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)

A Typology for Quantum Hall Liquids

Landau levels:1 state / flux quantum. 'filling factor' ν = Nelectrons. Nfluxes. = hc e ⇡2 ⇥ 10. 15. Tm. 2. = flux quantum. (gives electron Aharanov-Bohm phase 2π) ...

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A brand orientation typology for SMEs: a case research ...
2003). A firm can position its goods or services in a unique fashion ... 1996), or through the brand report card (Keller, 2000), or on the brand building .... Firm G (integrated). Retail. 30. Medium. Firm H (minimalist). Computer shop. 2. Low.

Ionic liquids as novel media for electrophilic/onium ion ... - Arkivoc
and metal-mediated reactions: a progress summary. Kenneth K. Laali .... accessible IL. EAN in combination with TFAA or Tf2O acts as an in-situ source of.

A Proposed Typology of Online Hate Crime.pdf
Page 1 of 26. A Proposed Typology of Online Hate Crime. Authors: William Jacks and Joanna R. Adler. Department of Psychology, Middlesex University, London, NW4 4BT, UK. Email: Professor Adler, [email protected] (Corresponding author). Abstract. Hate

Capturing complexity: a typology of reflective practice ...
designed to guide teacher educators in teaching reflection to pre-service ..... rather narrow view of the situation itself, calling ..... San Francisco: Jossey-Bass.

Towards a typology of Algonquian relative clauses ...
Participles combine characteristics of verbs and nominals to such a degree that these ..... Table 3: Participle vs. preverb relative clauses constructions .... PhD dissertation, University of ... In E. Doron, I. Sichel and M. Rappaport-Hovav, eds.