© Simon Trebst

Topological order and quantum criticality

Alexei KitaevAndreas Ludwig

Matthias TroyerZhenghan Wang

Charlotte Gils

Simon TrebstMicrosoft Station QUC Santa Barbara

Les Houches March 2010

© Simon Trebst

784 NATURE PHYSICS | VOL 5 | NOVEMBER 2009 | www.nature.com/naturephysics

news & views

e!ective ion concentration close to the surface, this crucially helps to transduce the ion channel’s sensitivity to the nanowire.

"is step having been made, a number of formidable challenges remain to make the ion-channel-based biointerface a reality. In their experiment, Misra and co-workers used two simple peptide pores, which are commonly used as ‘test devices’. It will be quite challenging to incorporate ‘real’ ion channels with more versatile functions into the lipid membrane coating the nanowires. "is would truly result in new ion-channel-based biosensors that

combine the extraordinary biochemical and electronic sensitivities of their components. For neuroprosthetics, however, it will also be important to make information transfer bidirectional, that is, to controllably transfer electronic to ionic or chemical signals. Furthermore, at present it is unclear how such devices would be made to function in vivo.

Friedrich C. Simmel is in the Physics Department at the Technical University Munich, James-Franck-Straße, D-854748 Garching, Germany. e-mail: simmel@ph.tum.de

References1. Gibson, W. Neuromancer (Ace, 1984).2. Gibson, W. Count Zero (Ace, 1986).3. Gibson, W. Mona Lisa Overdrive (Ace, 1988).4. Misra, N. et al. Proc. Natl Acad. Sci. USA

106, 13780–13784 (2009).5. Sigworth, F. J. Nature 423, 21–22 (2003).6. Fromherz, P., O!enhausser, A., Vetter, T. & Weis, J. Science

252, 1290–1293 (1991).7. Fromherz, P. & Stett, A. Phys. Rev. Lett. 75, 1670–1673 (1995).8. Peitz, I., Voelker, M. & Fromherz, P. Angew. Chem. Int. Ed.

46, 5787–5790 (2007).9. Cornell, B. A. et al. Nature 387, 580–583 (1997).10. Patolsky, F., Zheng, G. & Lieber, C. M. Nanomedicine

1, 51–65 (2006).11. Patolsky, F. et al. Science 313, 1100–1104 (2006).

There is no end to the list of remarkable quantum phases of matter that can emerge if simple building blocks are

endowed with just the right interactions and are cooled to low enough temperatures. In many cases, the essence of such phases can be captured by an order parameter. "ink of a quantum ferromagnet, for example, with the order parameter specifying the direction of the magnetization. More intricate quantum phases, which can arise in frustrated quantum magnets (which possess disordered ground states) or in systems of strongly correlated electrons or cold atoms, lack such telling diagnostics, and it becomes something of an art to characterize the quantum order that they hide. Charlotte Gils and colleagues, writing on page 834 of this issue1, show how particular quantum phases can be visualized as a quantum foam — a surface with topology-changing quantum #uctuations on all length scales.

"e logic that leads Gils et al. to their remarkable physical picture is best followed in two steps. First they consider so-called topological phases of matter in two-dimensional (2D) systems, following a microscopic description originally given by Michael Levin and Xiao-Gang Wen2. "ese topological phases are seemingly featureless quantum liquids, lacking any form of local order. But their special nature manifests itself at the edge of a system, where gapless edge modes arise, or, for systems de$ned on a surface with non-trivial topology, in characteristic

ground-state degeneracies. "e topological order also a!ects the quantum statistical properties of defects. Avoiding the

restrictions for three-dimensional systems, where bosonic or fermionic statistics are the rule, defects over 2D topological phases tend to be anyons. Interchanging anyons can lead not only to prefactors of ±1, as for bosons and fermions, respectively, but to ‘any’ phase or even to matrices acting on an internal quantum space. For the ‘Fibonacci anyons’ (which feature in the paper of Gils et al.1) the dimensionality of the n-particle internal space is precisely the nth element of the Fibonacci sequence (1, 1, 2, 3, 5, 8, …).

Topological phases are known to occur in fractional quantum Hall systems, where electrons are con$ned to a 2D Flatland, exposed to a strong perpendicular magnetic $eld (of many Teslas) and cooled to low temperatures (in the millikelvin range). "e magnetic $eld breaks the time-reversal symmetry, causing the characteristic edge modes to be chiral — that is, having a preferred direction and therefore one-way tra%c along the edges. But the topological phases of Levin and Wen2 are time-reversal invariant. It is then natural to visualize these states as anyonic quantum liquids living on a double sheet, with each sheet corresponding to a speci$c chirality of edge modes.

In the second step of their analysis, Gils et al.1 add a term to the Levin–Wen Hamiltonian, which lowers the energy of speci$c defects. "e defects can be visualized as ‘wormholes’ connecting the two sheets (Fig. 1). Driving the strength of the extra term to a critical value leads to

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 Schoutensa

b

c

Figure 1 | Topology matters. a, For the topology of a 2D surface, all that counts is the number of incontractible loops. b, A defect in a chiral 2D topological quantum liquid can be pictured as a tiny current circulating in an incontractible loop around a hole, creating a ‘chimney’. c, When there are two sheets connected by such chimneys, proliferation of these wormholes drives a topological phase to a critical point, visualized as a quantum foam.

nphys_N&Vs_NOV09.indd 784 23/10/09 11:08:13

784 NATURE PHYSICS | VOL 5 | NOVEMBER 2009 | www.nature.com/naturephysics

news & views

e!ective ion concentration close to the surface, this crucially helps to transduce the ion channel’s sensitivity to the nanowire.

"is step having been made, a number of formidable challenges remain to make the ion-channel-based biointerface a reality. In their experiment, Misra and co-workers used two simple peptide pores, which are commonly used as ‘test devices’. It will be quite challenging to incorporate ‘real’ ion channels with more versatile functions into the lipid membrane coating the nanowires. "is would truly result in new ion-channel-based biosensors that

combine the extraordinary biochemical and electronic sensitivities of their components. For neuroprosthetics, however, it will also be important to make information transfer bidirectional, that is, to controllably transfer electronic to ionic or chemical signals. Furthermore, at present it is unclear how such devices would be made to function in vivo.

Friedrich C. Simmel is in the Physics Department at the Technical University Munich, James-Franck-Straße, D-854748 Garching, Germany. e-mail: simmel@ph.tum.de

References1. Gibson, W. Neuromancer (Ace, 1984).2. Gibson, W. Count Zero (Ace, 1986).3. Gibson, W. Mona Lisa Overdrive (Ace, 1988).4. Misra, N. et al. Proc. Natl Acad. Sci. USA

106, 13780–13784 (2009).5. Sigworth, F. J. Nature 423, 21–22 (2003).6. Fromherz, P., O!enhausser, A., Vetter, T. & Weis, J. Science

252, 1290–1293 (1991).7. Fromherz, P. & Stett, A. Phys. Rev. Lett. 75, 1670–1673 (1995).8. Peitz, I., Voelker, M. & Fromherz, P. Angew. Chem. Int. Ed.

46, 5787–5790 (2007).9. Cornell, B. A. et al. Nature 387, 580–583 (1997).10. Patolsky, F., Zheng, G. & Lieber, C. M. Nanomedicine

1, 51–65 (2006).11. Patolsky, F. et al. Science 313, 1100–1104 (2006).

There is no end to the list of remarkable quantum phases of matter that can emerge if simple building blocks are

endowed with just the right interactions and are cooled to low enough temperatures. In many cases, the essence of such phases can be captured by an order parameter. "ink of a quantum ferromagnet, for example, with the order parameter specifying the direction of the magnetization. More intricate quantum phases, which can arise in frustrated quantum magnets (which possess disordered ground states) or in systems of strongly correlated electrons or cold atoms, lack such telling diagnostics, and it becomes something of an art to characterize the quantum order that they hide. Charlotte Gils and colleagues, writing on page 834 of this issue1, show how particular quantum phases can be visualized as a quantum foam — a surface with topology-changing quantum #uctuations on all length scales.

"e logic that leads Gils et al. to their remarkable physical picture is best followed in two steps. First they consider so-called topological phases of matter in two-dimensional (2D) systems, following a microscopic description originally given by Michael Levin and Xiao-Gang Wen2. "ese topological phases are seemingly featureless quantum liquids, lacking any form of local order. But their special nature manifests itself at the edge of a system, where gapless edge modes arise, or, for systems de$ned on a surface with non-trivial topology, in characteristic

ground-state degeneracies. "e topological order also a!ects the quantum statistical properties of defects. Avoiding the

restrictions for three-dimensional systems, where bosonic or fermionic statistics are the rule, defects over 2D topological phases tend to be anyons. Interchanging anyons can lead not only to prefactors of ±1, as for bosons and fermions, respectively, but to ‘any’ phase or even to matrices acting on an internal quantum space. For the ‘Fibonacci anyons’ (which feature in the paper of Gils et al.1) the dimensionality of the n-particle internal space is precisely the nth element of the Fibonacci sequence (1, 1, 2, 3, 5, 8, …).

Topological phases are known to occur in fractional quantum Hall systems, where electrons are con$ned to a 2D Flatland, exposed to a strong perpendicular magnetic $eld (of many Teslas) and cooled to low temperatures (in the millikelvin range). "e magnetic $eld breaks the time-reversal symmetry, causing the characteristic edge modes to be chiral — that is, having a preferred direction and therefore one-way tra%c along the edges. But the topological phases of Levin and Wen2 are time-reversal invariant. It is then natural to visualize these states as anyonic quantum liquids living on a double sheet, with each sheet corresponding to a speci$c chirality of edge modes.

In the second step of their analysis, Gils et al.1 add a term to the Levin–Wen Hamiltonian, which lowers the energy of speci$c defects. "e defects can be visualized as ‘wormholes’ connecting the two sheets (Fig. 1). Driving the strength of the extra term to a critical value leads to

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 Schoutensa

b

c

Figure 1 | Topology matters. a, For the topology of a 2D surface, all that counts is the number of incontractible loops. b, A defect in a chiral 2D topological quantum liquid can be pictured as a tiny current circulating in an incontractible loop around a hole, creating a ‘chimney’. c, When there are two sheets connected by such chimneys, proliferation of these wormholes drives a topological phase to a critical point, visualized as a quantum foam.

nphys_N&Vs_NOV09.indd 784 23/10/09 11:08:13

Physics in the plane:From condensed matter to string theory

© 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 sectorsNo local matrix element mixes the sectors

∆

eiθ

∆

€

0

€

1

H

© Simon Trebst

Topological order

quantum Hall liquids magnetic materials

2a

time reversal symmetry

broken invariant

© Simon Trebst

Quantum phase transitions

topological orderλ

λc

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

© Simon Trebst

The toric code in a magnetic fieldA first example

© Simon Trebst

The toric code

σi

As

Bp

As =�

j∈star(s)

σxj

Bp =�

j∈∂p

σzj

L2

L1

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

HTC = −Je

�

s

As − Jm

�

p

Bp +�

i

(hxσxi + hzσ

zi )

topological phase paramagnet

© Simon Trebst

Toric code and the transverse field Ising model

Bp = 2µzp

σxi = µx

pµxq

σi

Bp

Bp =�

j∈∂p

σzj

µp

HTFIM = −2Jm

�

p

µzp + hx

�

�p,q�

µxpµx

q

HTC = −Jm

�

p

Bp + hx

�

i

σxi

© Simon Trebst

Phase diagram

0

topological phase

vortex condensate

chargecondensate

self-d

uality

line

deconfined phase

paramagnethx

hz

continuoustransition

multicritical point?

first-ordertransition

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

© Simon Trebst

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

conf

inem

ent l

engt

h ξ

c

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

loop tension h

0

0.5

1

con

fin

emen

t le

ng

th

!c

L = 8L = 12L = 16

0.56 0.57 0.58 0.59 0.6

0.9

0.95

1

topological phase

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

loop tension h

0

0.5

1

con

fin

emen

t le

ng

th

!c

L = 8L = 12L = 16

0.56 0.57 0.58 0.59 0.6

0.9

0.95

1

topological phase

·6/(

L2

+2)

paramagnet

0 0.2 0.4 0.6 0.8 1loop tension h

0

0.2

0.4

0.6

0.8

1

gap

Δ /

2B

topological phasetopological phase paramagnet

σi

As

Bp

As =�

j∈star(s)

σxj

Bp =�

j∈∂p

σzj

L2

L1

magnetic vortices electric charges

plaquette excitations“magnetic vortices”

vertex excitations“electric charges”

© Simon Trebst

Splitting the topological degeneracy

δE ∝ L

0.54 0.55 0.56 0.57 0.58 0.59 0.6 0.61 0.62magnetic field hx

-8 -8

-6 -6

-4 -4

-2 -2

0 0

2 2

4 4

ener

gy sp

littin

g !E(L)

L = 48L = 40L = 32L = 24L = 16L = 12L = 8

topological phase paramagnet

δE ∝ exp(−αL)

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

© Simon Trebst

Phase diagram

0

topological phase

vortex condensate

chargecondensate

self-d

uality

line

deconfined phase

paramagnethx

hz

continuoustransition

multicritical point?

first-ordertransition

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

© Simon Trebst

Toric code: phase transitions

wavefunctiondeformations |ψ�

toric code paramagnet

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

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

z = 1

3D Isinguniversality

RG flow

HHamiltoniandeformations

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

Fendley (2008)

z ~ 2 2D Isinguniversality

“conformal QCP”

© Simon Trebst

Quantum double modelsfor time-reversal invariant systems

© Simon Trebst

Time-reversal invariant liquids

chiral excitations

✗

✗ “flux” excitationsnon-chiral

anyonic liquid L

anyonic liquid L̄

© Simon Trebst

The skeleton lattice

anyonic liquid

© Simon Trebst

The skeleton lattice

anyonic liquid

skeletonlattice

© Simon Trebst

Abelian liquids → loop gas / toric code

111

Semion fusion rulesloop gas

s× s = 1

1s s

Abelian liquids → loop gas / toric code

© Simon Trebst

Flux excitations

anyonic liquid

flux through “hole”

flux through “tube”

skeletonlattice

© Simon Trebst

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

© Simon Trebst

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.

© Simon Trebst

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

δφ(p),1 − Je

�

edges e

δ�(e),1

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

© Simon Trebst

The quantum phase transition

vary loop tension Je/Jp

“two sheets”topological order

“decoupled spheres”no topological order

© Simon Trebst

The quantum phase transition

“two sheets”topological order

“decoupled spheres”no topological order

topology fluctuationson all length scales

“quantum foam”

continuous transition

© Simon Trebst

Universality classes (semions)

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

Fendley (2008)

z ~ 2 2D Isinguniversality

wavefunctiondeformations |ψ�

toric code paramagnet

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

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

z = 1

3D Isinguniversality

RG flow

HHamiltoniandeformations

© Simon Trebst

The non-Abelian case

τ ττ

τ τ1 1

11

Fibonacci anyon fusion rules τ × τ = 1 + τ

© Simon Trebst

Non-abelian liquids → string net

τ ττ

τ τ1 1

11

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

Non-abelian liquids → string net

© Simon Trebst

The quantum phase transition

vary string tension Je/Jp

“two sheets”topological order

“decoupled spheres”no topological order

© Simon Trebst

One-dimensional analog

© Simon Trebst

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.

© Simon Trebst

Phase diagram

c = 14/15exactly solvable

c = 7/5exactly solvable

non-Abeliantopological phase

non-Abeliantopological phase

critical phasec = 7/5

Jp

Jr

© Simon Trebst

Gapless theory & exact solution

(1, 1)

(1, τ)

(τ, 1)(τ, τ)1 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

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

d =�

2 + φ

© Simon Trebst

c = 14/15exactly solvable

c = 7/5exactly solvable

non-abeliantopological phase

non-abeliantopological phase

“quantum foam”c = 7/5

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

Gapless theory & exact solution

© Simon Trebst

c = 14/15exactly solvable

c = 7/5exactly solvable

non-abeliantopological phase

non-abeliantopological phase

“quantum foam”c = 7/5

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

Gapless theory & exact solution

© Simon Trebst

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

c = 14/15exactly solvable

c = 7/5exactly solvable

non-abelian

topological phase

non-abelian

topological phase

“quantum foam”c = 7/5

Jp

Jr

© Simon Trebst

Back to two dimensions

wavefunctiondeformations |ψ�

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

© Simon Trebst

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 5, 834 (2009).