Xiao-Gang Wen- ‘Complete’ characterization of topological order: Modular trans., Pattern of zeros, Tensor net-work

8/3/2019 Xiao-Gang Wen- Complete characterization of topological order: Modular trans., Pattern of zeros, Tensor net-work

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Complete characterization of topological order:

Modular trans., Pattern of zeros, Tensor net-work

Xiao-Gang Wen, MIT

July, 2009; Dresden

Z.-C. Gu M. Levin M. Barkeshli Y.-M. Lu Z.-H. Wang

Xiao-Gang Wen, MIT Complete characterization of topological order: Modular tra
http://goforward/http://find/http://goback/


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How to characterize it? F. Wilczek

We used to believe that all phases and phases transition aredescribed by Landau theory in terms of symmetry breaking, order

parameters, long-range-correlations. People proposed chiral spin liquid (CSL) to explain high Tc

superconductor, and we found the CSL is characterized by T and Pbreaking S1 (S2 S3) = 0. Kalmeyer &Laughlin, 87; Wen & Wilczek & Zee, 89



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But this is not the end of story, but a begining.Many CSLs with the same symmetry new kind of order Wen, 89



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To describe/define the new order, we need to identifymeasurable universal quantities



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To describe/define the new order, we need to identifymeasurable universal quantities

Topology-dependent and topologically stable ground statedegeneracy can (partially) describe the new order. Wen 89, Wen & Niu 90(Motivate us to name the new order as topological order Wen 89)

Topological entanglement entropy and spectrum can describe(partially) the topological order. Kitaev & Preskill 06, Levin & Wen 06, Haldane 08

(Can be probed by quantum noise Klich & Levitov 08)Xiao-Gang Wen, MIT Complete characterization of topological order: Modular tra


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Topological order = pattern of long range entanglement

eiHlocal |topo. ordered = |direct product state = |1 |2 ...



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Lord Kelvins atoms (knotted tubes of ether) Kelvin, 1867 cannot befermions. (Knots are not topo. enough against local destruction.)



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Lord Kelvins atoms (knotted tubes of ether) Kelvin, 1867 cannot befermions. (Knots are not topo. enough against local destruction.)

|long range entangled =

|loops or string-nets

A modern picture of atoms (which unifies fermions and light): Electons/Quarks = ends of (long) tubes of ether.

Light = fluctuations of tubes of ether (density wave of tubes).



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How to find a complete description of topological orders?

Two issues:

What symbols/notions should we use to completely describetopological orders?

Knowing the symbols, how to calculate the symbol that describethe topological order in the ground state of a generic Hamiltonian.



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How to find a complete description of topological orders?

Two issues:

What symbols/notions should we use to completely describetopological orders?

Knowing the symbols, how to calculate the symbol that describethe topological order in the ground state of a generic Hamiltonian.

In this talk, I will describe three attempts to completely describe

topological orders



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An old attempt: Non-Abelian Berrys phases



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Projective representation of modular group

Consider a family of FQH Hamiltonian H(), = x + iy, with the

following inverse mass matrix: (m1)ij() =

y+

2x

y x

y xy

1y

.

+1 H( ) H( ) H( ) H( )

(x, y) (x + y, y) : + 1; (x, y) (y, x) : 1/.

Non-Abelian Berry phase of deg. ground states: U( ) = e

Non-Abelian Berry phase T = U( + 1) and

S = U( 1/) of the degenerate ground state projective representation of modular group, which maycompletely (?) describe the topological order. Wen 89

Eigenvalues of T quasiparticle statistics. Checked for Abelian FQH states described by the K-matrix.



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Topological order in FQH states and Pattern of zeros

We can use FQH wavefunction ({zi}) to completely characterizetopological order in ({zi}).



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({zi}) contain to too much useless information and the label ismany-to-one.



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The key is to extract universal information from ({zi}).



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The key is to extract universal information from ({zi}).

Filling fraction = 1/m Laughlin state 1/m = (zi zj)m

is characterized by the mth

zero as we bring two electrons together. Generalizing that, we bring a electrons together in a wave function:

Let zi = i + z(a), i = 1, 2, , a

({zi}) = SaP(1,...,a; z(a), za+1, za+2, ) + O(

Sa+1)

The sequence of integers {Sa} characterizes the FQH states in thefirst Landau level and is called the pattern of zeros (POZ).

The pattern of zeros {Sa} is a quantitative symbol todescribe and characterize the (chiral) topological order in

FQH states Wen & Wang 08, Barkeshli & Wen 08Xiao-Gang Wen, MIT Complete characterization of topological order: Modular tra

T d l ifi i f T l i l d i FQH


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Towards a classification of Topological order in FQH states

Not all sequences of integers {Sa} can correspond to a FQHwavefunction. Only those that satisfy, for any a, b, c,

Sa+b Sa Sb 0

Sa+b+c Sa+b Sb+c Sa+c + Sa + Sb + Sc = even 0

can correspond to FQH wavefunctions.Finding all those sequences may lead to a classification oftopological orders in FQH states

Relation to 1D CDW picture: Seidel & Lee 06, Bergholtz etal 06, Bernevig & Haldane 07,Seidel & Yang 08, Ardonne etal 08

view la = Sa Sa1 as the orbital occupied by ath electron

occupation distribution nl=number of la = l.Z2 : (S2, S3,...) = (0, 2, 4, 8,...), (nl) = (20|20|20|...), n = 2

Z(2)5 : (S2, S3,...) = (0, 2, 6,...), (nl) = (20102000|20102000|...), n = 5

n-cluster structure: n = number of particles in one period of nl.Xiao-Gang Wen, MIT Complete characterization of topological order: Modular tra

A li i f {S } h i i


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An application of {Sa} characterization

A quasiparticle in a FQH state can also be quan-titatively characterized by pattern of zeros {S;a}:

S ;a

Let (, zi) be a FQH wavefunction with a quasiparticle at ,then S;a is the order of zero of (, zi) when we bring a electronsto . ((, zi) =

(zi )1/m for a Laughlin quasiparticle.)

{S;a} must satisfies:

S;a+b S;a Sb 0,

S;a+b+c S;a+b S;a+c Sb+c + S;a + Sb + Sc 0

The above equations have many solutions, and each solution

correspond to a type of quasiparticle.{Sa} and {S;a} are quantitative characterizations of FQHstate and their quasiparticles. Such quantitativecharacterizations allow us to calculation topologicalproperties quantitatively.


T l i l i f h f


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Topological properties from the pattern of zeros

From {Sa} we can get (with the help of vertex algebra): n-cluster structure: wave function for n-electron bound states

reduce to Laughlin wavefunction ODLRO for n-electron clusters.ODLRO only contains cluster condition very limited info.

filling fraction = lima2a2

Sa= n

Sn+1Sn ground state degeneracy on genus g surfaces number of quasiparticle types quasiparticle charges Q = S,nSn

Sn+1Sn quasiparticle statistics/scaling-dim, and quantum dimensions the fusion algebra of quasiparticles the central charge of edge excitations

;a 1 2 3S ;a S ;a S

Quasiparticle fusionXiao-Gang Wen, MIT Complete characterization of topological order: Modular tra

Ch t i t d th h fi d i t L i


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Characterize topo. orders through fixed-point Lagrangian

We may want to use Lagrangian to characterize phases, but- some times similar Lagrangian correspond to the same phase- some times similar Lagrangian correspond to the different phases

cgg

Again the Lagrangian is a many-to-one label of topological orders(and symmetry breaking orders)


Ch t i t d th h fi d i t L i


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Characterize topo. orders through fixed-point Lagrangian

We may want to use Lagrangian to characterize phases, but- some times similar Lagrangian correspond to the same phase- some times similar Lagrangian correspond to the different phases

cgg

Again the Lagrangian is a many-to-one label of topological orders(and symmetry breaking orders)

A RG idea: under the RG transformation, a Lagrangian flows tofixed-point Lagrangian. It is the fixed-point Lagrangian that

characterizes phase.

cgg


Limitations of standard RG approach


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But the RG approach appear not to apply to topological phases:If a bosonic system is in a topological phase, then its low energyeffective Lagrangian (the fixed-point Lagrangian) can be

A pure gauge theory with G = Z2, Zn, U(1), SU(2),... A Chern-Simons gauge theory with any G A QED (U(1) gauge theory + massless fermions) A QCD (SU(2) gauge theory + massless fermions) A gravity theory with gapless gravitions




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But the RG approach appear not to apply to topological phases:If a bosonic system is in a topological phase, then its low energyeffective Lagrangian (the fixed-point Lagrangian) can be

A pure gauge theory with G = Z2, Zn, U(1), SU(2),... A Chern-Simons gauge theory with any G A QED (U(1) gauge theory + massless fermions) A QCD (SU(2) gauge theory + massless fermions) A gravity theory with gapless gravitions

How can the RG flow of a bosonic Lagrangian L() with a scaler

field produces such rich class of fixed-point Lagrangian withgauge fields and fermionic fields?


Tensor renormalization group: a new RG approach


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After a discretization, we can rewrite any space-time path integralat a tensor-trace over a tensor network

D(x, t)eL() ={i}

Tijkl tTr T

T

T

T

T J

T

T

T

T

T

T

T

TT

T

T

T

J

If we know how to calculate tensor-trace, we can solve any thing.




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D(x, t)eL() ={i}

Tijkl tTr T

T

T

T

T J

T

T

T

T

T

T

T

TT

T

T

T

J


Unfortunately, according to the principle of no-free lunch,calculating the tensor-trace is an NP hard problem. Schuch etc 07




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D(x, t)eL() ={i}

Tijkl tTr T

T

T

T

T J

T

T

T

T

T

T

T

TT

T

T

T

J


Unfortunately, according to the principle of no-free lunch,calculating the tensor-trace is an NP hard problem. Schuch etc 07

But Levin and Nave discovered a principle of free lousy lunch: ifyou are willing to accept some errors, calculating the tensor-tracehas only polynomial complexity.


Filtering out local entanglements


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T

T

t T

T T

T

T

T

T

T

T T T T

TTTT

SS

SS

T

TT

TT

T

4

3

2

1

T

T

repeat the above iteration to reduce the number of tensors to one,then take the trace of that tensor to calculate the path integral.


http://goforward/http://goforward/http://find/http://goback/


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T

T

t T

T T

T

T

T

T

T

T T T T

TTTT

SS

SS

T

TT

TT

T

4

3

2

1

T

T


Levin and Naves implementation of TRG flow T T

has a smallproblem: the resulting fixed-point tensor is not isolated.Local non-universal entanglements are not completely removed




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T

T

t T

T T

T

T

T

T

T

T T T T

TTTT

SS

SS

T

TT

TT

T

4

3

2

1

T

T


Levin and Naves implementation of TRG flow T T

has a smallproblem: the resulting fixed-point tensor is not isolated.Local non-universal entanglements are not completely removed

T

TTTT

T

TT

TT

T

T

T

T

T T

T T

T

T T T T

TTT

T

T

T

T

T T

S2

S4

S4S2

2S

4S

2S4S

S

S

3S

1S

3

1

S

S

S

S

1

3

1

3

Tensor entanglement filtering renormalization (TEFR): Gu & Wen 09Filter out local entanglement but keep long range entanglement


Application of TEFR to 2D statistical Ising model


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Using TRG/TEFR and the resulting fixed-point tensor, we cancalculate free energy, ground state energy, low energy spectrum,

central charge and scaling dimensions, entanglement entropy,correlation functions, etc Zhengcheng Gu & Wen 09

2

T

FF"

X1X2

=(b)/(a)2X1

T

c X =(c)/(a)2

(c)

T TTTT

(a) (b)

0.2

0.4

0.6

0.8

1

2 2.4 2.6 2.82.20

1.2

2.82.62.42.221.8

1

1.2

1.4

0.5

1

1.5

2

2.5

1.8

Fixed-point tensors: Thigh1111 = 1 and Tlow1111 = T

low2222 = 1.

Symmetry breaking as direct sum: Tlow = Thigh Thigh




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Using TRG/TEFR and the resulting fixed-point tensor, we cancalculate free energy, ground state energy, low energy spectrum,

central charge and scaling dimensions, entanglement entropy,correlation functions, etc Zhengcheng Gu & Wen 09

2

T

FF"

X1X2

=(b)/(a)2X1

T

c X =(c)/(a)2

(c)

T TTTT

(a) (b)

0.2

0.4

0.6

0.8

1

2 2.4 2.6 2.82.20

1.2

2.82.62.42.221.8

1

1.2

1.4

0.5

1

1.5

2

2.5

1.8

Fixed-point tensors: Thigh1111 = 1 and Tlow1111 = T

low2222 = 1.

Symmetry breaking as direct sum: Tlow = Thigh Thigh

c h1 h2 h3 h4

0.49942 0.12504 0.99996 1.12256 1.12403

1/2 1/8 1 9/8 9/8

Computation time: 10 hours on a desktop.Xiao-Gang Wen, MIT Complete characterization of topological order: Modular tra

Application of TEFR to spin-1 chain


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pp p

H =i

Si Si+1 + U(S

zi )

2

+ Bi

Sxi

y

2z

B

U

2.0

0

Haldane

TRI

Z

0.6

Z 2

y2

TRI

Haldane

0

2.0

Z

Zz

0.6

2

B

U


Application of TEFR to spin-1 chain


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pp p

H =i

Si Si+1 + U(S

zi )

2

+ Bi

Sxi

y

2z

B

U

2.0

0

Haldane

TRI

Z

0.6

Z 2

y2

TRI

Haldane

0

2.0

Z

Zz

0.6

2

B

U

H =i

Si Si+1 + U(S

zi )

2

+B

2

i

(Sxi (Szi+1)

2 + Sxi+1(Szi )

2 + 2Szi )

0.5

z

y

B

U

2.0

0

Haldane

TRI

Z

Z 2

2

0.5

y

TRI

Haldane

0

2.0

Z

Zz2

2

B

U


Haldane phase is a symmetry protected topological phase


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p y y p p g p

Fixed-point tensor for Haldane phase:HT

2 2

22

Tabcd = Ta1a2,b1b2,c1c2,d1d2 = 2a1b22b1c22c1d22d1a2a = 1, 2, 3, 4; a1, a2 = 1, 2.

TH + T TH if T has time-reversal, parity and translationsymmetry.

Haldane phase is a symmetry protected topological phase. Fixed-point wavefunction:

)1 2a a1 2 1 2c c( b b


Haldane phase is a symmetry protected topological phase


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p y y p p g p

Fixed-point tensor for Haldane phase:HT

2 2

22

Tabcd = Ta1a2,b1b2,c1c2,d1d2 = 2a1b22b1c22c1d22d1a2a = 1, 2, 3, 4; a1, a2 = 1, 2.

TH + T TH if T has time-reversal, parity and translationsymmetry.

Haldane phase is a symmetry protected topological phase. Fixed-point wavefunction:

)1 2a a1 2 1 2c c( b b

The boundary spin-1/2 and string order parameter are not goodways to characterize the Haldane phase.


Theory of topological order


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Topological

Order of zerosNetworkPatternTensor

TransformationModular


Theory of topological order


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Emergent

photons & electrons

Spin

liquid

Long range

entanglement

Lattice

gauge theory

Statistics

NonAbelian

Vertex Algebra

(CFT)

Topological

quantum computation

Classification

of knots

Classification

of 3manifoldsADS/CFT

Order

Stringnet

condensation

Herbertsmithite

Topological

High Tcsuperconductor

Emergent

gravity

of zeros

symm. breaking

Pattern

Network

Tensor

Category

Tensor

Numerical

Approach FQH

Edge state

TransformationModular

Cont. trans. without


Documents

Xiao-Gang Wen- ‘Complete’ characterization of topological order: Modular trans., Pattern of zeros, Tensor net-work