Download Probing order beyond the Landau paradigm

Real space RG and the
emergence of topological order
Michael Levin
Harvard University
Cody Nave
MIT
Basic issue

Consider quantum spin system in
topological phase:
Topological order
Fractional statistics
Ground state deg.
Lattice scale
Long distances
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:
Symmetry breaking
Sz
a
Topological

Topological order is an
emergent phenomena

No signature at lattice scale

Contrast with symmetry breaking order:
Symmetry breaking
Sz
a
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 models
A. Describe basic method
B. Explain physical picture (and relation to
DMRG)
C. Classify fixed points
II. Suggest a generalization to (2+1)D
A. 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
T
T
T
T
i
T
j
T
k
T
T
T
Z = ijk Tij Tjk …= Tr(TN)
T
One dimensional case
T
T
T
T
T
T
T
T
T
T
One dimensional case
T
T
T
T
T
T
T
T
T
T
One dimensional case
T
T
T’
T
T
T’
T
T
T
T’
T’ik = Tij Tjk
T
T’
T
T
T’
Higher dimensions
Naively:
T
T
T
T
T
T
T’
Higher dimensions
Naively:
T
T
T
T
T
T’
T
But tensors grow with each step
Tensor renormalization group
Tensor renormalization group

First step: find a tensor S such that
i
S
l
j
S
k
i

l
T T
j
k
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
i
S
l
j
S
k

i T T l
j
k
Physical picture

First step of TRG: find S such that
i
S
l
j
S
k

i T T l
j
k
Physical picture

First step of TRG: find S such that
i
S
l
j
S
k
??


i T T l
j
k
Physical picture

First step of TRG: find S such that
i
S
l
j
S
k

=
i T T l
j
k
Physical picture

First step of TRG: find S such that
i
S
l
j
S
k

i T T l
j
k
=
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 tensor
Tijk 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}.
i
k
j

TRG is iterative construction of Tijk for larger
and larger triangles

T* = limL   Tijk
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:
i
S*
l
j
S* k
i T* T* l
=
j
i
i
T*
j
k
k
S*
=
S*
j
S*
k
Physical derivation


Assume no long range order
Recall physical interpretation of T*:
k
i
j
   T*ijk i j k
Physical derivation


Assume no long range order
Recall physical interpretation of T*:
k
i1
i1 i2
i2
j
   T*ijk i j k
Physical derivation


Assume no long range order
Recall physical interpretation of T*:
k2
k1
i1
i2
j2
j1
   T*ijk i j k
Physical derivation


Assume no long range order
Recall physical interpretation of T*:
k2
k1
i1
i2
j2
j1
T*ijk = i2j1 j2k1 k2i1
Physical derivation


Assume no long range order
Recall physical interpretation of T*:
T*

=


T*ijk = i2j1 j2k1 k2i1
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 j1j2 k1k2 i1 
T*ijk = 1
Only one (trivial) universality class
Classification of fixed surfaces
2. Impose some symmetry (invariance under |i> 
Oij|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?
(1+1)D
Regular triangular
lattice
i
k
Tijk
j
(2+1)D
Generalization to (2+1)D?
(1+1)D
Regular triangular
lattice
i
k
Tijk
(2+1)D
Regular
triangulation of R3
j
Tijkl
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 by
indices corresponding to boundaries:

i
Fixed point ansatz in (2+1)D?
Expect that faces can be labeled by
indices corresponding to boundaries:

i1
a
i3
b
c
i2
Fixed point ansatz in (2+1)D?
Expect that faces can be labeled by
indices corresponding to boundaries:

i1
a
i3
f
c
e
b
i2
d
Fixed point ansatz in (2+1)D?
Expect that faces can be labeled by
indices corresponding to boundaries:

i1
a
i3
f
c
e
b
i2
d
T*ijkl = Fabcdef  i1 j1 k1 i2 j2 l2…
Fixed point solutions in
(2+1)D?

Substituting into RG transformation
gives fixed point constraints of form
n Fmlqkpn Fjipmns Fjsnlkr = FjipqkrFriqmls
etc.
(but no constraint on )
Fixed point solutions in
(2+1)D?

Substituting into RG transformation
gives fixed point constraints of form
n Fmlqkpn Fjipmns Fjsnlkr = FjipqkrFriqmls
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 points
3. Powerful numerical method in (1+1)D
Does it work in (2+1)D?