1 − z) dx + x dz , (1 − z)2

p ∗ db =

(1 − z) dy + y dz , (1 − z)2

and consequently £ ¤ (1 − z)2 [(1 − z) dx + x dz] ∧ (1 − z) dy + y dz p ωP1 = 2 = C 4 (1 − z)4 1 (1 − z)2 dx ∧ dy + (1 − z)y dx ∧ dz + x(1 − z) dz ∧ dy = = 2 (1 − z)2 i 1h y x = dx ∧ dy − dz ∧ dx + dz ∧ dy = 2· 1−z 1−z ¸ y(1 + z) x(1 + z) 1 dx ∧ dy − dz ∧ dx + dz ∧ dy = = 2 1 − z2 1 − z2 i 1h xz yz = dz ∧ dx + dz ∧ dy ωS 2 + dx ∧ dy − 2 1 − z2 1 − z2 ∗

(we used again the relation 1 − z 2 = x 2 + y 2 valid on S 2 ). We will now show that dx ∧ dy −

¯ yz xz ¯ dz ∧ dx + dz ∧ dy ¯ 2 2 2 = 0. x +y +z =1 1 − z2 1 − z2

This is clearly equivalent to ¡

¯ ¢ ¯ x 2 + y 2 dx ∧ dy + y z dx ∧ dz + xz dz ∧ dy ¯

x 2 +y 2 +z 2 =1

= 0.

(3.17)

In fact, by applying the exterior differential to both sides of the relation x 2 + y 2 + z 2 = 1, we get x2 + y 2 + z2 = 1

d

=⇒

x dx + y dy + z dz = 0.

The wedge products of both sides of the second equality by x dy on the right and by y dx on the left yield x 2 dx ∧ dy + xz dz ∧ dy = 0,

y 2 dx ∧ dy + y z dx ∧ dz = 0.

The sum of these two equalities is exactly (3.17). Thus we have proved that 1 p ∗ ωP1 = ωS 2 . C 2

60

3. Bloch bundles in graphene µ

We are now ready to compute explicitly v ± : B R (0) → S 2 , when µ 6= 0. Clearly we have µ µ Ran P ± (q) = Cφ± (q), where the latter is to be understood as an element in P1C . Explicitly, these read ³ϕ´ ³ϕ´ ³ ϕ ´i h ³ϕ´ µ − iαµ (q) sin ; sin + iαµ (q) cos Cφ+ (q) = cos 2 2 2 2 while

h ³ϕ´ ³ϕ´ ³ϕ´ ³ ϕ ´i Cφµ− (q) = − sin + iαµ (q) cos ; cos + iαµ (q) sin 2 2 2 2 p iϕ/2 2 (the phase e and the normalisation factor 1/ 1 + α are inessential because coordinates in P1C are homogeneous). We want to intepret these as points on S 2 via p −1 , using (3.15). In order to do so, we need the second coordinate to be real: so, using homogeneity, we multiply both coordinates by the complex conjugate of the second. This leads to the same calculations we have already done when computing the second column of the µ eigenprojector P ± (q) – see Equations (3.4a) and (3.4c). Consequently, we have ¸ · q µ Cφ± (q) = sin ϕ + iηµ (q); − cos ϕ ± 1 + ηµ (q)2 . Set ζ := sin ϕ + iη and ξ := − cos ϕ ± (and hence η 6= 0). We have

p

1 + η2 ; notice that ξ is always non-zero when µ 6= 0

|ζ|2 + ξ2 = sin2 ϕ + η2 + cos2 ϕ + 1 + η2 ∓ 2 cos ϕ µ ¶ q q 2 2 = ±2 1 + η − cos ϕ ± 1 + η ; 2

2

2

2

2

2

|ζ| − ξ = sin ϕ + η − cos ϕ − 1 − η ± 2 cos ϕ µ ¶ q 2 = ±2 cos ϕ − cos ϕ ± 1 + η ; µ ¶ q 2 ξℜζ = sin ϕ − cos ϕ ± 1 + η ; ¶ µ q 2 ξℑζ = η − cos ϕ ± 1 + η .

q

1 + η2 =

q

1 + η2 =

µ

Thus substituting these in (3.15) we get that4 v ± : B R (0) → S 2 is given by Ã ! Ã ! sin ϕ η cos ϕ q2 µ q1 µ v ± (q) = ± p ,p ,p =± p ,p ,p . 1 + η2 1 + η2 1 + η2 |q|2 + µ2 |q|2 + µ2 |q|2 + µ2 (3.18) From (3.18) we deduce that5 " Ã !# ¡ µ ¢∗ ¡ µ ¢∗ sin ϕ cos ϕ 1 v± x = ± p =⇒ v ± dx = ± p dϕ + sin ϕ d p ; 1 + η2 1 + η2 1 + η2 Ã ! ¡ µ ¢∗ ¡ µ ¢∗ η η v± y = ± p =⇒ v ± dy = ± d p ; 1 + η2 1 + η2 " Ã !# ¡ µ ¢∗ ¡ µ ¢∗ 1 cos ϕ sin ϕ v± z = ± p =⇒ v ± dz = ± − p dϕ + cos ϕ d p . 1 + η2 1 + η2 1 + η2 4 Strictly speaking, the map v µ is defined only for (q , q , µ) belonging to the punctured cylinder C• – see 1 2 ±

also Section 3.3. 5 Recall that ηµ (q) = µ/|q| does not depend on ϕ.

3.2 Geometric identification of the composite Bloch bundle

Notice that Ã d p

!

1 1 + η2

61

Ã ! 1 η η 1 =− dη = −η d p . dη = −η p 1 + η2 1 + η2 (1 + η2 )3/2 1 + η2

Moreover ¡

µ ¢∗

v±

(x 2 + y 2 ) =

sin2 ϕ + η2 . 1 + η2

Using (3.16), we can compute ¡

µ ¢∗ v ± ωS 2

= ±p

η

1 + η2

"

Ã ! sin2 ϕ η −η p dϕ ∧ d p + 1 + η2 1 + η2 ! #

2 1 + η2 sin ϕ + η2 Ã η cos2 ϕ ∧ dϕ ± d p + ηp 2 1+η 1 + η2 " Ã !# η sin ϕ 1 + η2 sin ϕ ±p −p dϕ ∧ d p = 2 1 + η2 sin ϕ + η2 1 + η2 1 + η2 Ã " ! # p η 1 + η2 η η =± 2 d p ∧ dϕ ± p sin ϕ + η2 1 + η2 1 + η2 " Ã ! # p sin ϕ 1 + η2 sin ϕ η d p ∧ dϕ = ± p sin2 ϕ + η2 1 + η2 1 + η2 ! Ã sin2 ϕ + η2 η ∧ dϕ = =± 2 d p sin ϕ + η2 1 + η2 ! Ã µ ∧ dϕ. = ±d p |q|2 + µ2

Comparing this with (3.7b), we see that ¡ µ ¢∗ µ v ± ωS 2 = 2ω± or equivalently (see (3.10)) ¡

µ ¢∗

u±

µ

ωP1 = ω± . C

(3.19)

µ

Thus we see that the pullback via u ± – the map disguising all the physics of the composite Bloch bundle under a geometric mask – of the curvature form of the tautological bundle on P1C is precisely the Berry curvature form. Hence we deduce that the composite µ µ Bloch bundle relative to the Bloch band E ± (q) is the pullback via u ± of the tautological µ bundle on P1C , or equivalently the pullback via v ± of the Hopf bundle on S 2 .

3.2.1 Detecting conical intersections: a geometric criterion In this section, we will state and prove the main result of this thesis. We will call smoothed Bloch bundle a complex line bundle L with base space £ ¤ £¡ ¢ ¤ £ ¤ B := T∗ × {−µ0 } ∪ T∗ \U × (−µ0 , µ0 ) ∪ T∗ × {µ0 } , µ0 > 0, where T∗ is a 2-torus and U is the image under the modΓ∗ projection map of a ball B R (k 0 ) ⊂ B; the fibre of L over a point ([k], µ) ∈ B is the range of the Hamiltonian eigenprojection relative to a single non-degenerate Bloch band, “perturbed” by the parameter

62

3. Bloch bundles in graphene

µ. This means that, if we denote by Lµ the vector bundle (ιµ )∗ L on T∗ \U , where ιµ is the inclusion of (T∗ \U ) × {µ} in B (i.e. the restriction of L on (T∗ \U ) × {µ}), then Lµ is a Bloch bundle relative to some effective Hamiltonian H µ (k). For example, this could be the data emerging from the numerical solution of the periodic Schrödinger equation, and µ may be thought as the regularising parameter produced by the discretisation process. We are interested in knowing if the line bundle L0 is extendable on the whole torus T∗ defining a trivial Bloch bundle, or if it is non-extendable, because a conical intersection occurs at [k 0 ], i.e. because [k 0 ] is a Dirac point. The following Theorem states that to answer this question we just have to compute an integral, and thus it can be also implemented in a computational model. © ª Theorem 9. Let L = Lµ µ∈[−µ0 ,µ0 ] be a smoothed Bloch bundle, defined as above. Denote by u : B → P1C , ([k], µ) 7→ u µ (k) the smooth map associating to each ([k], µ) ∈ B the fibre of L upon ([k], µ) (i.e. the fibre of Lµ upon [k]). Let C be the cylindric surface £ © ª¤ £ ¤ £ © ª¤ C := U × −µ0 ∪ ∂U × (−µ0 , µ0 ) ∪ U × µ0 , and compute

where ωP1

C

Ï 1 n=− u ∗ ωP1 C 2π C is the Fubini-Study form, defined as in (A.7). Then:

(3.20)

1. if n = 0, the Bloch bundle L0 extends to a (necessarily trivial) bundle on the whole T∗ ; 2. if n = ±1, the Bloch bundle L0 cannot be extended to the whole T∗ . Moreover, if n = ±1, there exists a C ∞ isomorphism between the restriction of the smoothed Bloch bundle L on C and the restriction of the composite Bloch bundle Pcomp,± on C, and the maps u|C and u ± |C are smoothly homotopic. Proof. This proof essentialy consists in a recollection of results from the classification theory of fibre bundles. Precise statements and references for their proofs may be found in Appendix A. We divide the proof in a few steps. Step 1: The Berry curvature ωµ of the Bloch bundle Lµ is the pullback of ωP1 via the C map u µ . Recall that on the affine line {ζ1 6= 0} ⊂ P1C , with coordinate ζ = ζ0 /ζ1 , the FubiniStudy form is given by dζ ∧ dζ ωP1 = i ¡ ¢2 . C 1 + |ζ|2 Fix a Bloch function φ = (φ1 , φ2 ) (we drop its arguments k and µ) satisfying kφk2Hf = |φ1 |2 + |φ2 |2 = 1 (see Equation (2.8)). Then the Berry curvature associated to L is given by ³ ´ ® ωµ (k) = i dφ, dφ L = i dφ1 ∧ dφ1 + dφ2 ∧ dφ2 .

(3.21)

(3.22)

3.2 Geometric identification of the composite Bloch bundle

Lastly, the map u is given by

63

£ ¤ u µ (k) = φ1 ; φ2 .

Assume that φ2 6= 0, so that we may use the above expression for ωP1 (where this C

condition is not satisfied, one has to use another local chart in P1C , but this amounts just in exchanging the rôles of ζ0 and ζ1 ). Then we have Ã ! µ ¶ φ1 φ1 d ∧d φ2 φ2 u ∗ ωP1 = i Ã . ! ¯ ¯ C ¯φ1 ¯2 2 1 + ¯ ¯2 ¯φ2 ¯ By (3.21) we deduce that Ã

#2 ¯ ¯ 2 !2 " ¯φ1 ¯ 1 ³¯¯ ¯¯2 ¯¯ ¯¯2 ´ 1 = ¯ ¯4 1 + ¯ ¯2 = ¯ ¯2 φ1 + φ2 ¯φ2 ¯ ¯φ2 ¯ ¯φ2 ¯

so that ¯ ¯4 φ2 dφ1 − φ1 dφ2 φ2 dφ1 − φ1 dφ2 u ∗ ωP1 = i ¯φ2 ¯ = ∧ 2 C φ22 φ2 ¯ ¯4 ´ ¯φ2 ¯ ³¯ ¯ ¯ ¯2 2 = i ¯ ¯4 ¯φ2 ¯ dφ1 ∧ dφ1 − φ1 φ2 dφ2 ∧ dφ1 − φ1 φ2 dφ1 ∧ dφ2 + ¯φ1 ¯ dφ2 ∧ dφ2 . ¯φ2 ¯ (3.23) Taking the differential of both sides of Equation (3.21) gives φ1 φ1 + φ2 φ2 = 1

d

=⇒

φ1 dφ1 + φ1 dφ1 + φ2 dφ2 + φ2 dφ2 = 0.

Taking the exterior product of the above equation on the left with φ j dφ j , j = 1, 2, yields ¯ ¯2 −φ1 φ2 dφ1 ∧ dφ2 = ¯φ1 ¯ dφ1 ∧ dφ1 + φ1 φ2 dφ1 ∧ dφ2 , ¯ ¯2 −φ1 φ2 dφ2 ∧ dφ1 = ¯φ2 ¯ dφ2 ∧ dφ2 + φ1 φ2 dφ2 ∧ dφ1 . Plugging both these equalities into (3.23), we get ³¯ ¯ ¯ ¯2 2 u ∗ ωP1 = i ¯φ2 ¯ dφ1 ∧ dφ1 + ¯φ2 ¯ dφ2 ∧ dφ2 + φ1 φ2 dφ2 ∧ dφ1 + C ´ ¯ ¯2 ¯ ¯2 + ¯φ1 ¯ dφ1 ∧ dφ1 + φ1 φ2 dφ1 ∧ dφ2 + ¯φ1 ¯ dφ2 ∧ dφ2 = ³¯ ¯ ¯ ¯ ´ ³ ´ 2 2 = i ¯φ1 ¯ + ¯φ2 ¯ dφ1 ∧ dφ1 + dφ2 ∧ dφ2 = ³ ´ = i dφ1 ∧ dφ1 + dφ2 ∧ dφ2

(3.24)

again by (3.21). Comparing (3.22) with (3.24), we deduce the desired equality between ωµ (k) and u ∗ ωP1 . C

Step 2: If n = 0, then the Bloch bundle L0 on T∗ \ U can be extended to the trivial bundle on the whole T∗ .

64

3. Bloch bundles in graphene

From Step 1 we deduce that n equals the only relevant Chern number associated to the smoothed Bloch bundle L, namely ch1 (C, L). In fact, notice that in the present case the boundary C = ∂B ≈ S 2 is a deformation retract of the base space B . Since complex line bundles on S 2 ' P1C are determined up to isomorphism by their first Chern number (compare Example 6 in Appendix A), and by hypothesis n = 0, this implies that the first Chern class of the line bundle L (and consequently that of L0 ) vanishes. Recall that the first Chern class of a complex vector bundle on a CW-complex equals the primary obstruction to the extension of the bundle from the 1-skeleton to the 2skeleton (see Appendix A, Section A.2.3). The base space T∗ \ U of the bundle L0 is a deformation retract of the 1-skeleton of T∗ . Thus, the vanishing of the first Chern class of L0 implies that there is no obstruction to the extension of L0 to the whole T∗ ; moreover, this extension is necessarily trivial. Step 3: If n = ±1, the restrictions on C of the smoothed Bloch bundle L and of the composite Bloch bundle Pcomp,± are isomorphic. This again follows by the one-to-one correspondence between (isomorphism classes of ) complex line bundles and integer numbers given by the first Chern number. Firstly, from Step 1 we deduce that the restriction of the complex line bundle L over C ≈ S 2 is the pullback of the bundle O(−1) on P1C , both if n = +1 or n = −1, via the map u (actually via its restriction u|C , but we will continue denoting this restriction by u for the sake of legibility). In fact, we have the following morphism of fibre bundles: E (L )

C

/ E (O(−1))

u˜

u

/ P1 C

where E (L) is the total space of the smoothed Bloch bundle (see Equation (1.27)) while E (O(−1)) is the total space of the tautological bundle (defined in (A.4)), vertical arrows are projection maps, and ue is defined by ¡£ ¤ ¢ ¡ ¢ ue k, φ τ = Cφ, φ with the same notations as in the above-mentioned Equations. As from Step 1 we have u ∗ ωP1 = ωµ (k) we also have C

u ∗ Ch1 (O(−1)) = Ch1 (L), because Ch1 (O(−1)) (respectively Ch1 (L)) is just (−1/2π) times the Fubini-Study 2-form ωP1 (respectively the Berry curvature ωµ (k)). On the other hand, naturality of the Chern C classes (see Equation (A.12)) gives u ∗ Ch1 (O(−1)) = Ch1 (u ∗ O(−1)) and hence the smoothed Bloch bundle and the pullback of the tautological bundle via the map u have the same first Chern class. As both these bundle have rank 1, the equality of their first Chern class implies that they are isomorphic. The same argument, together with Equation (3.19), gives that also Pcomp,± is isomorphic to the pullback via the map u ± of the tautological bundle O(−1) on P1C . Now we use the hypothesis that n = ±1; the restriction of L to C has the same first Chern number of

3.2 Geometric identification of the composite Bloch bundle

65

the composite Bloch bundle Pcomp,± , restricted to C (compare (3.8)). As C ≈ S 2 ' P1C , by the above-mentioned Example 6 of Appendix A we get that L and Pcomp,± are isomorphic line bundles on C. Step 4: If n = ±1, the restrictions to C of the maps u, defined in the statement of the Theorem, and u ± , defined in (3.9), are homotopic. As C is a 2-dimensional manifold, there is a pointed-sets isomorphism (see (A.3))

KU (1) (C) ' C, P1C £

¤

between the set KU (1) (C) of isomorphism classes of fibre bundles over C having U (1) as their structure group (this is in particular the case for complex line bundles), with the £ ¤ (isomorphism class of the) trivial bundle as its basepoint, and the set C, P1C of homotopy classes of maps from C to P1C , where the basepoint is given by the (homotopy class of the) constant map. The conclusion follows, as L ' u ∗ O(−1) and the composite Bloch ∗ bundle Pcomp,± ' u ± O(−1) are isomorphic (Step 3). Step 5: The homotopy deforming (the restriction to C of ) u into (the restriction to C of ) u ± , as in Step 4, may be chosen to be smooth. This is a classic approximation theorem: whenever two smooth maps between manifolds are homotopic, there exists a smooth homotopy between them – this is [16, Proposition 10.22]. Remark 8. In the case when n = ±1, let u t : C → P1C , t ∈ [0, 1], be a smooth homotopy between the two maps u, u ± : C → P1C , as in Step 5 of the preceding proof. Then this can be used to give an explicit isomorphism between the restrictions to C of the two bundles L and Pcomp,± , whose existence was proved in Step 3. This is given on the respective fibres as follows. Let p = ([k], µ) ∈ C and denote by L(p) (respectively Pcomp,± (p)) the fibre of L (respectively Pcomp,± ) upon p. As L ' ∗ u ∗ O(−1) and Pcomp,± ' u ± O(−1), we can identify L(p) with the fibre of O(−1) upon u(p), and Pcomp,± (p) with the fibre of O(−1) upon u ± (p). Now, the curve t ∈ [0, 1] 7→ u t (p) ∈ P1C is a smooth path in P1C from u(p) to u ± (p). Choose a connection ∇ on O(−1) (for example the Levi-Civita connection); then the parallel transport ∇u˙t (p) defines an isomorphism between the fibres of O(−1) on u(p) and u ± (p), and hence, with the above identifications, between L(p) and Pcomp,± (p). An important consequence of the above Theorem 9, which can be useful in computational physics, is the following. Corollary 2. Let L be a smoothed Bloch bundle, and let n be as in (3.20). If n = ±1, then e , defined on there exist a bundle L ¡ ¢ Be := T∗ × [−µ0 , µ0 ] \ {([k 0 ] , 0)} , such that its restriction to B is L and its restriction to the punctured cylinder

C• := U × [−µ0 , µ0 ] \ {([k0 ] , 0)} ¡

¢

is Pcomp,± . In particular, [k 0 ] is a Dirac point, in the sense of Section 2.3. Proof. Vector bundles are given by “gluing” together (trivial) bundles defined on the open sets covering the base space, so we can expect that we can glue L and Pcomp,± together along C provided that their restrictions on C are isomorphic (this is Step 3 in

66

3. Bloch bundles in graphene

the previous proof). The only difficulty we have to overcome is that C is a closed subset of Be. We argue as follows. Let T be a tubular open neighbourhood of C in Be, and let ρ : T → C be a retraction of T on C. For example, let 0 < r < R be so small that B R+r (k0 ) is all e contained in B (this r exists because B R (k 0 ) ⊂ B is open while B is closed) and define U to be the image under the mod Γ∗ projection map of the set B R+r (k 0 ) \ B R−r (k 0 ); then e × [−µ0 , µ0 ] and take T = U ρ : T → C,

¸ ¶ µ· k − k0 ,µ . ([k], µ) 7→ k 0 + R |k − k 0 |

As T ∩ C• is a deformation retract of C via the map ρ, we may extend the definition of L to T ∩ C• by letting ¡ ¢ L|T ∩C• = ρ ∗ L|C . Similarly, we can extend Pcomp,± outside C• setting

Pcomp,± |T ∩B = ρ ∗ Pcomp,± |C . ¡

¢

With these definitions, we can prove that

L | T ' ρ ∗ L |C ¡

¢

(3.25a)

and similarly

Pcomp,± |T ' ρ ∗ Pcomp,± |C . ¡

¢

(3.25b)

In fact, let V denote either L or Pcomp,± . By Remark 10 ³in Appendix A ´ we know that π ∞ all line bundles admit a morphism of bundles on UU (1) = EU (1) − → PC , the tautological bundle on P∞ , and that their isomorphism classes are uniquely determined by the C homotopy class of the map between the base spaces. Let / EU (1)

¡ ¢ E V|T T

fT

/ EU (1)

¡ ¡ ¢¢ E ρ ∗ V|C

and

/ P∞ C

T

fC

/ P∞ C

be the two morphisms just described; then (3.25a) and (3.25b) will hold as long as we prove that f T and f C are homotopic. Now, the following diagram fC

T ρ

/ P∞ C O

1T

C

≈ ι

fT

/T

where ι : C ,→ T denotes the inclusion map, is clearly commutative. By definition of deformation retract, the maps ι ◦ ρ and 1T are homotopic: hence f C = f T ◦ ι ◦ ρ ≈ f T ◦ 1T = f T as was to prove.

3.3 Berry curvature for conical intersections

67

By Theorem 9 we have L|C ' Pcomp,± |C , and hence also ¡ ¢ ¡ ¢ ρ ∗ L|C ' ρ ∗ Pcomp,± |C . Equations (3.25a) and (3.25b) thus give

L|T ' Pcomp,± |T . Thus the two line bundle L and Pcomp,± are isomorphic on the open set T , and this allows us to glue them. The above Corollary can be used in the following context. Suppose that L arises from the numerical solution of a periodic Schrödinger equation, so that µ can be thought of as the parameter controlling the error given by knowing the numerical values of the solution only on a discrete mesh. Thus the Bloch bundle has been constructed only away from a certain region of space, and we want to investigate the presence of a conical intersection in this region. By the above results, we have to calculate the integral (3.20) defining n; if the result is 0, then the bundle is extendable to a trivial bundle inside the region, and Bloch bands stay separated. On the other hand, if n = ±1, then there is a Dirac point in the region, and the Bloch functions can be interpolated using the canonical form φ± defined in (2.9).

3.3 Berry curvature for conical intersections Is there a way to define the Berry curvature also in the case µ = 0? Rigorously speaking, notions like “connection” and “curvature” make sense only in the case of C ∞ vector bundles, as they are defined through differential forms. The Bloch bundle for a conical intersection, on the other hand, is singular at the Dirac point q = 0 – see for example Equation (2.12). Thus, in order to define them also in the latter case, we have to “pay a toll”: this amounts to use differential forms intepreted in a distributional sense. In fact, as µ → 0, we obtain from (3.18) µ ¶ ¡ ¢ q2 q1 0 v ± (q) ≡ v ± (q) = ± sin ϕ, 0, cos ϕ = ± . , 0, |q| |q| Notice that this map is defined only on B R (0) \ {0}, and thus is not smooth on the whole ball. In order to “isolate” the regularity problems of v ± , we consider it as the composition of three maps: 1. the “inversion” map ι : B R (0) → B R (0) mapping (q 1 , q 2 ) to (q 2 , q 1 ); ˆ 2 → S 1 , sending the vector k = (k 1 , k 2 ) to k/|k|, re2. the “normalisation” map ν : R stricted to the ball of radius R centered at the origin; ˆ 2 with the equatorial circle 3. the “immersion” map j : S 1 → S 2 , identitifing S 1 ⊂ R 2 {y = 0} ⊂ S . Clearly, the map ν is singular at q = 0, while ι and j are “well-behaved” functions. ˆ 2 ) for all 1 ≤ p < 2, while it is not in W 1,p (R ˆ 2) Moreover, ν is in the Sobolev space W 1,p (R for all p ≥ 2, as simple calculations show (or just noticing that ν is an homogeneous map of degree 0). These maps do not possess enough regularity to define a pullback

68

3. Bloch bundles in graphene

of differential forms in the classic sense; however, one may generalise this theory using differential forms with distributional coefficients. In particular, the map ν is thouroghly studied in [8, Chap. 3, Secs. 2.2 and 2.4], where the authors show that it possesses a distributional Jacobian determinant proportional to the Dirac delta mass at q = 0. This means that the Berry curvature in the case of the conical crossing is “concentrated” at the Dirac point. We make this last statement more rigorous. We want to recover the conical intersection from the avoided crossing, defining for all test functions f ∈ C 0∞ (B R (0)) (i.e., infinitely many times differentiable functions with compact support in the ball B R (0)) µ

lim ω± [ f ] =: ω± [ f ],

(3.26)

µ→0

where T [ f ] denotes the action of the distribution T on the test function f . The distributions on the left-hand side of (3.26) are the ones obtained from the Berry curvature (3.7a) when µ is a fixed non-zero constant between −µ0 and µ0 (thus dµ = 0). Explicitly, these act on test functions as µ ω± [ f

1 ]=± 2

Ï B R (0)

d|q| ∧ dϕ ∂|q| p

µ |q|2 + µ2

f (|q|, ϕ).

(3.27)

We may assume that f is a radial-symmetric test function6 , i.e. f = f (|q|). To avoid any possible problem with the ill-posed definition of the polar coordinates at q = 0, we integrate in (3.27) over the annulus D r,R := B R (0) \ B r (0),

with 0 < r < R,

and then take the limit r → 0. Moreover, as we are more interested in computing Chern numbers rather than in the curvature itself, we get rid of the integration on the angle variable by dividing by a factor −2π. Hence we have 1 1 µ − ω± [ f ] = lim ∓ r →0 2π 2 6 If fe is in C ∞ (B (0)) and R 0

R

Z r

Ã d|q| ∂|q| p

!

µ

f (|q|).

|q|2 + µ2

1 µ µ , T± (|q|) := ± ∂|q| q 2 |q|2 + µ2

then we have µ£ ¤ ω± fe =

Ï

µ

B R (0)

d|q| ∧ dϕ T± (|q|) fe(|q|, ϕ) =

Set

Z 2π f (|q|) := 0

Z R 0

µ

d|q| T± (|q|)

Z 2π 0

dϕ fe(|q|, ϕ).

dϕ fe(|q|, ϕ).

Then f is a C ∞ function, because fe is smooth and integration is performed on the compact set S 1 ; moreover, the support of f is contained in B Re (0) where Re :=

max |q|. q∈supp fe

By definition Re < R, because fe has compact support in B R (0); hence also f is compactly supported in the same ball.

3.3 Berry curvature for conical intersections

69

Integration by parts gives R

Z r

Ã ∂|q| p

µ |q|2 + µ2

!

" f (|q|) d|q| = f (|q|) p

µ |q|2 + µ2

#|q|=R Z − |q|=r

µ

R r

d|q| p

|q|2 + µ2

∂|q| f (|q|).

In the boundary term, the evaluation at |q| = R of f gives zero, because f is supposed to be compactly supported. By taking the limit r → 0 we get " # Z R 1 µ 1 µ − ω± [ f ] = ∓ − f (0) − d|q| p ∂|q| f (|q|) . 2π 2 0 |q|2 + µ2 Now we notice that ¶ µ Z R Z R ¤ 1 1 1 £ ∂|q| f (|q|) = d|q| ∂|q| f (|q|) = p.v. lim d|q| p ∂|q| f = µ→0 0 |q| |q| 0 |q|2 + µ2 µ ¶ µ ¶ 1 1 = −∂|q| p.v. [ f ] = − F.p. 2 [ f ] |q| |q| where p.v.(1/|q|) is the Cauchy principal value, while F.p.(1/|q|2 ) is its distributional derivative, the Hadamard finite part. Hence this limit has a perfectly defined sense as a distribution and we deduce that Z R 1 1 1 1 1 µ − lim ω [ f ] = ± f (0) ± lim µ d|q|∂|q| f (|q|) = ± f (0). p 2 2 2π µ→0 ± 2 2 µ→0 0 2 |q| + µ In conclusion, ω± (q), as a distribution, is a multiple of the Dirac delta “function”: −

1 1 ω± = ± δ0 . 2π 2

70

3. Bloch bundles in graphene

Appendix A

Fibre bundles and Chern classes Obstruction theory [. . . ] is rather like being in a complicated labyrinth with only a weak torch. The obstructions per se only tell if our progress can be extended a little farther while allowing only small corrections. D ENNIS S ULLIVAN On the other hand, if you were a miner would you care to be without your light? Likewise, no topologist would forsake obstruction theory. P HILLIP A. G RIFFITHS J OHN W. M ORGAN

This Appendix is meant to include all definitions and results from differential geometry that were needed in this thesis. We start from the general theory of fibre bundles in the topological, smooth and analytic frameworks, then we specialise to complex vector bundles and illustrate the theory of Chern’s characteristic classes. The main references will be [25] and [2], for the theory of fibre bundles, and [9], for the theory of characteristic classes.

A.1 Fibre bundles In this section, C will denote one of the following categories: • the category Top of topological spaces, where morphisms are continuous maps between these spaces; • the category Man of (paracompact) smooth manifolds, where morphisms are C ∞ maps between these spaces; • the category Hol of complex analytic varieties, where morphisms are holomorphic maps between these spaces. Moreover, a group in the category C will mean: a topological group, if C = Top; a Lie group, if C = Man; an analytic group, if C = Hol. 71

72

A. Fibre bundles and Chern classes

A.1.1 Definition and first examples of fibre bundles Definition 3. A fibre bundle F in the category C is the collection of the following elements: 1. an object E of the category C, called the total space; 2. an object B of the category C, called the base space; 3. a surjective morphism π ∈ HomC (E , B ), called the projection; 4. an object F of the category C, called the typical fibre; 5. a group G in the category C, called the structure group of the bundle, together with a free action1 of G on F (equivalently, G is a subgroup of HomC (F, F )); 6. the equivalence class of a set of local trivialisations. The latter is a collection of isomorphisms ¡ ¢ Φ j ∈ HomC U j × F, π−1U j , © ª where U j j ∈J is an open covering of B , satisfying ¡ ¢ π Φ j (x, v) = x

for all x ∈ U j and all v ∈ F,

and such that, if we define Φ j ,x (v) := Φ j (x, v), then for all i , j ∈ J and for all x ∈ Ui ∩U j the isomorphism Φ−1 j ,x Φi ,x ∈ HomC (F, F ) coincides with the action of an element g j ,i (x) ∈ G on F . Moreover, it is required that the map g j ,i : Ui ∩U j → G just defined is a morphism in C. n o © ª Two such sets of local trivialisations Φ j j ∈J and Φ0j 0 0 0 are said to be equivalent j ∈J

if their union is again a set of local trivialisations. The set π−1 (x) ⊂ E , for x ∈ B , is called the fibre over x; by our hypotheses, if x ∈ U j then Φ j ,x is an isomorphism in C between the typical fibre F and the fibre over x. ³ ´ π The fibre bundle F will also be denoted by F = E − →B . © ª An important consequence of the definition of a fibre bundle is that if Φ j j ∈J is a set of local trivialisations, then for all i , j , k ∈ J and for all x ∈ Ui ∩U j ∩Uk the following cocycle condition holds: g i ,i (x) = g i ,k (x)g k, j (x)g j ,i (x) = e,

the identity of the group G.

(A.1)

In particular, if i = k we get g i , j (x) = g j ,i (x)−1 . 1 A (left) action of the group G on the space F is a morphism in Hom (G × F, F ), denoted by C

G × F → F,

(g , v) 7→ g · v,

such that if e ∈ G is the identity of the group then e · v = v for all v ∈ F , and moreover g · (h · v) = g h · v for all g , h ∈ G and for all v ∈ F . The action is called free if g · v = v for all v ∈ F implies g = e.

A.1 Fibre bundles

73

³ ´ π Definition 4. A (cross) section of the bundle F = E − → B is a morphism s ∈ HomC (B, E ) such that p ◦ s = 1B , or equivalently s(x) ∈ π−1 (x) for all x ∈ B. The set of all sections of the bundle F will be denoted by Γ(F) or Γ(B ; E ). Example 1. Let F and B be arbitrary objects in C and define E := B × F, π := π1 : B × F → B, (x, v) 7→ x. ´ Then T = B × F −→ B defines a fibre bundle, with structure group the trivial group G = {e} and (the equivalence class of ) the only local trivialisation ³

π1

Φ = 1B ×F : B × F → π−1 1 B = B × F. This bundle is called the trivial (or product) bundle over B with fibre F . All sections of the trivial bundle are of the form s(x) = (x, f (x)), where f is a morphism in HomC (B, F ). Our next example is the Möbius band. Consider the product [0, 1] × R of the unit interval and the real line, and visualise it as many copies of R each “sitting” at one point of [0, 1]. Identify the two end-points {0} and {1} of this interval, but when doing so twist the fibres, reflecting one of them around its origin. What you should get is something like this.

Figure A.1. The Möbius band

This is an example of the total space a non-trivial fibre bundle on the circle S 1 , represented by the unit interval with its end-points glued together. Here the structure group is Z2 , the cyclic group of order two, generated by the reflection of R around the origin. Sections of the Möbius band are best described if we think of S 1 as R/Z: then a section is the image under the quotient map R → R/Z of an antiperiodic function on the real line of period 1, i.e. of a function such that f (x) = − f (x + 1) for all x ∈ R. ³ ´ ³ ´ π1 π2 Definition 5. Given two bundles F1 = E 1 −→ B 1 and F2 = E 2 −→ B 2 such that their respective typical fibres and structure groups coincide (and are denoted by F and G, respectively), a morphism between F1 and F2 is a commutative diagram E1

fe

π1

B1

/ E2 π2

f

/ B2

74

A. Fibre bundles and Chern classes

where all arrows are morphisms in C. (This condition may be expressed as ¡ ¢ ¡ ¢ −1 fe π−1 f (x) , for all x ∈ B 1 1 (x) ⊂ π2 i.e. fe is “fibre-preserving”: it maps the fibre over the point x ∈ B 1 on the fibre over the (1) −1 −1 (2) point f (x) ∈ B 2 .) Moreover, denoting by fex : π−1 Uj , 1 (x) → π2 ( f (x)), for x ∈ U j 1 ∩ f 2 e the map induced by f by restriction, then ³ ´−1 fex Φ(1) gej 2 , j 1 (x) := Φ(2) j ,x j , f (x) 1

2

is a morphism in HomC (F, F ); it is required that it coincide with the action of an element in G, and that the map gej 2 , j 1 : U j(1) ∩ f −1U j(2) → G be a morphism in C. 1

2

Clearly there exists an identity morphism 1F : F → F of a fibre bundle F into itself; moreover, bundle morphisms can be composed and this composition is associative, so that fibre bundles form a category Bun. Notice that the commutativity of the above diagram defining a morphism of bundles uniquely determines the map f between the base spaces, once the map fe of the total spaces is given: this is due to the surjectivity of projections. However, in most cases it is more convenient to work on the base spaces, so that the map f is given, and the question of wheter there exists a map fe lifting f onto the total spaces is non-trivial. A particular case of the above definition is when B 1 = B 2 = B and f is the identity morphism of B . In this case, we obtain the category BunB of fibre bundles over B . ³ ´ π Definition 6. Let F = E − → B be a fibre bundle, and let B 0 be an object in C. Given a morphism f ∈ HomC (B 0 , B ), define © ª E 0 := (x, p) ∈ B 0 × E : f (x) = π(p)

and π0 : E 0 → B 0 ,

(x, p) 7→ x.

Also let U j0 := f −1U j and ¡ ¢−1 Φ0j : U j0 × F → π0 U j0 ,

(x, v) 7→ (x, Φ j ( f (x), v))

where F is the typical fibre of the bundle F. One checks (see [25, Sec. 10.2]) that the above data define a fibre bundle on B 0 , called the induced or pullback bundle; this bundle will be denoted by f ∗ F. Moreover, there is a natural morphism of bundles f ∗ F → F, given by E0

fe

π0

B0

f

/E /B

fe(x, p) := p. π

Let s ∈ Γ(F) be a section of the fibre bundle F. Then we can define the pullback section f ∗ s ∈ Γ( f ∗ F) as f ∗ s(x) := (x, s( f (x))),

for all x ∈ B 0 .

The right-hand side of the above equality is indeed an element of E 0 because π(s( f (x))) = f (x), as s is a section of F. Moreover π0 ( f ∗ s(x)) = x so that f ∗ s is indeed an element of Γ( f ∗ F).

A.1 Fibre bundles

75

Remark 9. If F is a fibre bundle over B , then one has 1∗B F = F. In addition to this, if ¡ ¢∗ f ∈ HomC (B 0 , B ) and g ∈ HomC (B 00 , B 0 ), then f ◦ g F = g ∗ f ∗ F. Example 2. Let c : B 0 → B be a constant map, and let F be a fibre bundle over B . Then c ∗ F is isomorphic to the trivial bundle over B 0 with the same typical fibre of F. Recall that two morphism f 0 , f 1 ∈ HomC (B 0 , B ) are called homotopic if there exists a morphism h ∈ HomC (B 0 × [0, 1], B ) such that h(x, 0) = f 0 (x) and h(x, 1) = f 1 (x) for all x ∈ B 0 . Homotopy defines an equivalence relation in the set HomC (B 0 , B ); we will denote the quotient space by [B 0 , B ]. Two spaces B and B 0 are said to be homotopically equivalent if there exists two morphisms f ∈ HomC (B, B 0 ) and g ∈ HomC (B 0 , B ) such that f ◦g is homotopic to 1B 0 and g ◦ f is homotopic to 1B . A space B is said to be contractible if it is homotopically equivalent to a point. Theorem 10. Let f 0 , f 1 ∈ HomC (B 0 , B ) be homotopic, where B 0 is a paracompact space, and let F be a bundle on B . Then the pullback bundles f 0∗ F and f 1∗ F are isomorphic in the category BunB 0 . Proof. See [25, Thm. 11.5]. Corollary 3. If B is paracompact and contractible, then ³ ´ all fibre bundles over B with fibre π1

F are isomorphic to the trivial bundle T = B × F −→ B .

Proof. Let f : B → {∗} and g : {∗} ³→ B be´maps such that f ◦ g is homotopic to 1{∗} and π g ◦ f is homotopic to 1B . Let F = E − → B be a fibre bundle with typical fibre F . Then

F = 1∗B F = g ◦ f ¡

¢∗

F = f ∗ g ∗ F.

Now g ∗ F is a bundle over the point {∗} having the same typical fibre F of F, while f is constant map, and hence by Example 2 f ∗ g ∗ F is the trivial bundle with typical fibre F.

A.1.2 Principal bundles and universal bundles Definition 7. A fibre bundle F is said to be principal if the typical fibre F coincides with the structure group G, and the action of G (the structure group) on itself (the typical fibre) is given by left translations L g : G ×G → G,

(g , v) 7→ g v.

Principal bundles are the key tool in the classification of fibre bundles on a given space B with given structure group G. In fact, every fibre bundle is completely specified, up to isomorphism, by its equivalence class of local trivialisations [25, Thm. 3.3], or © ª equivalently by a set of coordinate functions g j ,i : Ui ∩U j → G i , j ∈J satisfying the cocycle condition (A.1): in the definition of the latter only the base space B and the group structure G are involved, while the fibre F plays no rôle. In fact, if the coordinate func© ª tions g j ,i i , j ∈J are given, then one can set Ee =

[ j ∈J

Uj ×F

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A. Fibre bundles and Chern classes

and define the equivalence relation on Ee by (x, v) ∈ Ui × F ∼g (x 0 , v 0 ) ∈ U j × F

if and only if x = x 0 ∈ Ui ∩U j and v 0 = g j ,i (x)v. © ª The total space of the bundle associated to g j ,i i , j ∈J is then E = Ee/ ∼g with the obvious projection. ³ ´ π

Hence, for a given fibre bundle F = E − → B with typical fibre F , we can always cone which has the same coordinate functions of F struct the associated principal bundle F

but has G as its fibre, acting on itself by left translations. Then one gets that F1 and F2 f1 and F f2 are isomorphic objects of are isomorphic in the category BunB if and only if F the same category. We saw in Theorem 10 that homotopic maps of a paracompact space B 0 to a space B induce isomorphic pullback bundles over B 0 , whenever a bundle on B is given. We would like the converse of this fact to hold, that is, that if two bundles F0 and F1 over B 0 are isomorphic then there exists some “magic space” B , some “magic bundle” F and homotopic maps f 0 , f 1 ∈ HomC (B 0 , B ) such that f j∗ F = F j , j = 0, 1 – actually, it would suffice that this hold true only when F0 and F1 are principal bundles, due to the preceeding observations. As one should expect, this is not true for all base spaces B 0 and all groups G. Thus, we are lead to the following definition. Recall that a CW-complex is, roughly speaking, a topological space which has been constructed by gluing together d -dimensional cells, i.e. spaces which are homeomorphic to a closed disk in Rd , for various d ’s (possibly infinite), by a suitable identification of their boundaries; the precise definition may be found in [12, Appendix]. The n-skeleton of a CW-complex K is the sub-complex of K formed by the union of all its d -dimensional cells for d ∈ {0, . . . , n}. We say that a CW-complex is n-dimensional if it coincides with its n-skeleton, or equivalently if there are no cells in dimensions strictly greater than n. For example, the sphere S n is a n-dimensional CW-complex, obtained by gluing a 0-cell (i.e. a point) and an n-cell, collapsing the whole boundary of this n-disk on the point. Definition 8. Let KG (X ) denote the set of all isomorphism classes of principal bundles over X with structure group G. ³ ´ πG

A principal bundle UG = EG −−→ BG with structure group G is called n-universal if for any n-dimensional CW-complex K the correspondence [K , BG] → KG (K ),

homotopy class of f 7→ isomorphism class of f ∗ UG

is one-to-one2 . ³ ´ πG A principal bundle UG = EG −−→ BG with structure group G is called universal if it is n-universal for all n ∈ N. In this case, BG is sometimes called the classifying space for the group G. We restict our attention on CW-complexes because of their “combinatorial” topological nature. As any smooth manifold is homotopically equivalent to a CW-complex (this is a basic result in Morse theory: see [18, Thm. 3.5 and Cor. 6.7]), when C = Man this will not be an actual restriction. Now we can reformulate our previous question as follows: Which groups G admit universal bundles? The answer is provided by the following theorem. 2 In the book of Steenrood, this is called (n + 1)-universal.

A.1 Fibre bundles

77

Theorem 11. Let G be a connected Lie group. Then, for all n ∈ N, there exists a n-universal bundle with structure group G. Proof. This is [25, Thm. 19.6 and Rmk. 19.8]. The proof is given for a compact Lie group G, for which an immersion (or rather a faithful representation) in O(N ), the group of N × N orthogonal matrices, is known to exists for sufficiently large N : see for example [14, Cor. 4.22]. The result may be extended to connected Lie groups by virtue of the following fact: Any connected Lie group G is the product of a maximal compact subgroup K (isomorphic to a torus) and an abelian subgroup E (isomorphic to a Euclidean space). (The same result holds more generally for a semisimple Lie group with a finite center, see [14, Thm. 6.31]. If the group is connected these hypotheses are not necessary, as was proved by Iwasawa: see the reference in [25, Rmk. 12.14].) Then one can apply a procedure called reduction of the structure group [25, Cor. 12.8]: isomorphism classes of bundles with structure group G over a paracompact space are in one-to-one correspondence with isomorphism classes of bundles with structure group K over the same space. © ª In other words, the coordinate functions g j ,i i , j ∈J of a bundle with structure group G may be chosen to lie in the maximal compact subgroup of G. The following result (which is valid for all groups G) gives us a way to understand if a given principal bundle with structure group G is universal, in terms of topological properties of its total space. Recall that πn (X ) := [S n , X ], the set of homotopy classes of maps from the n-sphere to the space X , has a natural group structure (and is actually abelian if n > 1), and it is hence called the n-th homotopy group of X . With a slight abuse of notation, we will denote by π0 (X ) the set of path-connected components of X . A space X is said to be n-connected if π j (X ) is trivial for all j = 0, 1, . . . , n. ³ ´ πG Theorem 12. A principal bundle UG = EG −−→ BG with structure group G is n-universal if and only if the total space EG is n-connected. Moreover π j +1 (BG) ' π j (G),

for all j = 0, 1, . . . , n − 1.

(A.2)

³ ´ πG A principal bundle UG = EG −−→ BG with structure group G is universal if and only if EG is contractible. Moreover, the isomorphism in (A.2) holds for all j ∈ N. Proof. See [25, Thms. 19.4 and 19.9] for the first part. The second part follows from the following fact: A connected CW-complex X is contractible if and only if all its homotopy groups are trivial (this is an application of Whitehead’s theorem, see [12, Thm. 4.5]). Example 3. (See [2].) The complex projective line may be realised as the space of equivalence classes defined by the following equivalence relation on C2 \ {0}: (z 1 , z 2 ) ∼ (w 1 , w 2 ) if there exists λ ∈ C \ {0} such that (w 1 , w 2 ) = λ (z 1 , z 2 ) . Consider the projection π : C2 \ {0} → P1C ,

(z 1 , z 2 ) → [z 1 ; z 2 ]

and restrict it to the 3-sphere © ª S 3 = (z 1 , z 2 ) ∈ C2 : |z 1 |2 + |z 2 |2 = 1 .

78

A. Fibre bundles and Chern classes

Compose it with the inverse stereographic projection p −1 : P1C → S 2 defined in (3.14) to obtain ¡ ¢ πHopf : S 3 → S 2 , πHopf = p −1 ◦ π|S 3 . If [z 1 ; z 2 ] ∈ P1C , we may choose a representative in this class satisfying |z 1 |2 + |z 2 |2 = 1, because the coordinates in P1C are homogeneous. Thus the inverse image of [z 1 ; z 2 ] via the map π|S 3 is given by those (w 1 , w 2 ) ∈ C2 satisfying (w 1 , w 2 ) = λ (z 1 , z 2 ) ,

|w 1 |2 + |w 2 |2 = 1 = |z 1 |2 + |z 2 |2

¡ ¢−1 which implies |λ| = 1. Hence the set π|S 3 ([z 1 ; z 2 ]) consists of the great circle on S 3 passing from (z 1 , z 2 ). ³ ´ πHopf

Actually more can be proved, namely that H = S 3 −−−→ S 2 is a principal fibre bundle with structure group U (1) [25, Sec. 20], called the Hopf bundle. By the above theorem, we deduce that this bundle is 2-universal because π0 (S 3 ) = π1 (S 3 ) = π2 (S 3 ) = {0}. Hence for all 2-dimensional CW-complex Σ we have

KU (1) (Σ) ' Σ, S 2 ' Σ, P1C .

(A.3)

© ª E := (`, v) ∈ P1C × C2 : v ∈ `

(A.4)

£

¤

£

¤

Consider now (we think of P1C as the set of lines ` in C2 ), and call π1 : E → P1C the restriction to E of ³ π ´ 1 the projection map from P1C × C2 on the first factor. It may be proved that U = E −→ P1C is a fibre bundle (see also Example 6), called the tautological bundle over P1C : the fibre over the point in P1C represented by the line ` ⊂ C2 is just the line itself, and the structure group is the group of linear automorphisms of a complex vector space of dimension 1. We want to show that the pullback via the stereographic projection map p : S 2 → P1C of this tautological bundle U is closely related to the Hopf bundle, defined above. In ³ ´ π0 order to do so, let p ∗ U = E 0 −→ S 2 where (compare Definition 6) © ª E 0 = (x, (`, v)) ∈ S 2 × E : p(x) = π(`, v) = ` ,

π0 (x, (`, v)) = x.

Moreover, let s 0 ∈ Γ(U) be the zero section, whose value at the point ` ∈ P1C is the zero vector 0 ∈ `; thus we also have the associated pullback section p ∗ s 0 : S 2 → E 0 . Define the map ¡ ¢ fe: S 3 → E 0 \ p ∗ s 0 (S 2 ), fe(z 1 , z 2 ) = p −1 ([z 1 ; z 2 ]) , ([z 1 ; z 2 ] , (z 1 , z 2 )) . One immediately checks that the diagram S3

fe

/ E 0 \ p ∗ s 0 (S 2 )

πHopf

S2

π0

p

/ P1 C

is commutative, and thus fe defines an isomorphism of bundles: its inverse is given by © ª fe−1 : E 0 \ p ∗ s 0 (S 2 ) = (x, (`, v)) ∈ S 2 × E : p(x) = `, v 6= 0 → S 3 ,

(x, (`, v)) 7→

v . |v|

A.2 Chern classes. . .

79

A.2 Chern classes. . . An important class of fibre bundles is represented by vector bundles: these are just bundles where the typical fibre F carries a vector-space structure, and the structure group G is (a subgroup of ) GL(F ) with the natural action on F . We will be primarly interested in complex vector bundles, namely the case where the base field of the fibre vector spaces is C. The complex dimension of the typical fibre will then be called the rank of the vector bundle; vector bundles with rank equal to 1 will be called line bundles. The reduction process described in the proof of Theorem 11 tells us that the transition function of a rank-m complex vector bundle may be chosen to lie in U (m), the group of m × m unitary matrices; in fact, this is the maximal complex subgroup of GL(m, C) (this is basically due to the fact that every invertible matrix A has a polar decomposition A = U e X , where U is unitary and X is Hermitian). Remark 10. We know from Theorem 11 that U (m) admits n-universal principal bundles for every n ∈ N. Actually, it can be shown (see [19, Thm. 14.6]) that the classifying space for U (m) is the infinite Grassmannian BU (m) := G(m, C∞ ) of m-dimensional vector spaces in an “infinite-dimensional” ambient space C∞ . The universal bundle UU (m) = (EU (m) → G(m, C∞ )) is the so-called tautological bundle, whose fibre over the point of G(m, C∞ ) represented by the m-dimensional space V is just V itself. Thus, this is a rank-m bundle. In the case m = 1, the infinite Grassmannian G(1, C∞ ) is also called the infinite projective space, and denoted by P∞ C . Notice that nearly every operation that makes sense for vector space carries over ³ ´ ³ ´ π π0 to vector bundles: for example, if V = E − → B and V0 = E 0 −→ B are complex vector bundles over the same space having ranks m and m 0 , respectively, then we may define: 1. the direct sum vector bundle V ⊕ V0 whose fibre over the point x ∈ B is just π−1 (x)⊕ ¡ 0 ¢−1 π (x); the rank of this vector bundle is m + m 0 ; 2. the dual vector bundle V∗ whose fibre over the point x ∈ B is just the dual space to π−1 (x), namely π−1 (x)∗ ≡ HomVect (π−1 (x), C) (Vect denotes the category whose objects are vector spaces and morphisms are given by linear maps between these spaces); the rank of this vector bundle is m; 3. the tensor product vector bundle V ⊗ V0 whose fibre over the point x ∈ B is just ¡ ¢−1 π−1 (x) ⊗C π0 (x); the rank of this vector bundle is mm 0 ; V 4. the k-th exterior power vector bundle k V Ãwhose fibre over the point x ∈ B is just ! n Vk −1 π (x); the rank of this vector bundle is . k Another important feature of vector spaces that, ³ ´ in some way, carries over to vector π

bundles, is the concept of a basis. If V = E − → B is a vector bundle of rank m, we say that a collection of sections {s 1 , . . . , s m } of V over an open set U ⊂ B is a local frame for V if {s 1 (x), . . . , s m (x)} is a basis for π−1 (x) for all x ∈ U . Local frames are very much like local trivialisations. In fact, if ΦU : U × Cm → π−1U is a isomorphism in the category C and if {e1 , . . . , em } denotes the canonical basis of Cm , then setting ¡ ¢ s j (x) := ΦU x, e j ,

j = 1, . . . , m,

80

A. Fibre bundles and Chern classes

we get a local frame of V over U . Conversely, if {s 1 , . . . , s m } is a local frame over U , then we may define ! Ã m m X X v j s j (x) v j e j := ΦU x, j =1

j =1

and obtain a local trivialisation of V over U . Notice that if s is any section of V over U , then we may write s(x) =

m X

¡ ¢ σ j (x)ΦU x, e j ,

x ∈U,

j =1

and thus represent s as σ = (σ1 , . . . , σm ) ∈ HomC (U , Cm ). If V ⊂ B is another open subset ¡ ¢ and σ0 = σ01 , . . . , σ0m is the representation of s on U ∩ V , then we have m X j =1

m ¡ ¢ ¡ ¢ X σ0j (x)ΦV x, e j σ j (x)ΦU x, e j =

=⇒

m X

σ j (x)e j =

j =1

j =1

m X j =1

¡ ¢ −1 σ0j (x)ΦU ΦV x, e j

and consequently σ = gUV σ0 . (A.5) ª Conversely, if g j ,i : Ui ∩U j → GL(m, C) i , j ∈J is a set of transition functions for the vector © ª bundle V, and if σ j j ∈J is a collection of morphisms σ j ∈ HomC (U j , Cm ), satisfying the condition σ j = g j ,i σi on Ui ∩U j for all i , j ∈ J , ©

then there exists a global section s ∈ Γ(V) such that s|U j = σ j . (This is called the sheaf condition.) Example 4. Let M be a m-dimensional smooth manifold. We may define the tangent bundle T M as follows: if T x M denotes the tangent space at x to M , consider [ TM = T x M , and π : T M → M , v ∈ T x M 7→ x. x∈M

© ª If Ψ j : U j → Rm j ∈J is an atlas for the manifold structure of M , define Φ j : U j × Rm → π−1U j =

[ x∈U j

Tx M ,

¡ ¢ (x, v) 7→ DΨ−1 Ψ j (x) v j

where DΨ−1 denotes the Jacobian matrix of the diffeomorphism Ψ−1 : Rm → U j . As is j j © ª easily checked, the collection Φ j j ∈J defines a set of local trivialisations for the bundle ³ ´ π TM = T M − → M . This is an example of a real vector bundle, whose rank is the dimension of the manifold m. Sections of the tangent bundle are just vector fields. Now let M be a d -dimensional complex manifold. We can look at M as a (real) smooth manifold of dimension m = 2d , and thus consider the (real) tangent bundle TR M we just defined. If we repeat the construction above starting from the complexified tangent spaces T xC M := T x M ⊗R C, we would get the complexified tangent bundle TC M . Explicitly, if (z 1 , . . . , z d ) are holomorphic local coordinates in M around x, and if we write z j = v j +iu j for the real and imaginary parts of these coordinates, then we have T x M = R∂v 1 ⊕ R∂u1 ⊕ · · · ⊕ R∂v d ⊕ R∂ud , T xC M = C∂v 1 ⊕ C∂u1 ⊕ · · · ⊕ C∂v d ⊕ C∂ud = = C∂z1 ⊕ C∂z1 ⊕ · · · ⊕ C∂zd ⊕ C∂zd .

A.2 Chern classes. . .

81

© ª (Actually the set ∂v 1 , ∂u1 , . . . , ∂v d , ∂ud defines a smooth local frame for TR M .) Set T x(1,0) M := C∂z1 ⊕ · · · ⊕ C∂zd ,

T x(0,1) M := C∂z1 ⊕ · · · ⊕ C∂zd = T x(1,0) M .

The d -dimensional space T x(1,0) M will be called the holomorphic tangent space of M at x, while T x(0,1) M will be called the antiholomorphic tangent space of M at x. We clearly have T xC M = T x(1,0) M ⊕ T x(0,1) M , and as a change of local coordinates of a complex manifold is defined by an holomorphic function we have a corresponding splitting of the complexified tangent bundle into its holomorphic and antiholomorphic part:

TC M = T (1,0) M ⊕ T (0,1) M . We also define 1. TR∗ M := (TR M )∗ , the cotangent bundle (denoted simply by T ∗ M if M is a smooth manifold, regardless of any complex structure); 2. TC∗ M := (TC M )∗ , the complexified cotangent bundle; we have an analogous splitting TC∗ M = T ∗(1,0) M ⊕ T ∗(0,1) M into the holomorphic cotangent bundle and antiholomorphic cotangent bundle. Tensor, symmetric and exterior powers of the (complexified) tangent and cotangent bundles are usually called the tensor bundles associated to the (complex) manifold M . In particular, we will make large use of the following: V 1. when M is a smooth manifold, sections of p T ∗ M will be called differential forms of degree p (or simply p-forms); the space of these sections will be denoted by Ap (M ); 2. when M is a complex manifold, sections of ^p ∗(1,0) ^q ∗(0,1) T ∗(p,q) M := T M⊗ T M will be called differential forms of bidegree (p, q) (or simply (p, q)-forms); the space of these sections will be denoted by A(p,q) (M ).

A.2.1 . . . of holomorphic vector bundles The main reference for this section will be [9, Secs. 1.1 and 3.3]. Let M be a compact complex manifold. Recall that an irreducible analytic hypersurface in M is an analytic submanifold of M of codimension 1, or equivalently a submanifold of M which is locally the zero locus of a single holomorphic (complex-valued) function on M not identically zero, such that its smooth locus is connected. In the following, “hypersurface” will always mean “irreducible analytic hypersurface”. The additive group formally generated by hypersurfaces in M is called the group of divisors of M , denoted Div(M ). Example 5. Let g be an holomorphic function on M and let V be an hypersurface in M . Choose local holomorphic coordinates around a point x ∈ V such that V = { f (z) = 0}. Expand g = f k h, where h is holomorphic around x and h(x) 6= 0. We define the order of g at p, denoted by ordV,p (g ), as the largest integer k such that such an expansion may

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A. Fibre bundles and Chern classes

be done. It may be proved that this integer does not depend on p (see [9, pg. 130]), so that we have a well-defined order of g on V , denoted by ordV (g ). From the definition follows that ordV (g h) = ordV (g ) + ordV (h), for all holomorphic functions g and h. Moreover, if f is a meromorphic function on M , i.e. a function which can locally be expressed as a ratio g /h of relatively prime holomorphic functions, we define ordV ( f ) := ordV (g ) − ordV (h). We say that f has a pole along V if ordV ( f ) < 0, and that it has a zero along V if ordV ( f ) > 0. Given the meromorphic function f on M , we define the divisor associated to f as ( f ) :=

X

ordV ( f )V

V

where the sum is extended over all hypersurfaces in M . Divisors of the form ( f ) for some meromorphic function on M will be called principal; as ( f 1 + f 2 ) = ( f 1 ) + ( f 2 ) and ( f −1 ) = −( f ), we see that principal divisors together with the null divisor form a subgroup Princ(M ) of Div(M ). As a partial converse construction, consider a divisor D=

X

a i Vi ,

a i ∈ Z.

i ∈I

© ª Choose an open cover U j j ∈J in M such that in each U j the hypersuface Vi can be expressed as the zero locus of some holomorphic function f i , j . Then the meromorphic functions Y a f j := f i , ji i ∈I

¡ ¢ are called the local defining functions of D, meaning that D ∩U j = f j . We now return to complex vector bundle to establish a deep connection between divisors and line bundles over the compact complex manifold M . We saw in the preceeding section that line bundles are uniquely determined, up to isomorphism, by their © ª transition functions g j ,i : Ui ∩U j → GL(1, C) ≡ C• i , j ∈J , identifying the fibres {x} × C for © ª x ∈ Ui ∩U j by the action of g j ,i (x). It is easy to see that two such cocycles g j ,i i , j ∈J and ½ n o ³ ´−1 ¾ 0 0 g j ,i induce the same (i.e. isomorphic) line bundle if and only if g j ,i g j ,i i , j ∈J iª, j ∈J © is a coboundary, namely if there exist non-zero holomorphic functions f j : U j → C j ∈J such that fj g 0j ,i = g j ,i for all i , j ∈ J . fi In particular, this implies that isomorphism classes of line bundles over M form a group, called the Picard group of M and denoted by Pic(M ), having the (isomorphism class of the) trivial bundle M × C as unity, and where product and inverses are defined by

L1 L2 := L1 ⊗ L2 , L−1 := L∗ . We are now ready to describe the correspondence between divisors and line bundles over M . If D ∈ Div(M ), choose an open covering {U j } j ∈J of M and, correspondingly, the

A.2 Chern classes. . .

83

© ª local defining functions f j j ∈J associated to D. Define g j ,i :=

fj fi

.

(A.6)

Then one immediately gets that on Ui ∩U j ∩Uk g i ,i =

fi = 1, fi

g i ,k g k, j g j ,i =

fi fk f j =1 fk f j fi

so that the cocycle condition is satisfied. Thus we may construct a line bundle [D] having © ª g j ,i i , j ∈J as transition functions, and it is readily checked that [·] : Div(M ) → Pic(M ) is a group homomorphism. Moreover, [D] is the trivial line bundle if and only if D = ( f ) where f is a meromorphic function on M ; hence, we may define the Chow group of M as CH1 (M ) := Div(M )/ Princ(M ) and obtain an injective quotient homomorphism, which we will continue to denote by [·] : CH1 (W ) → Pic(M ). We want to determine the image of the above homomorphism, in terms of sections © ª of a line bundle L. Using the sheaf condition (A.5), if g j ,i i , j ∈J is a set of transition functions for the line bundle, we may define a meromorphic section s of L to be a collection © ª of meromorphic functions s j j ∈J satisfying s j = g j ,i s i on Ui ∩U j , for all i , j ∈ J . From this we deduce that if s is a non-zero global meromorphic section of L, then s j /s i = g j ,i is holomorphic on Ui ∩U j , and hence for any hypersurface V ⊂ M ordV (s j ) = ordV (s i ). Hence if V ∩ U 6= ; we may define ordV (s) := ordV (s j ); the divisor (s) associated to the meromorphic section s is then X (s) := ordV (s)V V

where the sum is extended to all hypersurfaces in M . From the definition (A.6) of the transition functions of the line bundle L = [D], we © ª see that if f j j ∈J are the local defining functions then this glue together to give a meromorphic section s D of L, and conversely if s is a meromorphic section of L then L = [(s)]. Hence we see that a line bundle is associated to a divisor, L = [D], if and only if it has a non-zero global meromorphic section. 2k For N ⊂ M an analytic submanifold of codimension k, let η N ∈ HdR (M ) be the de Rham cohomology class of a 2k-form such that, if n is the dimension of M , then for all 2(n−k) ω ∈ HdR (M ) Z Z ω= ω ∧ ηN . N

M

This is well defined due to Stokes theorem and the compactness of M ; it is nothing but the Poincaré dual of the current associated to N , i.e. of the linear functional on 2(n−k) HdR (M ) Z 2(n−k) ω ∈ HdR (M ) 7→ ω. N

84

A. Fibre bundles and Chern classes

If D ∈ Div(M ), we define η D by ηD =

X

a i η Vi ,

if D =

i ∈I

X

a i Vi .

i ∈I

Definition 9. If L = [D] is the line bundle over M associated to D ∈ Div(M ), we define its first Chern class to be 2 Ch1 (L) := η D ∈ HdR (M ). Example 6. Let M = PnC ; then [·] : CH1 (PnC ) → Pic(M ) ³ ´ π is an isomorphism. In fact, if L = E − → PnC is a line bundle, then L admits a meromorphic section. This is true for every projective variety, i.e. any complex manifold which can be embedded in PCN , for sufficiently large N , as the zero locus of homogeneous polynomial equations. To give a plausibilty argument (rather than a complete proof, which would be much more complicated!) of this fact, let [ζ0 ; . . . ; ζn ] be homogeneous coordinates on PnC , and let U ' Cn be the open subset {ζn 6= 0} ⊂ PnC . Let ι : U → PnC be the inclusion map. Then ι∗ L is a line bundle over Cn ; as Cn is contractible, this bundle is trivial (see Corollary 3). This gives a local trivialisation ΦU : U ×C ' ι∗ L → π−1U . Define an holomorphic section s on U by s(x) = (x, 1) ∈ U × C; then s extends to a meromorphic section over the whole PnC , having poles along the hyperplane {ζn = 0}. The same kind of argument gives that, even if by definition each hypersurface V ⊂ © ª PnC is locally given by a single holomorphic equation f (z) = 0 , the function f may be chosen to be a homogeneous polynomial. Hence for (the class of ) a divisor D ∈ CH1 (PnC ) we have a well defined homomorphism X X deg : CH1 (PnC ) → Z, D = a i Vi 7→ a i deg( f i ) i ∈I

i ∈I

if each Vi is locally defined as the zero locus of the function f i . This is clearly surjective, but it is also injective: infact, if deg D = 0 and D = D 1 − D 2 ∈ Div(M ), where D 1 and D 2 have only positive coefficients, then deg D 1 = deg D 2 = d so that there exist two homoge¡ ¢ ¡ ¢ neous polynomials f 1 and f 2 of degree d such that D 1 = f 1 and D 2 = f 2 . Hence f 1 / f 2 is a well defined meromorphic function and clearly D = ( f 1 / f 2 ), so that D is principal. By composing the two isomorphism Z ' CH1 (PnC ) ' Pic(PnC ), we see that all isomorphism classes of line bundles over PnC may be labelled by just one integer d . We call O(d ) the line bundle corresponding to the integer d ∈ Z; this integer counts the difference between the number of zeroes and the number of poles of a section of the line bundle. For example, we have O(−1) = [−H ], where H is the hyperplane {ζn = 0}. In order to see this, we let U be the line bundle having © ª E := (`, v) ∈ PnC × Cn+1 : v ∈ ` (we think of PnC as the set of lines ` in Cn+1 ) as its total space, with projection on the first factor3 . The map µ µ ¶¶ ζ0 ζn−1 s n ([ζ]) = [ζ], ,..., , 1 , if [ζ] = [ζ0 ; . . . ; ζn ] , ζn ζn 3 For n = 1, this definition is compatible with the one already given in Example 3.

A.2 Chern classes. . .

85

is holomorphic on U = {ζn 6= 0}, and extends to a section of U which has a pole of order 1 over H . Hence

U = [(s n )] = [−H ] = O(−1). For this reason, O(−1) is called the tautological bundle over PnC . Its dual is clearly O(1) = [H ], the hyperplane bundle. All other line bundles over PnC can be constructed from these two, as ⊗d O(1)

if d > 0,

O(−1)

if d < 0.

O(d ) = PnC × C ⊗|d |

if d = 0,

The Chern class of O(1) is (1/2π) times the Fubini-Study form (see [27, Sec. 3.3.2]) de© ª fined on the open subset U j = ζ j 6= 0 , with coordinates z t = ζt /ζ j , by ωPnC ([ζ]) j = i∂∂ ln

1+

1 X

|z t |

Ã Ã ! ! X X 2 2 = −i∂∂ ln 1 + |z t | = i∂∂ ln 1 + |z t | 2 t 6= j

(A.7)

t 6= j

t 6= j

where the Dolbeaut operators ∂ and ∂ are defined on U j as ∂ f (z, z) =

X

∂z t f (z, z) dz t ,

t 6= j

∂ f (z, z) =

X

∂z t f (z, z) dz t .

t 6= j

³ ´ π We can give a definition of higher Chern classes for a vector bundle V = E − → M of rank m ≥ 1. Given a collection S = {s 1 , . . . , s m } of global holomorphic sections of V, we call the degeneracy set D j (S), for j = 1, . . . , m, the subset of M given by the points x ∈ M © ª such that s 1 (x), . . . , s j (x) are linearly dependent in π−1 (x): © ª D j (S) := x ∈ M : s 1 (x) ∧ · · · ∧ s j (x) = 0 . We say that the collection of sections S is generic if, for all j = 1, . . . , m − 1, the image of the section σ j +1 intersects transversely the subset of E which is given by the collection of all the vector spaces spanned by σ1 (x), . . . , σ j (x), for x ∈ M . With some choice of orientation (see [9, pg. 412]), one can see that if S is a generic collection of sections then its degeneracy locus D j (S) is a cycle in homology. Definition 10. If S is a generic collection of sections of the vector bundle V, the Poincaré dual of the degeneracy cycle D m− j +1 (S) is called the j -th Chern class of the bundle V, 2j

and denoted by Ch j (V) ∈ HdR (M ). Remark 11. If L is a line bundle (m = 1) over M and S = {s} is a non-zero global holomorphic section of L, then its degeneracy cycle coincides with its zero locus. As also the divisor (s) accounts for the zeroes of the holomorphic section, the two definitions of first Chern class of an holomorphic vector bundle coincide.

86

A. Fibre bundles and Chern classes

A.2.2 . . . of smooth vector bundles The Chern classes we defined for holomorphic vector bundles are de Rham cohomology classes of the base space of the bundle. However, de Rham cohomology only depends on the C ∞ structure of the complex manifold M , and has nothing to do with the holomorphicity. This suggests that Chern classes may be generalised to the category of C ∞ complex vector bundles over smooth (real) manifolds: this is exactly what we will do in this section. We will follow [9, Sec. 3.3] and [19, ³ App. ´C]. π

Let M be a smooth manifold and let V = E − → M be a vector bundle of rank m. We p will denote by A (M ; E ) the space of E -valued p-forms, namely of sections of the bundle Vp ∗ TC M ⊗ V. Clearly A0 (M ; E ) = Γ(V).

Definition 11. A connection on V is a C-linear mapping ∇ : A0 (M ; E ) → A1 (M ; E ) satisfying Leibnitz rule ∇( f s) = d f ⊗ s + f ∇s,

for all f ∈ C ∞ (M ; C) and s ∈ A0 (M ; E ).

Roughly speaking, a connection is a way to differentiate sections of the vector bundle

V; if s ∈ Γ(V) is a section of V and X ∈ Γ(T M ) is a vector field, then we may think of the contraction ∇ X s := ∇s · X as the derivative of s along the direction X . We want to give a local description of a connection on a fibre bundle V. Let ΦU : U × m C → π−1U be a local trivialisation for V; as was already noticed, we can equivalently choose a local frame {s 1 , . . . , s m } on U . As ∇s a is an E -valued 1-form for all a = 1, . . . , m, we may write4 m X ∇s a = iAab ⊗ s b (A.8) b=1

where A = (Aab )1≤a,b≤m is a matrix of complex-valued 1-forms on U , i.e. a section of the tensor bundle End (T ∗ M ) ' T ∗ M ⊗ T M . Leibnitz rule and C-linearity allow us to deduce that this matrix uniquely determines the connection ∇: in fact, as every section on U can be written as m X s= f a sa , a=1 4 The factor i in this formula is due to following convention. If V is a C ∞ vector bundle on the complex manifold M , an Hermitian metric on V is a smooth section h of the tensor bundle V∗ ⊗ V∗ such that h(x) is a scalar product on the fibre π−1 (x) for all x ∈ M . Let ∇ be a connection on V; as a consequence of the splitting TC∗ M = T ∗(1,0) M ⊕ T ∗(0,1) M of the complexified cotangent bundle into its holomorphic and antiholomor-

phic part, we may write ∇ as ∇ = ∇(1,0) ⊕ ∇(0,1) . We say that ∇ is compatible with the complex structure if ∇(0,1) coincides with the Dolbeaut operator ∂. We also say that ∇ is compatible with the Hermitian metric h if for all s, t ∈ Γ(V) we have dh(s, t ) = h(∇s, t ) + h(s, ∇t ).

Even if in general there is no “natural” connection on a generic vector bundle V on a complex manifold M , if V has an Hermitian metric then there exists one and only one connection which is compatible with the metric and with the complex structure; it is called the Levi-Civita connection (see [9, pg. 73] or [27, Prop. 3.12]). The Levi-Civita connection matrix A with respect to any orthonormal frame (i.e. a local frame such that h(s a , s b ) = δab ), with the factor i in our definition, is thus Hermitian; it would be antihermitian without it.

A.2 Chern classes. . .

87

we get that m X

∇s =

∇( f a s a ) =

a=1

m X

m ¡ X

d f a ⊗ s a + f a ∇s a =

a=1

¢ d f a + i f b Aba ⊗ s a .

a,b=1

Thus, sometimes the name “connection” is used to denote the matrix of 1-forms A. © ª Moreover, given an open cover U j j ∈J of M given by trivialising subsets and a col© ª lection A j j ∈J of arbitrary m ×m matrices, one can define the connections ∇ j on U j by the above formula and glue them together with a partition of unity subordinate to the © ª open cover U j j ∈J . Hence, connections associated to a given vector bundle V always exist. Remark 12. If (x 1 , . . . , x m ) are local coordinates on U ⊂ M , then we may write

Aab =

m X

Γabc dx c ,

for all a, b = 1, . . . , m.

c=1

Then the C ∞ function Γabc are called the Christoffel symbols of the connection ∇. Suppose now that V ⊂ M is another trivialising open subset for V, or equivalently that {s˜1 , . . . , s˜m } is a local frame on V . On U ∩ V we have sea =

m X

g ad s d ,

sd =

m ¡ X

g −1

b=1

d =1

¢

d b seb

where g = gUV is the transition function. Then we get ∇e sa = = =

m X d =1 m X

m ¡ ¢ X ∇ g ad s d = dg ad ⊗ s d + g ad ∇s d = d =1

µ dg ad ⊗ s d + g ad

m X

¶

iAd c ⊗ s c =

c=1

d =1 m X

¡

¢ dg ad + ig ac Acd ⊗ s d =

c,d =1

= =

m ¡ X

" dg ad + ig ac Acd ⊗

c,d =1 m X b,c,d =1

¢

m ¡ X b=1

# g

−1

¢

d b seb

=

£ ¡ ¢ ¡ ¢ ¤ dg ad g −1 d b + ig ac Acd g −1 d b ⊗ seb .

Now notice that m X d =1 m X c=1

¡ ¢ ¡ ¢ dg ad g −1 d b = dg · g −1 ab ,

¢ g ac Acd = g · A ad ¡

=⇒

m ¡ X d =1

¢ ¡ ¢ ¡ ¢ g · A ad g −1 d b = g · A · g −1 ab ,

and hence ∇e sa =

m ¡ X b=1

dg · g −1 + ig · A · g −1

¢

ab ⊗ seb

=

m ¡ X ¢ i −i dg · g −1 + g · A · g −1 ab ⊗ seb .

b=1

88

A. Fibre bundles and Chern classes

Comparing this with the definition (A.8) of the connection matrix on V m X

∇e sa =

b=1

eab ⊗ seb iA

we deduce that, in matrix notation, e = g · A · g −1 − i dg · g −1 . A

(A.9)

We may define C-linear operators ∇ = ∇(k) : Ak (M ; E ) → Ak+1 (M ; E ) by “forcing” Leibnitz rule, i.e. requiring that ∇(0) = ∇ is the prescribed connection on V and that ∇(k+1) (η ⊗ s) = dη ⊗ s + (−1)k η ∧ ∇s,

for all η ∈ Ak (E ) and s ∈ A0 (M ; E ).

The operator K ∇ := ∇(1) ◦ ∇ : A0 (M ; E ) → A2 (M ; E ) is called the curvature tensor associated to the connection ∇. It may be proved that ¡ V ¢ K ∇ is C ∞ (M ; C)-linear, and thus defines a section of the bundle Hom V, 2 T ∗ M ⊗ V ' V2 ∗ T M ⊗ End(V) (see [19, App. C, Lemma 5]). If {s 1 , . . . , s n } is a local frame on U for the vector bundle V, then we have K∇ sa = = =

m X b=1 m X

∇ (iAab ⊗ s b ) =

m X

i dAab ⊗ s b − iAab ∧ ∇s b =

b=1

i dAab ⊗ s b − iAab ∧

µ

m X

¶

iAbc ⊗ s c =

c=1

b=1 m µ X

m X

b=1

c=1

i dAab ⊗ s b +

¶

Aac ∧ Acb ⊗ s b .

Thus we see that the curvature K ∇ is completely determined by the matrix of 2-forms ωA = (ωab )1≤a,b≤m satisfying the relation K∇ sa =

m X

iωab ⊗ s b ,

b=1

and that ωA is given in terms of the connection matrix A by ωA = dA − iA ∧ A.

(A.10)

One can check that if {s˜1 , . . . , s˜m } is another local frame on V ⊂ M , then on U ∩V one has ωAe = g ωA g −1

(A.11)

where g = gUV is a transition function. The curvature tensor K ∇ allows us to construct characteristic classes associated to the vector bundle V. Let gl(m, C) denote the algebra of all m × m matrices. We say that a polynomial P : gl(m, C) → C is invariant if for all A, B ∈ gl(m, C) we have P (AB ) = P (B A)

A.2 Chern classes. . .

89

or equivalently if for all A ∈ gl(n, C) and T ∈ GL(n, C) we have ¡ ¢ P (A) = P T AT −1 . The basic invariant polynomials are the elementary symmetric functions of the eigenvalues of a matrix A ∈ gl(n, C), namely those defined by the relation det(A + λ1) =

m X

σm−n (A)λn .

n=0

In particular, σ1 (A) = tr A and σm (A) = det A. These are really “basic”, because every other invariant polynomial can be expressed as a polynomial in σ1 , . . . , σm (see [19, App. C, Lemma 6]). Since the wedge product is commutative on differential forms of even degree, we may evaluate any homogeneous invariant polynomial P of degree j on the curvature matrix ωA . Due to the invariance property and to Equation (A.11), we see that this values does not depend on the trivialising open subset U ⊂ M , and hence we have a well defined E -valued 2 j -form P (K ∇ ) ∈ A2 j (M ). The fundamental properties of this form are listed in the following Lemma. Lemma 2. If P is an homogeneous invariant polynomial of degree j , then: 1. the 2 j -form P (K ∇ ) is closed, i.e. dP (K ∇ ) = 0; 2j

2. the de Rham cohomology class [P (K ∇ )]dR ∈ HdR (M ) does not depend on the connection ∇, but only on the isomorphism class of the vector bundle V. Proof. See [9, pg. 403] or [19, App. C, pgg. 296 and 298]. Thus we may define Definition 12. The j -th Chern class of the C ∞ complex vector bundle V is · µ ¶ ¶¸ µ ¤ i 1 j£ 2j Ch j (V) := σ j K∇ = − σ j (ωA ) dR ∈ HdR (M ) 2π 2π dR where ∇ is an arbitrary connection on V.

A.2.3 . . . of topological vector bundles In this section, we generalise the definition of Chern classes to vector bundles over a topological space B . In the previous sections, these were de Rham cohomology classes, but this cohomology makes no sense for topological spaces which are not smooth manifolds. However, we still have the singular homology groups and their “duals”, the singular cohomology groups. These are defined as follows. Let I denote the unit interval [0, 1] and consider continuous maps I n → X (by definition I 0 consists of just one point); the latter are called n-cubes. We say that an n-cube is degenerate if it does not depend on some of the coordinates in I n . Formal linear combination of n-cubes with coefficients in Z form a group, and degenerate cubes form a subgroup. We denote the quotient group by C n (X ; Z), and call it the group of singular n-chains in X . If (x 1 , . . . , x n ) are coordinates on I n , we call I nj,0 := I n ∩ {x j = 0} and I nj,1 := I n ∩ {x j = 1}, for j = 1, . . . , n, the faces of

90

A. Fibre bundles and Chern classes

I n . If f : I n → B is continuous, we let f j ,0 (respectively f j ,1 ) be the restriction of f to I nj,0 (respectively I nj,1 ). We define the boundary of f as ∂ f :=

n X

¡ ¢ (−1) j f j ,1 − f j ,0 .

j =1

The boundary operator ∂ can be extended a Z-linear operator ∂ ≡ ∂n : C n (X ; Z) → C n−1 (X ; Z), and it is easily shown by direct computation that ∂n ◦ ∂n+1 = 0. Hence if we define Zn (X ; Z) = ker ∂n

the group of singular n-cycles in X , and

B n (X ; Z) = im ∂n+1

the group of singular n-boundaries in X ,

then B n (X ; Z) is a subgroup of Zn (X ; Z), which in turn is a subgroup of C n (X ; Z). The n-th homology group of X is defined as Hn (X ; Z) := Zn (X ; Z)/B n (X ; Z). The n-th cohomology group of X is defined as H n (X ; Z) := HomZ (Hn (X ; Z), Z) i.e. as the set of Z-linear maps (i.e. group homomorphisms) on Hn (X ; Z) with values in Z. Clearly, by defining accordingly C n (X ; Z) (the group of singular n-cochains), Z n (X ; Z) (the group of singular n-cocycles) and B n (X ; Z) (the group of singular n-coboundaries), one has H n (X ; Z) = Z n (X ; Z)/B n (X ; Z). Every continuous map g : X → Y induces group homomorphisms g ∗ : Hn (X ; Z) → Hn (Y ; Z) and g ∗ : H n (Y ; Z) → H n (X ; Z); these are defined on n-cubes by composition, i.e. if f : I n → X is continuous then g ∗ f = g ◦ f , while if η : Hn (X ; Z) → Z is a group homomorphism then g ∗ η = η ◦ g ∗ . It can be proved that if g 0 , g 1 : X → Y are homotopic ¡ ¢ ¡ ¢∗ ¡ ¢ ¡ ¢∗ maps then g 0 ∗ (respectively g 0 ) and g 1 ∗ (respectively g 1 ) coincide (see [7, Thm. 5.5]). In particular, if X and Y are homotopically equivalent spaces then they have the same singular homology and cohomology groups. Remark 13. In all the above definitions, the coefficients group Z may be substituted with any (additive) abelian group A; in this case we write Hn (X ; A) and H n (X ; A) for homology and cohomology groups with coefficients in A, respectively. As an abelian group A is just a Z-module, we have natural inclusions Hn (X ; Z) ,→ Hn (X ; A) and H n (X ; Z) ,→ H n (X ; A) for all n ∈ N. We now come back to Chern classes. Before specialising to the case of complex vector bundles, ³ ´ we state the problem in the more general context of fibre bundles. Let π

F= E− → B be a fibre bundle over the finite-dimensional CW-complex B , with typi-

cal fibre F . (There are some technical hypotheses to be made, regarding the triviality of some canonically-defined actions of the fundamental groups π1 (B ) and π1 (F ) on the homotopy groups π j (F ) of the fibre space. We will not dwell on them: the reader is addressed to [25, Part III].) We want to study the existence of a cross section of the bundle F, by defining it “inductively” on its skeleton. Suppose we have a map s : B j −1 → E , defined over the ( j − 1)-skeleton of B , which satisfies π ◦ s = 1B j −1 ; we may call s a “partial

A.2 Chern classes. . .

91

cross section”. We would like to extend s to a partial section on the j -skeleton of B , or at least to measure the obstruction to such an extension. As we want to define s on the j -cells, let e j be one of them. A cell is nothing but a space homeomorphic to a closed disk in R j , and hence it is contractible; consequently, the part of F which lies above e j (defined as ι∗ F, were ι : e j ,→ B is the inclusion map) is trivial: π−1 e j ' e j × F. On the boundary ∂e j ' S j −1 the cross section s : ∂e j → π−1 ∂e j = ∂e j ×F is already given. Identifying e j with the unit disc centered at the origin D ⊂ R j , the composition of s|∂e j with the projection of ∂e j × F on the second factor gives rise to a map S j −1 → F ; this defines an element c(s, e j ) ∈ π j −1 (F ). Clearly the vanishing of this element is a necessary and sufficient condition for s to extend to the whole cell e j . In fact, if c(s, e j ) = 0, we have that s is homotopic to a constant map, i.e. there exists a continuous function h : S j −1 × I → B such that h(p, 0) = x 0 ∈ B and h(p, 1) = s(p) for all p ∈ S j −1 . Then if we define µ ¶ x se: D → B, se(x) := h , |x| |x| we get that se is an extension of s on the whole cell D ≡ e j . Conversely, if se: D → B is given, then h(p, t ) := se(t p) defines an homotopy between s = se|∂D and the map constantly equal to se(0). In particular, we get that if F is n-connected (i.e. π j (F ) = {0} for all j = 0, 1, . . . , n) then every partial section defined on the (n − 1)-skeleton of B extends to a partial section defined on the n-skeleton of B . Thus we see that the first interesting case is when we want to extend a section s : B n → E , defined on the n-skeleton of B , to one defined on its (n + 1)-skeleton, where n is the least integer such that πn (F ) is non-trivial. The map c s : {(n + 1)-cells of B } → πn (F ),

e n+1 7→ c(s, e n+1 )

can thus be viewed as an element of C n+1 (B ; πn (F )), the space of (n + 1)-cochains with coefficients in the group πn (F ); we call it the primary obstruction cochain. Lemma 3.

1. The primary obstruction cochain is actually a cocycle.

2. Suppose that c s is a coboundary. Then we can alter the definition of s on the nskeleton of B , without changing its definition on the (n − 1)-skeleton of B , so that the partial section s 0 we obtain satisfies c s 0 = 0. Proof. The first assertion is [25, Thm. 32.4], while the second is [25, Thm. 33.5]. In view of this Lemma, we may consider c s , the first obstruction to extending a partial section s : B n → E to a partial section on the (n + 1)-skeleton, as an element of the group H n+1 (B ; πn (F )); its vanishing is a necessary and sufficient condition of the section to be extendable. ³ ´ π We now turn to complex vector bundles. Let V = E − → B be such a bundle and let m © ª be its rank. Recall that we can use the transition functions g j ,i : Ui ∩U j → U (m) i , j ∈J to construct bundles associated to V by changing the typical fibre Cm to any space endowed with a continuous action of the group U (m). In particular, we are interested in changing the fibre to the complex Stiefel manifold Vm,r , consisting of orthonormal r frames in Cm (i.e. sets of r orthonormal vectors); we will denote by V(r ) this associated bundle. The Stiefel manifold Vm,r can also be described as the homogeneous space

92

A. Fibre bundles and Chern classes

U (m)/U (m − r ), so that it has a natural action of the unitary group U (m): this groups acts transitively on orthormal r -frames in Cm , and the space that leaves fixed a particular frame is the subgroup U (m − r ) acting on the orthogonal complement of the space spanned by the frame itself. For example Vm,1 ' S 2m−1 while Vm,m ' U (m); thus V(m) is the principal bundle associated to V. It is known (see [25, Sec. 25.7]) that ¡ ¢ ¡ ¢ πn Vm,r = {0} for all n ≤ 2(m − r ), while π2m−2r +1 Vm,r ' Z. Definition 13. The singular cohomology class of the first obstruction to the existence of a section of the bundle V(m− j +1) is called the j -th Chern class of the vector bundle V, and is denoted by ¡ ¡ ¢¢ Ch j (V) ∈ H 2 j B ; π2 j −1 Vm,m− j +1 = H 2 j (B ; Z), j = 1, . . . , m.

A.2.4 Equivalence of the definitions and general properties We have now given three different definitions of Chern classes, for three different classes of vector bundles – and actually more could be given! Of course, we require some consistency to hold between these definitions. This is achieved through the next result. Theorem 13. 1. Let M be a complex manifold of dimension n and V be an holomorphic complex vector bundle on M ; V may also be seen as a smooth complex vector bundle over the real manifold M of dimension 2n. Then the holomorphic Chern classes of V defined via Definition 10 coincide with the smooth Chern classes of V defined via Definition 12. 2. Let M be a smooth manifold and V be a C ∞ complex vector bundle on M of rank m; V may also be seen as a topological complex vector bundle over the topological space M . Then the images of the smooth Chern classes of V defined via Definition 12 2j through the inclusions HdR (M ) ,→ H 2 j (M ; R), j = 1, . . . , m, coincide with the images of the continuous Chern classes of V defined via Definition 13 through the inclusions H 2 j (M ; Z) ,→ H 2 j (M ; R), j = 1, . . . , m. Proof. The first claim is proved in [9, pg. 413], while the second is [7, Ex. 20, pg. 117]. We now list some properties of the Chern classes. In the next Theorem, C denotes the category Top, Man or Hol. Theorem 14.

1. Chern classes have integer-valued classes over integer cycles.

2. Chern classes are natural, meaning that if f : N → M is a morphism in C and V is a complex vector bundle on M , then Ch j ( f ∗ V) = f ∗ Ch j (V),

for all j.

(A.12)

for all j.

(A.13)

3. If V∗ denotes the dual bundle of V, then Ch j (V∗ ) = (−1) j Ch j (V),

4. If V is a complex vector bundle of rank m and L is a line bundle, then Ch1 (V ⊗ L) = Ch1 (V) + mCh1 (L).

A.2 Chern classes. . .

93

5. Define the total Chern class of the complex vector bundle V of rank m on M as the formal sum Ch(V) = 1 + Ch1 (V) + · · · + Chm (V). Then the Whitney product formula Ch(V ⊕ W) = Ch(V)Ch(W), where V and W are complex vector bundles, holds. 6. Complex line bundles are uniquely determined, up to C ∞ isomorphism, by their first Chern class. 7. If V is a complex vector bundle on a d -dimensional CW-complex (or smooth manifold, or analytic manifold) and d ≤ 3, then V is determined, up to isomorphism, by its first Chern class. Proof.

1. This follows from Definition 13 and the equivalence of all definitions.

2. This is [9, Item 1, pg. 407]. 3. This is [9, Item 4, pg. 408]. 4. This is [9, Item 3, pg. 407]. 5. This is [9, Item 2, pg. 407]. 6. This is [9, pg. 140]. 7. This follows from the fact that H n (M ; Z) is trivial for n > d , where d is the dimension of the CW-complex (or smooth manifold, or analytic manifold); see [7, pg. 77]. Thus the only a priori non-trivial Chern class on a space of dimension d ≤ 3 is the first Chern class Ch1 (V) ∈ H 2 (M ; Z), and consequently the same argument used to prove the previous property holds.

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A. Fibre bundles and Chern classes

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