Real space RG and the emergence of topological order
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Real space RG and the emergence of topological order
Michael LevinHarvard University
Cody NaveMIT
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Basic issue
Fractional statisticsGround state deg.
Topological order
Lattice scale Long distances
Consider quantum spin system in topological phase:
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Topological order is an emergent phenomena No signature at lattice scale Contrast with symmetry breaking order:
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Topological order is an emergent phenomena No signature at lattice scale Contrast with symmetry breaking order:
Sz
a
Symmetry breaking Topological
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Topological order is an emergent phenomena No signature at lattice scale Contrast with symmetry breaking order:
Sz
a
Symmetry breaking Topological
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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
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One approach: Real space renormalization group
Generic models flow to special fixed points:
Expect fixed points are string-net (e.g. Turaev-Viro) models
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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)
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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
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Tensor network models
Very general class of lattice models
Examples:- Ising model- Potts model - Six vertex model
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Definition
Need: Tensor Tijk, where i,j,k=1,…,D.
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Definition Define: e-S(i,j,k,…) = Tijk Tilm Tjnp Tkqr …
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Definition Define: e-S(i,j,k,…) = Tijk Tilm Tjnp Tkqr …
Partition function:
Z = ijk e-S(i,j,k,…)
= ijk Tijk Tilm Tjnp …
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One dimensional case
TT TT TT TT TTi j
Z = ijk Tij Tjk …= Tr(TN)
k
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One dimensional case
TT TT TT TT TT
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One dimensional case
TT TT TT TT TT
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One dimensional case
TT TT TT TT TT
T’ T’ T’ T’ T’
T’ik = Tij Tjk
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Higher dimensions
T T
T
TT
TT’
Naively:
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Higher dimensions
T T
T
TT
TT’
Naively:
But tensors grow with each step
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Tensor renormalization group
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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
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Tensor renormalization group
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Tensor renormalization group
Second step:
T’ijk = pqr SkpqSjqr Sirp
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Tensor renormalization group
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Tensor renormalization group
Iterate: T T’ T’’ …
Efficiently compute partition function Z
Fixed point T* captures universal physics
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Physical picture
Consider generic lattice model:
Want: partition function ZR
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Physical picture
Partition function for triangle:
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Physical picture
Think of (a,b,c) as a tensor
Then: ZR = …
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Physical picture
Think of (a,b,c) as a tensor
Then: ZR = …
Tensor network model!
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Physical picture
First step of TRG: find S such that
j k
i
j k
l T TS
S
i l
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Physical picture
First step of TRG: find S such that
j k
i
j k
l T TS
S
i l
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Physical picture
First step of TRG: find S such that
j k
i
j k
l T TS
S
i l
??
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Physical picture
First step of TRG: find S such that
j k
i
j k
l T TS
S
i l
=
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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 !
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Physical picture
Second step:
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Physical picture
Second step:
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Physical picture
TRG combines small triangles into larger triangles
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Physical picture
But the indices of tensor have larger and larger ranges: 2L 23L …
How can truncation to tensorTijk possibly be accurate?
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Physical interpretation of
is a quantum wave function
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Non-critical case
System non-critical is a ground state of gapped Hamiltonian
is weakly entangled: as L , entanglement entropy S const.
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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
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Critical case
is a gapless ground state as L , S ~ log L
Method breaks down at criticality
Analogous to breakdown of DMRG
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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.
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Example: Triangular lattice Ising model
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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
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Physical derivation
Assume no long range order Recall physical interpretation of T*:
i
j
k
T*ijk i j k
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Physical derivation
Assume no long range order Recall physical interpretation of T*:
j
k
T*ijk i j k
i1
i2
i1 i2
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Physical derivation
Assume no long range order Recall physical interpretation of T*:
T*ijk i j k
i1
i2k1
k2
j2 j1
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Physical derivation
Assume no long range order Recall physical interpretation of T*:
i1
i2k1
k2
j2 j1
T*ijk = i2j1
j2k1 k2i1
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Physical derivation
Assume no long range order Recall physical interpretation of T*:
T*ijk = i2j1
j2k1 k2i1
T*
=
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Fixed point solutions Are these actually solutions? Yes.
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Fixed point solutions Are these actually solutions? Yes. But we have too many solutions! What’s going on?
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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
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Fixed points
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Fixed surfaces
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Fixed surfaces
The points on each surface differ in short distance physics
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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
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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
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Generalization to (2+1)D?
(1+1)D (2+1)D
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Generalization to (2+1)D?
Tijk
Regular triangular lattice
(1+1)D (2+1)D
i jk
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Generalization to (2+1)D?
Tijk Tijkl
Regular triangular lattice
Regular triangulation of R3
(1+1)D (2+1)D
i jk
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Generalization to (2+1)D?
(1+1)D (2+1)D
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Generalization to (2+1)D?
(1+1)D (2+1)D
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Fixed point ansatz in (2+1)D? Expect that faces can be labeled byindices corresponding to boundaries:
i
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Fixed point ansatz in (2+1)D? Expect that faces can be labeled byindices corresponding to boundaries:
i1
i2i3
b
c
a
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Fixed point ansatz in (2+1)D? Expect that faces can be labeled byindices corresponding to boundaries:
i1
i2i3
b
c
ad
e
f
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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
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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 )
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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!
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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?