A Typology for Quantum Hall Liquids (and other topological tales) Siddharth A. Parameswaran University of California, Berkeley
Physics Colloquium, IIT, Chennai, September 27, 2013
Collaborators
Steve Kivelson Stanford
Shivaji Sondhi Princeton
Boris Spivak U. Washington
+ related work with Steve Simon, Ed Rezayi and R. Shankar
Overview Topological Phases & Quantum Hall Physics Classification of Quantum Hall liquids into ‘Type I’ and ‘Type II’ in analogy to SCs, but fractional charges instead of vortices
Identification of Type I Quantum Hall liquids entirely new class of topologically ordered quantum phases!
Experiments and Implications [S.P., et.al., Phys. Rev. Lett.106, 236801(‘11); Phys. Rev. B 85, 241307R (’12); Phys. Rev. Lett.109, 237004 (‘12)]
Setting: 2DEG in high magnetic field B
E
j ~!c
Marcus group, Copenhagen
quantized cyclotron orbits: Landau levels Quantum Hall Regime low temperature
kB T
very clean samples !c ⌧coll ⌧ 1
c
Hall Resistance Ix
Vy Vx
Resistivity
⇢xy
[data: Willett et. al., ‘89]
2 Nobel Prizes!
⇢xx
von Klitzing ‘85 Tsui, Störmer, Laughlin ‘98
5 Buckley Prizes Tsui, Störmer, Gossard ‘84, Laughlin ‘86, Haldane ‘93, Read, Jain, Willett ‘02, Macdonald, Girvin, Eisenstein ‘07
Magnetic Field
‘Typology’ of Quantum Hall Liquids Quantum Hall liquids: featureless (at center of steps~‘commensurate’ density)
‘Quasiparticle’ excitations: fractional electric charge New classification of QH liquids based on quasiparticle behavior
[S.P., et.al., Phys. Rev. Lett.106, 236801 (2011); Phys. Rev. B 85, 241307 (R)(2012)]
‘Typology’ of Quantum Hall Liquids Quantum Hall liquids: featureless (at center of steps~‘commensurate’ density)
‘Quasiparticle’ excitations: fractional electric charge New classification of QH liquids based on quasiparticle behavior ‘Type I’ quasiparticles attract
‘Type II’ quasiparticles repel
[S.P., et.al., Phys. Rev. Lett.106, 236801 (2011); Phys. Rev. B 85, 241307 (R)(2012)]
‘Typology’ of Quantum Hall Liquids Quantum Hall liquids: featureless (at center of steps~‘commensurate’ density)
‘Quasiparticle’ excitations: fractional electric charge New classification of QH liquids based on quasiparticle behavior ‘Type II’
‘Type I’ quasiparticles attract
quasiparticles repel
So far: ‘Type II’ behavior implicitly assumed
Does Type I exist?
[S.P., et.al., Phys. Rev. Lett.106, 236801 (2011); Phys. Rev. B 85, 241307 (R)(2012)]
‘Typology’ of Quantum Hall Liquids Quantum Hall liquids: featureless (at center of steps~‘commensurate’ density)
‘Quasiparticle’ excitations: fractional electric charge New classification of QH liquids based on quasiparticle behavior ‘Type II’
‘Type I’
quasiparticles repel
quasiparticles attract
So far: ‘Type II’ behavior implicitly assumed
Does Type I exist? short answer:Yes. long answer: this talk! [S.P., et.al., Phys. Rev. Lett.106, 236801 (2011); Phys. Rev. B 85, 241307 (R)(2012)]
Step back: two basic questions
Where does classification fit on the ‘map’ of condensed matter?
FQ HE
What’s the language?
Until ~1980, broken symmetries crystals
magnets
superfluids
Building blocks: Order parameter + Landau theory
Until ~1980, broken symmetries crystals
magnets
superfluids
Building blocks: Order parameter + Landau theory
1980s-now: featureless ‘topological liquids’ (patterns of ‘long-range’ entanglement) No symmetry breaking! no order parameter, no Landau theory
Until ~1980, broken symmetries crystals
magnets
superfluids
Building blocks: Order parameter + Landau theory
1980s-now: featureless ‘topological liquids’ (patterns of ‘long-range’ entanglement) No symmetry breaking! no order parameter, no Landau theory instead:
Topological Order (quantum number fractionalization) FQHE
[Wen]
+ also, spin liquids...
Low-Energy Theory of Topological Liquids Analog of Landau theory? For QH: Chern-Simons theory:
LCS
k µ⌫⇢ = " aµ @ ⌫ a⇢ 4⇡
(microscopic: Laughlin wavefunction) ‘Topological’ Field Theory ground-state degeneracy fractional charge fractional statistics ‘braiding’
[Witten; applied to CM by Wen & others]
Low-Energy Theory of Topological Liquids Analog of Landau theory? For QH: Chern-Simons theory:
LCS
k µ⌫⇢ = " aµ @ ⌫ a⇢ 4⇡
(microscopic: Laughlin wavefunction) ‘Topological’ Field Theory ground-state degeneracy fractional charge fractional statistics ‘braiding’
[Witten; applied to CM by Wen & others]
BUT: asymptotically long wavelength, no energetics...
Energy Scales
Energy Scales ‘high’
~1 eV
microscopic lattice + models symmetries B
E
j
fractionalized quasiparticles
‘low’
~10-4 eV
Model Hamiltonians materials numerics
emergent gauge field
Topological Field Theory braiding, statistics, topological order
Energy Scales ‘high’
~1 eV
microscopic lattice + models symmetries B
E
Model Hamiltonians materials numerics
j
Intermediate Scales More accessible to experiment Can we still ‘see’ topological effects?
fractionalized quasiparticles
‘low’
~10-4 eV
emergent gauge field
???
Topological Field Theory braiding, statistics, topological order
Landau Levels and Degeneracies Landau levels: 1 state / flux quantum hc ⇡ 2 ⇥ 10 = flux quantum = e
15
Tm2
(gives electron Aharanov-Bohm phase 2π)
‘filling factor’ ν =
Nelectrons Nfluxes
Landau Levels and Degeneracies Landau levels: 1 state / flux quantum hc ⇡ 2 ⇥ 10 = flux quantum = e
15
Tm2
(gives electron Aharanov-Bohm phase 2π)
‘filling factor’ ν =
ν =1
e
e
e
e
e
e
e
e
e
Nelectrons Nfluxes
unique state, with or without interactions
Landau Levels and Degeneracies Landau levels: 1 state / flux quantum hc ⇡ 2 ⇥ 10 = flux quantum = e
15
Tm2
(gives electron Aharanov-Bohm phase 2π)
‘filling factor’ ν =
ν =1
e
e
e
e
e
e
e
e
e
e
ν =1/3
or e
Nfluxes
unique state, with or without interactions
e
e
Nelectrons
e
e
e
or
e
or... e
many ways, same energy!
Landau Levels and Degeneracies Landau levels: 1 state / flux quantum hc ⇡ 2 ⇥ 10 = flux quantum = e
15
Tm2
(gives electron Aharanov-Bohm phase 2π)
‘filling factor’ ν =
ν =1
e
e
e
e
e
e
e
e
e
e
ν =1/3
or e
Nfluxes
unique state, with or without interactions
e
e
Nelectrons
e
e
e
or
e
or... e
many ways, same energy! need interaction to pick which one
‘Flux Attachment’: Composite Boson Mean-Field Theory ν =1/3: too many fluxes, too few electrons
‘Flux Attachment’: Composite Boson Mean-Field Theory ν =1/3: too many fluxes, too few electrons ‘Attach’ flux to electrons: create composite bosons [Girvin & Macdonald ’87; Read ’89; Zhang, Hansson & Kivelson, ’89]
electron + 3 fluxes
CB
e
composite boson
‘Flux Attachment’: Composite Boson Mean-Field Theory ν =1/3: too many fluxes, too few electrons ‘Attach’ flux to electrons: create composite bosons [Girvin & Macdonald ’87; Read ’89; Zhang, Hansson & Kivelson, ’89]
electron + 3 fluxes
CB
composite boson
e
Composite bosons at B=0, T=0: Bose condense superfluid ‘Gluing’ flux: superfluid has QH physics
Quantum Hall Physics from Superfluids
1 ⌫= 3
Quantum Hall Physics from Superfluids Superfluid 3 fluxes glued to each charge e
1 ⌫= 3
Quantum Hall Physics from Superfluids Superfluid 3 fluxes glued to each charge
flux expulsion
e
1 ⌫= 3
Quantum Hall Physics from Superfluids Superfluid 3 fluxes glued to each charge
flux expulsion
e
1 ⌫= 3
Quantum Hall Physics from Superfluids Superfluid 3 fluxes glued to each charge
flux expulsion
e
charge gap
1 ⌫= 3
Quantum Hall Physics from Superfluids Superfluid 3 fluxes glued to each charge e
flux expulsion quantized vortices
charge gap fractional charges e/3
1 ⌫= 3
(translation via flux attachment)
Flux
Meissner Effect (Higgs mechanism)
vortices
F QH E
Superfluid-Quantum Hall Dictionary
⇄
Charge
⇄
Incompressibility (charge gap)
⇄
fractionally charged quasiparticles
Superfluid bosons (e.g.,
4He):
vortices always repel!
Superconductors (Cooper pairing of fermions) have two types! Type I ‘vortices’ attract
Type II vortices repel
phase separation
Abrikosov lattice
Why?
Superfluid bosons (e.g.,
4He):
vortices always repel!
Superconductors (Cooper pairing of fermions) have two types! Type I ‘vortices’ attract
Type II vortices repel
phase separation
Abrikosov lattice
Why? SC phase gains ‘pairing’ energy, costs ‘diamagnetic energy’ Competition controlled by two independent length scales: ξ, λ
Why two types of superconductors? Compute energy of N-S boundary (bulk energy equal)
Type I: ⌧ ⇠ flux
Type II:
pairing
N
S
flux N
⇠ pairing S
Flux expelled quickly Pairing grows slowly
Flux expelled slowly Pairing grows quickly
Emag
Emag
Epair > 0
Minimize boundary, ⟹phase separate
Epair < 0
Maximize boundary, ⟹form vortices
Why two types of superconductors? Compute energy of N-S boundary (bulk energy equal)
Type I: ⌧ ⇠ flux
Type II:
pairing
N
S
flux
⇠ pairing
N
S
Flux expelled quickly Pairing grows slowly
Flux expelled slowly Pairing grows quickly
Emag
Emag
Epair > 0
Minimize boundary, ⟹phase separate
Epair < 0
Maximize boundary, ⟹form vortices
Cooper Pairing: independence of ξ and λ
Cooper Pairing in Quantum Hall systems?
Cooper Pairing in Quantum Hall systems? ν=1/2:
composite fermions
[Jain ’89; Lopez & Fradkin ’91]
two possibilities at B=0, T=0 e
Cooper Pairing in Quantum Hall systems? ν=1/2:
composite fermions
[Jain ’89; Lopez & Fradkin ’91]
two possibilities at B=0, T=0 e
‘normal’ metal
[Halperin, Lee, Read ’93]
pairing
superconductor
[Moore, Read ’91; Greiter et.al. ’91]
Cooper Pairing in Quantum Hall systems? ν=1/2:
composite fermions
[Jain ’89; Lopez & Fradkin ’91]
two possibilities at B=0, T=0 e
‘normal’ metal pairing
superconductor
pairing
[Halperin, Lee, Read ’93] [Moore, Read ’91; Greiter et.al. ’91]
same dictionary, but vortices carry half-flux charge e/4
Cooper Pairing in Quantum Hall systems? ν=1/2:
composite fermions
[Jain ’89; Lopez & Fradkin ’91]
two possibilities at B=0, T=0 ‘normal’ metal
e
[Halperin, Lee, Read ’93]
pairing
superconductor
pairing
[Moore, Read ’91; Greiter et.al. ’91]
same dictionary, but vortices carry half-flux charge e/4
Near pairing transition: universality!
Cooper Pairing in Quantum Hall systems? 1 = 2
Half filling is special
[Willett et. al., ’90]
1 = (+2) 2 [Pan et. al., ’99]
composite fermions
metal
pairing
e
superconductor ‘Pfaffian’
Cooper Pairing in Quantum Hall systems? 1 = 2
Half filling is special
[Willett et. al., ’90]
1 = (+2) 2 [Pan et. al., ’99]
composite fermions
metal
pairing
e
superconductor ‘Pfaffian’
critical point
Field theory near Critical Point kF
Composite Fermi Liquid
e
(Fermi surface + Chern Simons field)
S=
Z
Z
dtdr
0
drdr |
†
{i@t
r|
2
V (r
µ
e
e(a0 + A0 )} 0
r )|
r0 |
2
1 |(r 2m
e µ⌫⇢ " aµ @ ⌫ a⇢ 4⇡ · 2
e(a + A)) |
2
Field theory near Critical Point kF
Composite Fermi Liquid
e
(Fermi surface + Chern Simons field)
S=
Z
Z
dtdr
0
drdr |
†
{i@t
r|
2
V (r
µ
e
e(a0 + A0 )} 0
r )|
r0 |
2
1 |(r 2m
e µ⌫⇢ " aµ @ ⌫ a⇢ 4⇡ · 2
e(a + A)) |
2
[Halperin, Lee, Read ’93; Nayak & Wilczek ’94; Foster et.al. ’03]
Field theory near Critical Point kF
Composite Fermi Liquid (Fermi surface + Chern Simons field)
S=
Z
Z
dtdr
0
drdr |
†
{i@t
r|
2
V (r
µ
+ pairing
e
r )|
e
e
e(a0 + A0 )} 0
e
r0 |
2
1 |(r 2m
e(a + A)) |
e µ⌫⇢ " aµ @⌫ a⇢ +Spairing 4⇡ · 2
2
[Halperin, Lee, Read ’93; Nayak & Wilczek ’94; Foster et.al. ’03]
E Δ kF
k
critical point: tune pairing strength (measured in terms of gap, Δ)
Towards Type I: Length scales Superconductors are characterized by two independent scales
⇠
~ E Δ
✓
1 ↵
1 C ne
◆1/2
electronic spacing
√fine structure const.
k
kF
[S.P., et.al., Phys. Rev. Lett.106, 236801 (2011); Phys. Rev. B 85, 241307 (R)(2012)]
Towards Type I: Length scales Superconductors are characterized by two independent scales vF
Coherence length order parameter variation
Penetration depth flux screening
⇠
~ E Δ
✓
1 ↵
1 C ne
◆1/2
electronic spacing
√fine structure const.
k
kF
[S.P., et.al., Phys. Rev. Lett.106, 236801 (2011); Phys. Rev. B 85, 241307 (R)(2012)]
Towards Type I: Length scales Paired QH states are characterized by two independent scales Coherence length order parameter variation
Penetration depth charge screening
vF
⇠ `B
=
✓
~ eB
◆1/2
(electronic spacing ~ `B “fine structure const.”~1)
E Δ
k
kF
[S.P., et.al., Phys. Rev. Lett.106, 236801 (2011); Phys. Rev. B 85, 241307 (R)(2012)]
Towards Type I: Length scales Paired QH states are characterized by two independent scales vF
Coherence length order parameter variation
Penetration depth charge screening
⇠ `B
=
✓
~ eB
◆1/2
(electronic spacing ~ `B “fine structure const.”~1)
E Type I if Δ kF
k
`B ⌧ ⇠
(equivalently,
⌧ ✏F ) Always true near critical point!
[S.P., et.al., Phys. Rev. Lett.106, 236801 (2011); Phys. Rev. B 85, 241307 (R)(2012)]
Quasiparticle Structure in Type I Regime
⇠
`B `B
⇥
r
Label
Flux attachment + pairing: charge e/4 [S.P., et.al., Phys. Rev. Lett.106, 236801 (2011); Phys. Rev. B 85, 241307 (R)(2012)]
Type I Quasiparticles Attract Quasiparticle costs pairing energy F 2 and Coulomb energy ⇡ e /`B
[S.P., et.al., Phys. Rev. Lett.106, 236801 (2011); Phys. Rev. B 85, 241307 (R)(2012)]
Type I Quasiparticles Attract Quasiparticle costs pairing energy F 2 and Coulomb energy ⇡ e /`B
2
Two well separated quasiparticles cost ⇡ 2("F + e /`B )
[S.P., et.al., Phys. Rev. Lett.106, 236801 (2011); Phys. Rev. B 85, 241307 (R)(2012)]
Type I Quasiparticles Attract Quasiparticle costs pairing energy F 2 and Coulomb energy ⇡ e /`B
2
Two well separated quasiparticles cost ⇡ 2("F + e /`B ) By combining, gain ⇠ "F 2 but Coulomb goes up by e /
[S.P., et.al., Phys. Rev. Lett.106, 236801 (2011); Phys. Rev. B 85, 241307 (R)(2012)]
Type I Quasiparticles Attract Quasiparticle costs pairing energy F 2 and Coulomb energy ⇡ e /`B
2
Two well separated quasiparticles cost ⇡ 2("F + e /`B ) By combining, gain ⇠ "F 2 but Coulomb goes up by e / quasiparticles bind when ⇠
e2 /"F ⇠ `B
weak coupling regime is Type I! [S.P., et.al., Phys. Rev. Lett.106, 236801 (2011); Phys. Rev. B 85, 241307 (R)(2012)]
The First Type I Quantum Hall liquid Paired QH state: near pairing transition, quasiparticles always attract
Coulomb repulsion stabilizes mesoscopic ‘bubble’ N≫1 quasiparticles per bubble
Lots of new physics to explore: distinct class of topological liquids! Quasiparticle attraction: bad news for computing schemes? ...good news for new physics! [S.P., et.al., Phys. Rev. Lett.106, 236801 (2011); Phys. Rev. B 85, 241307 (R)(2012)]
Phase diagram near half-filling `B fixed, vary
ignoring long-range interactions
⇥ 1
⇥c2
2
density of added quasiparticles
⇤c
1 ⇥ Type I no added quasiparticles (commensurate filling)
Type II (`B /⇠)c ⇠ 1
`B /⇠ (/
[S.P., et.al., Phys. Rev. Lett.106, 236801 (2011); Phys. Rev. B 85, 241307 (R)(2012)]
/"F )
Phase diagram near half-filling `B fixed, vary
ignoring long-range interactions
⇥ 1
⇥c2
phase separation (~Landau)
2
quasiparticle crystal (~Abrikosov)
density of added quasiparticles
⇤c
1 ⇥ Type I no added quasiparticles (commensurate filling)
Type II (`B /⇠)c ⇠ 1
`B /⇠ (/
[S.P., et.al., Phys. Rev. Lett.106, 236801 (2011); Phys. Rev. B 85, 241307 (R)(2012)]
/"F )
Phase diagram near half-filling `B fixed, vary
⇥
✗ phase separation (~Landau)
density of added quasiparticles
ignoring long-range interactions
⇤c
Long-range interactions change the story
1
⇥c2
2
quasiparticle crystal (~Abrikosov)
1 ⇥
Type I
no added quasiparticles (commensurate filling)
Type II (`B /⇠)c ⇠ 1
`B /⇠ (/
[S.P., et.al., Phys. Rev. Lett.106, 236801 (2011); Phys. Rev. B 85, 241307 (R)(2012)]
/"F )
Phase diagram near half-filling long-range interactions forbid macroscopic phase sep.
`B fixed, vary
⇥
[Spivak & Kivelson ’05]
⇥c2
1 2
quasiparticle crystal (~Abrikosov)
density of added quasiparticles
⇤c
1 ⇥ Type II
Type I no added quasiparticles (commensurate filling)
1/ 2
`B /⇠ (/
[S.P., et.al., Phys. Rev. Lett.106, 236801 (2011); Phys. Rev. B 85, 241307 (R)(2012)]
/"F )
Phase diagram near half-filling long-range interactions forbid macroscopic phase sep.
`B fixed, vary
⇥
[Spivak & Kivelson ’05] frustrated phase separation droplets
stripes
⇥c2
bubbles
1 2
quasiparticle crystal (~Abrikosov)
density of added quasiparticles
⇤c
1 ⇥ Type II
Type I no added quasiparticles (commensurate filling)
1/ 2
`B /⇠ (/
[S.P., et.al., Phys. Rev. Lett.106, 236801 (2011); Phys. Rev. B 85, 241307 (R)(2012)]
/"F )
Have Isolated Quasiparticles been Observed? [Venkatachalam, et. al., Nature 469, 187 (2011)]
1. Harvard Group
Measures: capacitance - charging with e* quasiparticles ⇤ ⇤ e5/2 ⇡ .76e7/3 = .254e
e*
- large ‘puddle’: 1/N effect - no info on stability of isolated quasiparticles! No Type I/II Discrimination!
Have Isolated Quasiparticles been Observed? 2. Weizmann Group
[Dolev, et. al., Nature 452, 829 (2008)]
Measures: quantum ‘shot noise’ - edge physics (from top. field theory) - no info on bulk quasiparticle energy - works even with zero bulk quasiparticle No Type I/II Discrimination!
Have Isolated Quasiparticles been Observed?
Also: experiments to probe ‘non-Abelian statistics’ (Bell Labs, Chicago groups...)
- interferometry/‘phase slips’ of quasiparticles - tricky to obtain info. on stability of isolated quasiparticles
No Type I/II Discrimination!
How to detect Type I? One idea: Microwave absorption
e.g. ν= 4+δ
[Exp’t: Engel, Tsui, ...; Theory: Fogler, Goerbig....]
N=1
bubble crystals + weak disorder + B-field pinning mode resonances frequency depends on charge/bubble! vs. Type I: N>1
Type II: N=1
N≳2
Topological quantum computation Mike Freedman Alexei Kitaev (Milner Prize ’12, $3,000,000!)
Station Q
Idea:‘braid’ QH quasiparticles
Topological quantum computation Mike Freedman Alexei Kitaev
Idea:‘braid’ QH quasiparticles
(Milner Prize ’12, $3,000,000!)
Station Q
Type I: ‘bubbles’ of many quasiparticles
Topological quantum computation Mike Freedman Alexei Kitaev
Idea:‘braid’ QH quasiparticles
(Milner Prize ’12, $3,000,000!)
Station Q
Type I: ‘bubbles’ of many quasiparticles
‘even bubble’
‘odd bubble’
Topological quantum computation Mike Freedman Alexei Kitaev
Idea:‘braid’ QH quasiparticles
(Milner Prize ’12, $3,000,000!)
Station Q
Type I: ‘bubbles’ of many quasiparticles
no good for computing!
✘
‘even bubble’
✔
‘odd bubble’
Stories for another time: Extending SC Dictionary Sharp thermal crossover from paired→compressible Neutral fermion physics Dipole moment tied to neutral fermions
[S.P., et. al., Phys. Rev. Lett. 109, 237004 (2012).]
Second sound-like finite-T collective modes
Interface and Weak-Link Physics Proximity/Josephson Effects
⇥ 1 r
g
g<0
Andreev scattering
g=0 d
E
j
Conclusions Two classes of QH liquids: Type I vs. Type II Type I: quasiparticles attract, phase separate Paired QH states: Type I at weak coupling
[S.P., Kivelson, Sondhi, Spivak, Phys. Rev. Lett.106, 236801 (2011)]
New set of phases with exotic particles! Complicates topological quantum computation proposals
‘... now Type I superconductors have become exotic.’ - Alexei Abrikosov, Nobel Lecture (’03)
Conclusions Two classes of QH liquids: Type I vs. Type II Type I: quasiparticles attract, phase separate Paired QH states: Type I at weak coupling
[S.P., Kivelson, Sondhi, Spivak, Phys. Rev. Lett.106, 236801 (2011)]
New set of phases with exotic particles! Complicates topological quantum computation proposals
Also: infinite number of other Type I examples (w/ fine-tuning) Field theory: ‘self-dual’ points of Chern-Simons Landau-Ginzburg theory Microscopic: perturb model Hamiltonians [S.P., Kivelson, Rezayi, Simon, Sondhi, Spivak, Phys. Rev. B 85, 241307 (R) (2012)]
‘... now Type I superconductors have become exotic.’ - Alexei Abrikosov, Nobel Lecture (’03)