real space rg and the emergence of topological order
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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?