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
![Page 1: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/1.jpg)
Real space RG and the emergence of topological order
Michael LevinHarvard University
Cody NaveMIT
![Page 2: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/2.jpg)
Basic issue
Fractional statisticsGround state deg.
Topological order
Lattice scale Long distances
Consider quantum spin system in topological phase:
![Page 3: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/3.jpg)
Topological order is an emergent phenomena No signature at lattice scale Contrast with symmetry breaking order:
![Page 4: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/4.jpg)
Topological order is an emergent phenomena No signature at lattice scale Contrast with symmetry breaking order:
Sz
a
Symmetry breaking Topological
![Page 5: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/5.jpg)
Topological order is an emergent phenomena No signature at lattice scale Contrast with symmetry breaking order:
Sz
a
Symmetry breaking Topological
![Page 6: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/6.jpg)
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
![Page 7: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/7.jpg)
One approach: Real space renormalization group
Generic models flow to special fixed points:
Expect fixed points are string-net (e.g. Turaev-Viro) models
![Page 8: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/8.jpg)
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)
![Page 9: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/9.jpg)
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
![Page 10: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/10.jpg)
Tensor network models
Very general class of lattice models
Examples:- Ising model- Potts model - Six vertex model
![Page 11: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/11.jpg)
Definition
Need: Tensor Tijk, where i,j,k=1,…,D.
![Page 12: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/12.jpg)
Definition Define: e-S(i,j,k,…) = Tijk Tilm Tjnp Tkqr …
![Page 13: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/13.jpg)
Definition Define: e-S(i,j,k,…) = Tijk Tilm Tjnp Tkqr …
Partition function:
Z = ijk e-S(i,j,k,…)
= ijk Tijk Tilm Tjnp …
![Page 14: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/14.jpg)
One dimensional case
TT TT TT TT TTi j
Z = ijk Tij Tjk …= Tr(TN)
k
![Page 15: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/15.jpg)
One dimensional case
TT TT TT TT TT
![Page 16: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/16.jpg)
One dimensional case
TT TT TT TT TT
![Page 17: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/17.jpg)
One dimensional case
TT TT TT TT TT
T’ T’ T’ T’ T’
T’ik = Tij Tjk
![Page 18: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/18.jpg)
Higher dimensions
T T
T
TT
TT’
Naively:
![Page 19: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/19.jpg)
Higher dimensions
T T
T
TT
TT’
Naively:
But tensors grow with each step
![Page 20: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/20.jpg)
Tensor renormalization group
![Page 21: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/21.jpg)
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
![Page 22: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/22.jpg)
Tensor renormalization group
![Page 23: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/23.jpg)
Tensor renormalization group
Second step:
T’ijk = pqr SkpqSjqr Sirp
![Page 24: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/24.jpg)
Tensor renormalization group
![Page 25: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/25.jpg)
Tensor renormalization group
Iterate: T T’ T’’ …
Efficiently compute partition function Z
Fixed point T* captures universal physics
![Page 26: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/26.jpg)
Physical picture
Consider generic lattice model:
Want: partition function ZR
![Page 27: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/27.jpg)
Physical picture
Partition function for triangle:
![Page 28: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/28.jpg)
Physical picture
Think of (a,b,c) as a tensor
Then: ZR = …
![Page 29: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/29.jpg)
Physical picture
Think of (a,b,c) as a tensor
Then: ZR = …
Tensor network model!
![Page 30: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/30.jpg)
Physical picture
First step of TRG: find S such that
j k
i
j k
l T TS
S
i l
![Page 31: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/31.jpg)
Physical picture
First step of TRG: find S such that
j k
i
j k
l T TS
S
i l
![Page 32: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/32.jpg)
Physical picture
First step of TRG: find S such that
j k
i
j k
l T TS
S
i l
??
![Page 33: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/33.jpg)
Physical picture
First step of TRG: find S such that
j k
i
j k
l T TS
S
i l
=
![Page 34: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/34.jpg)
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 !
![Page 35: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/35.jpg)
Physical picture
Second step:
![Page 36: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/36.jpg)
Physical picture
Second step:
![Page 37: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/37.jpg)
Physical picture
TRG combines small triangles into larger triangles
![Page 38: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/38.jpg)
Physical picture
But the indices of tensor have larger and larger ranges: 2L 23L …
How can truncation to tensorTijk possibly be accurate?
![Page 39: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/39.jpg)
Physical interpretation of
is a quantum wave function
![Page 40: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/40.jpg)
Non-critical case
System non-critical is a ground state of gapped Hamiltonian
is weakly entangled: as L , entanglement entropy S const.
![Page 41: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/41.jpg)
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
![Page 42: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/42.jpg)
Critical case
is a gapless ground state as L , S ~ log L
Method breaks down at criticality
Analogous to breakdown of DMRG
![Page 43: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/43.jpg)
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.
![Page 44: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/44.jpg)
Example: Triangular lattice Ising model
![Page 45: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/45.jpg)
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
![Page 46: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/46.jpg)
Physical derivation
Assume no long range order Recall physical interpretation of T*:
i
j
k
T*ijk i j k
![Page 47: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/47.jpg)
Physical derivation
Assume no long range order Recall physical interpretation of T*:
j
k
T*ijk i j k
i1
i2
i1 i2
![Page 48: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/48.jpg)
Physical derivation
Assume no long range order Recall physical interpretation of T*:
T*ijk i j k
i1
i2k1
k2
j2 j1
![Page 49: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/49.jpg)
Physical derivation
Assume no long range order Recall physical interpretation of T*:
i1
i2k1
k2
j2 j1
T*ijk = i2j1
j2k1 k2i1
![Page 50: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/50.jpg)
Physical derivation
Assume no long range order Recall physical interpretation of T*:
T*ijk = i2j1
j2k1 k2i1
T*
=
![Page 51: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/51.jpg)
Fixed point solutions Are these actually solutions? Yes.
![Page 52: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/52.jpg)
Fixed point solutions Are these actually solutions? Yes. But we have too many solutions! What’s going on?
![Page 53: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/53.jpg)
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
![Page 54: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/54.jpg)
Fixed points
![Page 55: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/55.jpg)
Fixed surfaces
![Page 56: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/56.jpg)
Fixed surfaces
The points on each surface differ in short distance physics
![Page 57: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/57.jpg)
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
![Page 58: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/58.jpg)
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
![Page 59: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/59.jpg)
Generalization to (2+1)D?
(1+1)D (2+1)D
![Page 60: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/60.jpg)
Generalization to (2+1)D?
Tijk
Regular triangular lattice
(1+1)D (2+1)D
i jk
![Page 61: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/61.jpg)
Generalization to (2+1)D?
Tijk Tijkl
Regular triangular lattice
Regular triangulation of R3
(1+1)D (2+1)D
i jk
![Page 62: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/62.jpg)
Generalization to (2+1)D?
(1+1)D (2+1)D
![Page 63: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/63.jpg)
Generalization to (2+1)D?
(1+1)D (2+1)D
![Page 64: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/64.jpg)
Fixed point ansatz in (2+1)D? Expect that faces can be labeled byindices corresponding to boundaries:
i
![Page 65: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/65.jpg)
Fixed point ansatz in (2+1)D? Expect that faces can be labeled byindices corresponding to boundaries:
i1
i2i3
b
c
a
![Page 66: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/66.jpg)
Fixed point ansatz in (2+1)D? Expect that faces can be labeled byindices corresponding to boundaries:
i1
i2i3
b
c
ad
e
f
![Page 67: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/67.jpg)
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
![Page 68: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/68.jpg)
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 )
![Page 69: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/69.jpg)
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!
![Page 70: Real space RG and the emergence of topological order](https://reader035.vdocument.in/reader035/viewer/2022062408/56813ad5550346895da30c50/html5/thumbnails/70.jpg)
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?