simon trebst et al- topological order and quantum criticality

8/3/2019 Simon Trebst et al- Topological order and quantum criticality

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© Simon Trebst

Topological order and quantum criticality

Alexei KitaevAndreas Ludwig

Matthias Troyer

Zhenghan Wang

Charlotte Gils

http://www.lptms.u-psud.fr/index.php?option=com_content&view=article&id=99

http://www.lptms.u-psud.fr/index.php?option=com_content&view=article&id=99

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

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

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




© Simon Trebst

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TOPOLOGICAL PHASES

Wormholes in quantum matterProliferation of so-called anyonic defects in a topological phase of quantum matter leads to a critical state that can

be visualized as a ‘quantum foam’, with topology-changing fluctuations on all length scales.

Kareljan Schoutens

news & views

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Physics in the plane:

Fr om condensed matter to string theory

http://www.nature.com/nphys/journal/v5/n11/full/nphys1439.html











© 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





© Simon Trebst

Topological quantum liquids

• Gapped spectrum• No broken symmetry

• Degenerate ground state on torus

• Fractional statistics of excitations

• Hilbert space split into topological sectors No local matrix element mixes the sectors

∆

eiθ

∆

0

1

H





© Simon Trebst

Topological order

quantum Hall liquids magnetic materials

time reversal symmetry

broken invariant





© Simon Trebst

Quantum phase transitions

topological order

λc

some other (ordered) state

no local order parameter ↓

no Landau-Ginzburg-Wilson theory

complex field theoretical framework ↓

doubled (non-Abelian) Chern-Simons theories

Liquids on surfaces can provide a “topological framework”.

time-reversal invariant systems





© Simon Trebst

The toric code in a magnetic fieldA first example





© Simon Trebst

The toric code

σi

As

B p

As =

�

j∈star(s)

σxj

B p =

�

j∈∂ p

σzj

L2

L1

A. Kitaev, Ann. Phys. 303, 2 (2003).

HTC = −J e

�

s

As − J m

�

p

B p +�

i

(hxσx

i+ hzσ

z

i)

topological phase paramagnet

http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WB1-47HJX86-1&_coverDate=01%252F31%252F2003&_alid=446651033&_rdoc=1&_fmt=&_orig=search&_qd=1&_cdi=6697&_sort=d&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=807499fb24543bc777400bff2aa41eed

http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WB1-47HJX86-1&_coverDate=01%252F31%252F2003&_alid=446651033&_rdoc=1&_fmt=&_orig=search&_qd=1&_cdi=6697&_sort=d&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=807499fb24543bc777400bff2aa41eed




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Toric code and the transverse field Ising model

B p = 2µz

p

xi = µ

x pµ

xq

σi

B p

B p =

�

j∈∂ p

σzj

µ p

HTFIM = −2J m�

p

µz

p+ hx�

� p,q

µx

pµx

q

HTC =−J m� p

B p + hx�i

σxi





Phase diagram

0

topological phase

vortex condensate

chargecondensate

s e l f -

d u a l i t y

l i n e

deconfined phase

paramagneth x

h z

continuoustransition

multicritical point?

first-ordertransition

I.S. Tupitsyn et al., arXiv:0804.3175

http://arxiv.org/abs/0804.3175






Excitations: condensation vs. confinement

topological phase

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1loop tension h

0

0.5

1

c o n f i n e m e n t l e n g t h

ξ c

L = 8 L = 12 L = 16

0.56 0.57 0.58 0.59 0.6

0.9

0.95

1

·

6 / ( L

+

2 )

paramagnet

0 0.2 0.4 0.6 0.8 1loop tension h

0

0.2

0.4

0.6

0.8

1

g a p

Δ /

2 B

topological phasetopological phase paramagnet

σi

As

B p

L2

L1

magnetic vortices electric charges

plaquette excitations“magnetic vortices”

vertex excitations“electric charges”




Splitting the topological degeneracy

δE ∝ L

0.54 0.55 0.56 0.57 0.58 0.59 0.6 0.61 0.62

magnetic field h x

-8 -8

-6 -6

-4 -4

-2 -2

0 0

2 2

4 4

e n e r g y

s p l i t t i n g

! E

( L )

L = 48 L = 40 L = 32 L = 24 L = 16 L = 12 L

= 8

topological phase paramagnet

δE ∝ exp(−αL)

ST et al., Phys. Rev. Lett. 98, 070602 (2007).

http://prl.aps.org/abstract/PRL/v98/i7/e070602

http://prl.aps.org/abstract/PRL/v98/i7/e070602





Phase diagram

0

topological phase

vortex condensate

chargecondensate

s e l f -

d u a l i t y

l i n e

deconfined phase

paramagneth x

h z

continuoustransition

multicritical point?

first-ordertransition

I.S. Tupitsyn et al., arXiv:0804.3175







Toric code: phase transitions

wavefunctiondefor mations

|ψ�

toric code paramagnet

Fradkin & Shenker (1979)

ST et al. (2007)Tupitsyn et al. (2008)

Vidal, Dusuel, Schmidt (2009)

z = 1

3D Ising

universality

R G f low

Hamiltoniandefor mations

Ardonne, Fendley, Fradkin (2004)

Castelnovo & Chamon (2008)

Fendley (2008)

z ~ 22D Ising

univer sality“ con formal QC P ”





Quantum double models

for time-reversal invariant systems





Time-reversal invariant liquids

chiral excitations

“flux” excitationsnon-chiral

anyonic liquid L

anyonic liquid L̄





The skeleton lattice

anyonic liquid





The skeleton lattice

anyonic liquid

skeleton

lattice





Abelian liquids→ loop gas / toric code

1

11

Semion fusion rules loop gas× s = 1

1

s

Abelian liquids → loop gas / toric code





Flux excitations

anyonic liquid

flux through

“hole”

flux through“tube”

skeleton

lattice





Extreme limits

no flux through “holes”↓

close holes

↓“two sheets”

The “two sheets” ground state exhibits topological order.

In the toric code model this is the flux-free, loop gas ground state





Extreme limits

no flux through “tubes”↓

pinch off tubes

↓“decoupled spheres”

The “decoupled spheres” ground state exhibits no topological order.

In the toric code model this is the paramagnetic state.





Semions (Abelian):

Toric code in a magnetic field / loop gas with loop tension .J e/J p

Connecting the limits: a microscopic model

H = −J p

�

plaquettes p

δφ( p),1 − J e

�

edges e

δ�(e),1

Varying the couplings we can connect the two extreme limits.J e/J p





The quantum phase transition

vary loop tension J e/J p

“two sheets”

topological order

“decoupled spheres”

no topological order






“two sheets”

topological order



topology fluctuations

on all length scales

“quantum foam”

continuous transition





Universality classes (semions)

Ardonne, Fendley, Fradkin (2004)

Castelnovo & Chamon (2008)

Fendley (2008)

z ~ 22D Ising

univer sality

wavefunctiondefor mations

|ψ�

toric code paramagnet

Fradkin & Shenker (1979)

STet al.

(2007)Tupitsyn et al. (2008)

Vidal, Dusuel, Schmidt (2009)

z = 1

3D Ising

universality

R G f low






The non-Abelian case

τ τ

τ

τ τ

1 1

11

Fibonacci anyon fusion rules τ × τ = 1 + τ





© Simon Trebst

Non-abelian liquids → string net

τ τ

τ

τ τ

1 1

11

Fibonacci anyon fusion rules τ × τ = 1 + τ string net

Non-abelian liquids → string net





© Simon Trebst

A continuous transition

• A continuous quantum phase transition connectsthe two extremal topological states.

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

• The gapless theory is exactly sovable.





© Simon Trebst

Phase diagram

c = 14/15exactly solvable


non-Abeliantopological phase

non-Abelian

topological phase

critical phase

c= 7

/5

J p

J r





© Simon Trebst

Gapless theory & exact solution

(1, 1)

(1, τ )

(τ , 1)(τ , τ ) τ

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

(X i)2 = d · X i X iX i±1X i = X i [X i, X j ] = 0

|i− j| ≥ 2for

=� 2 + φ





© Simon Trebst



non-abeliantopological phase

non-abelian

topological phase

“quantum foam”

c= 7

/5

J p

J r

0 2 4 6 8 10 12momentum k

x[2π/L]

0 0

0.5 0.5

1 1

1.5 1.5

2 2

r e s c a l e d

e n e r g y

E ( k

x ,

k y

)

L = 12, ky

= 0

L = 12, ky

= π

primary fields

descendant fields

02/452/15

4/15

2/3

14/15

56/45

8/5

1+2/451+2/15

1+4/15

1+2/3

1+14/15

τ-flux

τ-fluxno-flux

( τ+1)-flux

( τ+1)-flux

τ-flux

( τ+1)-flux

no-flux

τ-flux

τ-flux

τ-flux

τ-fluxno-flux

τ

-flux






© Simon Trebst




non-abelian

topological phase

“quantum foam”

c= 7

/5

J p

J r

0 2 4 6 8momentum k

x[2π/L]

0 0

0.5 0.5

1 1

1.5 1.5

2 2

r e s c a l e

d

e n e r g y

E ( k

x ,

k y

)

L = 8, ky = 0

L = 8, ky

= π

primary fields

0

3/201/5

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






© Simon Trebst

Dressed flux excitations

0 π /2 π 3π /2 2π

momentum K x

0 0

0.5 0.5

1 1

1.5 1.5

2 2

e n e r

g y

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

“quantum foam”

= 7/5

p

J r





© Simon Trebst

Back to two dimensions

wavefunction

def ormations

|ψ�

z = 2

c = 14/15

Fendley & Fradkin (2008)

str ing netclassicalstate

z = 1

continuous

tr ansition?


str ing net

z = 1

c = 14/15classicalstate

1+1 D

2+1 D

2+0 D





Summar y

• A “topological” framework for the descriptionof 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. Wang

arXi :0906 1579 N t Ph i 5 834 (2009)














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