Real space RG and the emergence of topological order

Michael LevinHarvard University

Cody NaveMIT

Basic issue

Fractional statisticsGround state deg.

Topological order

Lattice scale Long distances

Consider quantum spin system in topological phase:

Topological order is an emergent phenomena No signature at lattice scale Contrast with symmetry breaking order:

Topological order is an emergent phenomena No signature at lattice scale Contrast with symmetry breaking order:

Sz

a

Symmetry breaking Topological

Topological order is an emergent phenomena No signature at lattice scale Contrast with symmetry breaking order:

Sz

a

Symmetry breaking Topological

Problem

Hard to probe topological order- e.g. numerical simulations

Even harder to predict topological order- Very limited analytic methods- Only understand exactly soluble string-net

(e.g. Turaev-Viro) models where = a

One approach: Real space renormalization group

Generic models flow to special fixed points:

Expect fixed points are string-net (e.g. Turaev-Viro) models

Outline

I. RG method for (1+1)D modelsA. Describe basic methodB. Explain physical picture (and relation to DMRG)C. Classify fixed points

II. Suggest a generalization to (2+1)DA. Fixed points exactly soluble string-net models (e.g. Turaev-Viro)

Hamiltonian vs. path integral approach Want to do RG on (1+1)D quantum

lattice models

Could do RG on (H,) (DMRG)

Instead, RG on 2D “classical” lattice models

(e.g. Ising model) with potentially complex weights

Tensor network models

Very general class of lattice models

Examples:- Ising model- Potts model - Six vertex model

Definition

Need: Tensor Tijk, where i,j,k=1,…,D.

Definition Define: e-S(i,j,k,…) = Tijk Tilm Tjnp Tkqr …

Definition Define: e-S(i,j,k,…) = Tijk Tilm Tjnp Tkqr …

Partition function:

Z = ijk e-S(i,j,k,…)

= ijk Tijk Tilm Tjnp …

One dimensional case

TT TT TT TT TTi j

Z = ijk Tij Tjk …= Tr(TN)

k

One dimensional case

TT TT TT TT TT

One dimensional case

TT TT TT TT TT

One dimensional case

TT TT TT TT TT

T’ T’ T’ T’ T’

T’ik = Tij Tjk

Higher dimensions

T T

T

TT

TT’

Naively:

Higher dimensions

T T

T

TT

TT’

Naively:

But tensors grow with each step

Tensor renormalization group

Tensor renormalization group

i l

j k

i

j k

l T TS

S

First step: find a tensor S such that

n SlinSjkn m Tijm Tklm

Tensor renormalization group

Tensor renormalization group

Second step:

T’ijk = pqr SkpqSjqr Sirp

Tensor renormalization group

Tensor renormalization group

Iterate: T T’ T’’ …

Efficiently compute partition function Z

Fixed point T* captures universal physics

Physical picture

Consider generic lattice model:

Want: partition function ZR

Physical picture

Partition function for triangle:

Physical picture

Think of (a,b,c) as a tensor

Then: ZR = …

Physical picture

Think of (a,b,c) as a tensor

Then: ZR = …

Tensor network model!

Physical picture

First step of TRG: find S such that

j k

i

j k

l T TS

S

i l

Physical picture

First step of TRG: find S such that

j k

i

j k

l T TS

S

i l

Physical picture

First step of TRG: find S such that

j k

i

j k

l T TS

S

i l

??

Physical picture

First step of TRG: find S such that

j k

i

j k

l T TS

S

i l

=

Physical picture

First step of TRG: find S such that

j k

i

j k

l T TS

S

i l

=

S is partition function for !

Physical picture

Second step:

Physical picture

Second step:

Physical picture

TRG combines small triangles into larger triangles

Physical picture

But the indices of tensor have larger and larger ranges: 2L 23L …

How can truncation to tensorTijk possibly be accurate?

Physical interpretation of

is a quantum wave function

Non-critical case

System non-critical is a ground state of gapped Hamiltonian

is weakly entangled: as L , entanglement entropy S const.

Non-critical case (continued) Can factor accurately as

1D Tijk i j k

for appropriate basis states {i}.

TRG is iterative construction of Tijk for larger and larger triangles

T* = limL Tijk

i

j

k

Critical case

is a gapless ground state as L , S ~ log L

Method breaks down at criticality

Analogous to breakdown of DMRG

Example: Triangular lattice Ising model Z = exp(K i j)

Realized by a tensor network with D=2:

T111 = 1, T122 = T212 = T221 = , T112 = T121 = T211 = T222 = 0

where = e-2K.

Example: Triangular lattice Ising model

Finding the fixed points

Fixed point tensors S*,T* satisfy:

j k

i

j k

l T* T*S*

S*

i l

S* S*

S*

T*

i

j kkj

i

Physical derivation

Assume no long range order Recall physical interpretation of T*:

i

j

k

T*ijk i j k

Physical derivation

Assume no long range order Recall physical interpretation of T*:

j

k

T*ijk i j k

i1

i2

i1 i2

Physical derivation

Assume no long range order Recall physical interpretation of T*:

T*ijk i j k

i1

i2k1

k2

j2 j1

Physical derivation

Assume no long range order Recall physical interpretation of T*:

i1

i2k1

k2

j2 j1

T*ijk = i2j1

j2k1 k2i1

Physical derivation

Assume no long range order Recall physical interpretation of T*:

T*ijk = i2j1

j2k1 k2i1

T*

=

Fixed point solutions Are these actually solutions? Yes.

Fixed point solutions Are these actually solutions? Yes. But we have too many solutions! What’s going on?

Fixed point solutions Are these actually solutions? Yes. But we have too many solutions! What’s going on?

Coarse graining is incomplete!

Fixed point still contains some lattice scale physics

Fixed points

Fixed surfaces

Fixed surfaces

The points on each surface differ in short distance physics

Classification of fixed surfaces

Two cases:1. No symmetry:

- Can continuously change any T*

ijk = i2 j1j2 k1

k2 i1

T*ijk = 1

Only one (trivial) universality class

Classification of fixed surfaces

2. Impose some symmetry (invariance under |i> Oi

j|j>):

- Can classify possibilities for each group G

- Fixed surfaces {Proj. rep. of G such that is

a rep. of G}

- e.g., G = SO(3), = spin-1/2: Haldane spin-1 chain!

Only nontrivial possibilities are generalizations of spin-1 chain

Generalization to (2+1)D?

(1+1)D (2+1)D

Generalization to (2+1)D?

Tijk

Regular triangular lattice

(1+1)D (2+1)D

i jk

Generalization to (2+1)D?

Tijk Tijkl

Regular triangular lattice

Regular triangulation of R3

(1+1)D (2+1)D

i jk

Generalization to (2+1)D?

(1+1)D (2+1)D

Generalization to (2+1)D?

(1+1)D (2+1)D

Fixed point ansatz in (2+1)D? Expect that faces can be labeled byindices corresponding to boundaries:

i

Fixed point ansatz in (2+1)D? Expect that faces can be labeled byindices corresponding to boundaries:

i1

i2i3

b

c

a

Fixed point ansatz in (2+1)D? Expect that faces can be labeled byindices corresponding to boundaries:

i1

i2i3

b

c

ad

e

f

Fixed point ansatz in (2+1)D? Expect that faces can be labeled byindices corresponding to boundaries:

i1

i2i3

b

c

a

T*ijkl = Fabc

def i1 j1 k1 i2 j2 l2

…

d

e

f

Fixed point solutions in (2+1)D?

Substituting into RG transformation gives fixed point constraints of form

n Fmlqkpn Fjip

mns Fjsnlkr = Fjip

qkrFriqmls

etc.

(but no constraint on )

Fixed point solutions in (2+1)D?

Substituting into RG transformation gives fixed point constraints of form

n Fmlqkpn Fjip

mns Fjsnlkr = Fjip

qkrFriqmls

etc.

(but no constraint on )

Exactly constraints for Turaev-Viro (or string-net) models!

Conclusion TRG approach gives:

1. Understanding of emergence of topological order.2. Classification of fixed points3. Powerful numerical method in (1+1)D

Does it work in (2+1)D?