topology driven quantum phase transitions · ©simon trebst summary • a “topological”...

© Simon Trebst

Topology driven quantum phase transitions

Alexei KitaevAndreas Ludwig

Matthias TroyerZhenghan Wang

Charlotte Gils

Simon TrebstMicrosoft Station QUC Santa Barbara

Dresden July 2009

http://www.kitp.ucsb.edu/~trebst/


http://stationq.ucsb.edu/index.html

http://stationq.ucsb.edu/index.html

http://www.mpipks-dresden.mpg.de/~topo09/

http://www.mpipks-dresden.mpg.de/~topo09/

© Simon Trebst

Spontaneous symmetry breaking• ground state has less symmetry than high-T phase

• Landau-Ginzburg-Wilson theory

• local order parameter

Topological quantum liquids

Topological order• ground state has more symmetry than high-T phase

• degenerate ground states

• non-local order parameter

• quasiparticles have fractional statistics = anyons

cosmic microwave background



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Topological order

quantum Hall liquids magnetic materials

2a

time reversal symmetry

broken invariant



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Quantum phase transitions

topological order!

!c

some other (ordered) state

no local order parameter ↓

no Landau-Ginzburg-Wilson theory

How can we describe these phase transitions?

complex field theoretical framework ↓

doubled (non-Abelian) Chern-Simons theories

Liquids on surfaces can provide a “topological framework”.

time-reversal invariant systems



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Time-reversal invariant liquids

chiral excitations

✗

✗ “flux” excitationsnon-chiral

anyonic liquid L

anyonic liquid L̄



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Flux excitations

anyonic liquid

flux through “hole”

flux through “tube”



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Flux excitations

anyonic liquid

flux through “hole”

flux through “tube”

skeletonlattice



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Abelian liquids → loop gas / toric code

111

Semion fusion rulesloop gas

s! s = 1

1s s

Abelian liquids → loop gas / toric code



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Extreme limits

The “two sheets” ground state exhibits topological order. In the toric code model this is the flux-free, loop gas ground state

no flux through “holes”↓

close holes↓

“two sheets”



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Extreme limits

The “decoupled spheres” ground state exhibits no topological order. In the toric code model this is the fully polarized, classically ordered state.

no flux through “tubes”↓

pinch off tubes↓

“decoupled spheres”



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Semions (Abelian): Toric code in a magnetic field / loop gas with loop tension .Je/Jp

Connecting the limits: a microscopic model

H = !Jp

!

plaquettes p

!!(p),1 ! Je

!

edges e

!"(e),1

Varying the couplings we can connect the two extreme limits.Je/Jp



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The quantum phase transition

vary loop tension Je/Jp

“two sheets”topological order

“decoupled spheres”no topological order



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topology fluctuationson all length scales

“quantum foam”

continuous transition



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Universality classes (semions)

“gapless” toric code

Ardonne, Fendley, Fradkin (2004)Castelnovo & Chamon (2008)

Fendley (2008)

z = 2 2D Isinguniversality

wavefunctiondeformations |!!

toric codeclassical

(polarized)state

Fradkin & Shenker (1979)ST et al. (2007)

Tupitsyn et al. (2008)Vidal, Dusuel, Schmidt (2009)

z = 1

3D Isinguniversality

RG flow

HHamiltoniandeformations



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The non-Abelian case

! !!

! !1 1

11

Fibonacci anyon fusion rules ! ! ! = 1 + !



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Non-abelian liquids → string net

! !!

! !1 1

11

Fibonacci anyon fusion rules ! ! ! = 1 + !string net

Non-abelian liquids → string net



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vary string tension Je/Jp





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A continuous transition

• A continuous quantum phase transition connects the two extremal topological states.

• The transition is driven by topology fluctuations on all length scales.

• The gapless theory is exactly sovable.



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Gapless theory & exact solution

(1, 1)

(1, !)

(!, 1)(!, !)1 !

The gapless theory is a CFT with .c = 14/15

The Hamiltonian maps onto the Dynkin diagram D6

The operators in the Hamiltonian form aTemperley-Lieb algebra

(Xi)2 = d · Xi XiXi±1Xi = Xi [Xi, Xj ] = 0|i! j| " 2for

d =!

2 + !



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Jp

Jr

0 2 4 6 8 10 12momentum kx [2π/L]

0 0

0.5 0.5

1 1

1.5 1.5

2 2

resc

aled

ene

rgy

E(k

x , k y )

L = 12, ky = 0L = 12, ky = πprimary fieldsdescendant fields

02/452/154/15

2/3

14/15

56/45

8/5

1+2/451+2/151+4/15

1+2/3

1+14/15

τ-flux

τ-fluxno-flux

(τ+1)-flux

(τ+1)-fluxτ-flux(τ+1)-flux

no-flux

τ-fluxτ-flux

τ-flux

τ-fluxno-flux

τ-flux




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Jp

Jr

0 2 4 6 8momentum kx [2π/L]

0 0

0.5 0.5

1 1

1.5 1.5

2 2

resc

aled

ene

rgy

E( k

x, ky )

L = 8, ky = 0L = 8, ky = πprimary fields

03/20 1/5

τ-flux

(τ+1)-flux

(τ+1)-flux

(τ+1)-flux

τ-flux

2/5

19/20

6/5

7/4

7/5

τ-flux

τ-fluxno-flux




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Dressed flux excitations

0 π/2 π 3π/2 2πmomentum Kx

0 0

0.5 0.5

1 1

1.5 1.5

2 2en

ergy

E(K

x)

θ = 0.3π

θ = 0.4π

θ = 0.6π

θ = 0.7π

θ = 0.8π

θ = 0.6π



non-abelian

topological phase

non-abelian

topological phase


Jp

Jr



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Back to two dimensions

wavefunctiondeformations |!!

z = 2c = 14/15Fendley & Fradkin (2008)

string net classicalstate

z = 1

continuoustransition?

HHamiltoniandeformations

string netz = 1

c = 14/15classical

state1+1 D

2+1 D

2+0 D



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Summary

• A “topological” framework for the description of topological phases and their phase transitions.

• Unifying description of loop gases and string nets.

• Quantum phase transition is driven by fluctuations of topology.

• Visualization of a more abstract mathematical description, namely doubled non-Abelian Chern-Simons theories.

• The 2D quantum phase transition out of a non-Abelian phase is still an open issue: A continuous transition in a novel universality class?

C. Gils, ST, A. Kitaev, A. Ludwig, M. Troyer, and Z. WangarXiv:0906.1579 → Nature Physics (accepted)



http://arxiv.org/abs/0906.1579




topology driven quantum phase transitions · ©simon trebst summary • a “topological”...

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