1. Symmetric Tensor Networks and Topological Phases
Ying Ran (Boston College)
Quantum Entanglement 2017, Fudan Univ & NCTS, Taipei, Jan. 2017

2. Acknowledgement: Collaborators: References:
Shenghan Jiang (Boston College)
Panjin Kim, Hyungyong Lee, Jung Hoon Han (Sungkyunkwan University)
Brayden Ware, Chao-Ming Jian, Michael Zaletel (StationQ)
References:
arXiv: , S. Jiang, Y. Ran
arXiv: , P. Kim, H. Lee, S. Jiang, B. Ware, C. Jian, M. Zaletel, J. Han, Y. Ran
arXiv: , S. Jiang, P. Kim, J. Han, Y. Ran
arXiv: , S. Jiang, Y. Ran

3. Two types of symmetric topological phases
SPT --- symmetry protected topological phases: (Pollmann, Berg, Turner, Oshikawa, Chen, Liu, Gu, Wen…)
Example: AKLT spin chain, integer quantum Hall states, topological insulators…
Features:
no topological order
anomalous edge states protected by symmetry
SET --- symmetry enriched topological phases: (Wen, Essin, Hermele, Mesaros,YR, Barkeshli….)
Example: toric code, gapped quantum spin liquids, fractional quantum Hall states…
topological order (anyon excitations in 2d)
symmetry can be fractionalized (e.g. e/3 quasiparticle in Laughlin’s state).

4. Motivations
We focus on bosonic topological (SET or SPT) phases, which require strong interactions to realize.
Conceptual issues:
-- Classification problems
(SPT phases with spatial symmetries.)
(2) “Practical” issues: How to realize them?
-- Physical intuitions/guiding principles?
(Are there criteria like the band-inversion picture in topological insulators?)
-- Numerical methods suitable for searching for these topological phases in models?
(How to write down generic variational wavefunctions?)

5. (onsite and spatial ) symmetries of the system
Main result
Based on tensor-network formulation, we develop a machinery to: (1) systematically (but partially) classify topological phases (2) construct generic variational wavefunctions for these phases
(onsite and spatial ) symmetries of the system
(Partially) classification of topological phases and construction of generic wavefunctions for each class
1D-MPS, 2D-PEPS, and 3D generalizations
figures from R. Orus, Annals Phys. (2014)
This machinery answers:
How many classes of symmetric tensor- network wavefunctions that cannot be smoothly deformed into each other under certain assumptions?

6. Main result
Based on tensor-network formulation, we develop a machinery to: (1) systematically (but partially) classify topological phases (2) construct generic variational wavefunctions for these phases
(onsite and spatial ) symmetries of the system
(Partially) classification of topological phases and construction of generic wavefunctions for each class
Numerically simulating
Topological phases in practical models
Combining with tensor-based
variational numerical algorithms

7. Plan: Three applications of this machinery
(1) Classification and simulation of competing spin liquids in the spin-1/2 Heisenberg model on the kagome lattice

8. Plan: Three applications of this machinery
(1) Classification and simulation of competing spin liquids in the spin-1/2 Heisenberg model on the kagome lattice
(2) Classification of bosonic cohomological SPT: 𝐻 𝑑+1 (𝑆𝐺,𝑈 1 )
𝑆𝐺: on-site and lattice symmetries (onsite (Chen, Liu, Gu, Wen…), lattice (Chen, Hermele, Fu, Qi, Furusaki, Cheng…) )
𝑇 and 𝑃(mirror) should be treated as “anti-unitary”
Generic tensor wavefunctions for every class (if SG is discrete)

9. Plan: Three applications of this machinery
(1) Classification and simulation of competing spin liquids in the spin-1/2 Heisenberg model on the kagome lattice
(2) Classification of bosonic cohomological SPT: 𝐻 𝑑+1 (𝑆𝐺,𝑈 1 )
𝑆𝐺: on-site and lattice symmetries (onsite (Chen, Liu, Gu, Wen…), lattice (Chen, Hermele, Fu, Qi, Furusaki, Cheng…) )
𝑇 and 𝑃(mirror) should be treated as “anti-unitary”
Generic tensor wavefunctions for every class (if SG is discrete)
(3) A by-product: a general connection between “conventional” SET phases and SPT phases in 2D. For example:
SET: Toric code + 𝑍 2 Ising symmetry = 𝐼,𝑔 . 𝑔 𝑒 2 =−1, 𝑔 𝑚 2 =1
Trivial paramagnet
SET
Nontrivial SPT
condensing m
with 𝑔 𝑚 =1
condensing m
with 𝑔 𝑚 =−1

10. What are tensor networks?𝑖 𝛽 𝛾 𝛼
= 𝐴 𝛼𝛽𝛾𝛿 𝑖 ~ ∑ 𝐴 𝛼𝛽𝛾𝛿 𝑖 𝑖 ⊗|𝛼𝛽𝛾𝛿〉
tensor 𝐴 𝛿 𝑖 𝑖′ 𝛽 𝛽′ 𝐴 𝐵
tensor contraction 𝛼 𝛾 𝛾′
= 𝛾 𝐴 𝛼𝛽𝛾𝛿 𝑖 ⋅ 𝐵 𝛾𝛽′𝛾′𝛿′ 𝑖 ′ 𝛿 𝛿′ 𝐷 𝑑
𝜓 = 𝑖 𝑐 𝑖 1 𝑖 2 … 𝑖 𝑛 | 𝑖 1 , 𝑖 2 ,…, 𝑖 𝑛 〉
PEPS
⋯⋯⋯
Parameters:
𝑑 𝐷 4 /site
|𝜓〉

11. Why tensor networks?
1D-MPS, 2D-PEPS, and 3D generalizations …)
figures from R. Orus, Annals Phys. (2014)
In the past, indeed the first deep insight and systematic results of SPT phase were obtained by studying symmetry properties of MPS in 1d. (Pollmann, Berg, Turner, Oshikawa, Chen, Gu, Wen…)
Powerful numerical algorithms
1D MPS: DMRG (S. White …)
2D PEPS: iTEBD, CTM, TRG… (Cirac, Verstraete, Vidal, Gu, Levin, Wen, Xiang …)

12. Why tensor networks? = 𝐴 𝛼𝛽𝛾𝛿 𝑖 ~ ∑ 𝐴 𝛼𝛽𝛾𝛿 𝑖 𝑖 ⊗|𝛼𝛽𝛾𝛿〉
1D-MPS, 2D-PEPS, and 3D generalizations …)
figures from R. Orus, Annals Phys. (2014)
In particular, the tensor-network formulation is particularly suitable to understand the local symmetry properties of a global wavefunction, which are essential for symmetric topological phases. 𝑖 𝛽 𝛾 𝛼
= 𝐴 𝛼𝛽𝛾𝛿 𝑖 ~ ∑ 𝐴 𝛼𝛽𝛾𝛿 𝑖 𝑖 ⊗|𝛼𝛽𝛾𝛿〉 𝐴 𝛿
symmetry properties of these local quantum states (in enlarged Hilbert space)
 symmetry properties of the global physical state

13. How to implement global symmetries into tensor-network?
Basic assumption:
Symmetries on physical wavefunctions
Gauge transformation on internal legs ~

14. Gauge redundancy in a tensor-network𝐴 𝐵 𝐴 𝐵
𝑉 𝑉 −1 =
One internal bond ~ 𝐺𝐿(𝐷,𝑪) gauge redundancy 𝐷 𝑑
Gauge redundancy
~ 𝐺𝐿 𝐷,𝑪 𝑁 𝑏𝑜𝑛𝑑

15. Gauge redundancy & symmetry
𝜓 =𝑔|𝜓⟩
global symmetries on physical wavefunctions
gauge transformation on internal legs ~ 𝑔 ⋯⋯ 𝑔 ⋯⋯
𝑊 𝑔 2 ⋯⋯ ⋯⋯
𝑊 𝑔 5 =
𝑊 𝑔 1
𝑊 𝑔 3
𝑊 𝑔 3 −1 ⋯⋯ ⋯⋯ ⋯⋯
𝑊 𝑔 6 ⋯⋯ ⋯⋯ ⋯⋯
𝑊 𝑔 4
𝑊 𝑔 7 ⋯⋯ ⋯⋯
𝑇𝑁= 𝑊 𝑔 𝑔∘𝑇𝑁
Classify symmetric phases by different symmetry transformation rules of local tensors.
What are consistent conditions for 𝑊 𝑔 ?

16. Example: SPT phases: 𝐻 𝑑+1 𝑆𝐺,𝑈 1 (d=1)
Warm-up: d=1 (Pollmann, Berg, Turner, Oshikawa, Chen, Gu, Wen…)
The claim is that 𝐻 2 𝑆𝐺,𝑈 1 generally classifies d=1 bosonic SPT phase with full SG including both internal and spatial symmetries.
The main purpose here is to demonstrate the anti-unitary action of
mirror reflection in the tensor formulation.
𝑺=𝟏 AKLT chain
spin- 1 2
Famous example:
Protected by 𝑆𝑂 3
It turns out that

17. SPT phases: 𝐻 𝑑+1 𝑆𝐺,𝑈 1 (d=1)𝑔 =
Symmetry Condition:
𝑊 𝑔 (𝑎,𝑙)
𝑊 𝑔 (𝑎,𝑟)
𝑊 𝑔 1 𝑔 1 𝑊 𝑔 2 𝑔 2 = 𝑊 𝑔 1 𝑔 2 𝑔 1 𝑔 2 ?
It turns out that
Both sides of the transformations send the TN back to the same TN,
yet they do not have to be the same!

18. SPT phases: 𝐻 𝑑+1 𝑆𝐺,𝑈 1 (d=1)𝑔 =
Symmetry Condition:
𝑊 𝑔 (𝑎,𝑙)
𝑊 𝑔 (𝑎,𝑟)
𝑊 𝑔 1 𝑔 1 𝑊 𝑔 2 𝑔 2 = 𝑊 𝑔 1 𝑔 2 𝑔 1 𝑔 2 ?
Invariant gauge group:
IGG:
It turns out that 𝜒
𝜒 −1 𝜒
𝜒 −1 𝜒
𝜒 −1 𝜒
𝜒 −1
a single U(1) phase variable over the whole lattice.
Unlike arbitrary gauge transformations, an element in IGG leaves all local tensors invariant.

19. SPT phases: 𝐻 𝑑+1 𝑆𝐺,𝑈 1 (d=1)𝑔 =
Symmetry Condition:
𝑊 𝑔 (𝑎,𝑙)
𝑊 𝑔 (𝑎,𝑟)
⟹ 𝑊 𝑔 1 𝑔 1 𝑊 𝑔 2 𝑔 2 =𝜒( 𝑔 1 , 𝑔 2 ) 𝑊 𝑔 1 𝑔 2 𝑔 1 𝑔 2 , 𝜒 𝑔 1 , 𝑔 2 ∈𝐼𝐺𝐺
It turns out that
𝜒 −1 𝜒
𝜒 −1 𝜒
𝜒 −1 𝜒
𝜒 −1 𝜒
IGG:

20. SPT phases: 𝐻 𝑑+1 𝑆𝐺,𝑈 1 (d=1)𝑔 =
Symmetry Condition:
𝑊 𝑔 (𝑎,𝑙)
𝑊 𝑔 (𝑎,𝑟)
⟹ 𝑊 𝑔 1 𝑔 1 𝑊 𝑔 2 𝑔 2 =𝜒( 𝑔 1 , 𝑔 2 ) 𝑊 𝑔 1 𝑔 2 𝑔 1 𝑔 2 , 𝜒 𝑔 1 , 𝑔 2 ∈𝐼𝐺𝐺
Associativity ⇒ 𝜒( 𝑔 1 , 𝑔 2 ) 𝜒( 𝑔 1 𝑔 2 , 𝑔 3 )= 𝑔 1 𝜒 𝑔 2 , 𝑔 3 𝜒 𝑔 1 , 𝑔 2 𝑔 3 .
It turns out that 𝜒
𝜒 −1 𝜒
𝜒 −1 𝜒
𝜒 −1 𝜒
𝜒 −1
IGG: P
Just as time-reversal, mirror reflection send 𝜒→ 𝜒 −1 ,
so they should be treated as anti-unitary

21. SPT phases: 𝐻 𝑑+1 𝑆𝐺,𝑈 1 (d=1)𝑔 =
Symmetry Condition:
𝑊 𝑔 (𝑎,𝑙)
𝑊 𝑔 (𝑎,𝑟)
⟹ 𝑊 𝑔 1 𝑔 1 𝑊 𝑔 2 𝑔 2 =𝜒( 𝑔 1 , 𝑔 2 ) 𝑊 𝑔 1 𝑔 2 𝑔 1 𝑔 2 , 𝜒 𝑔 1 , 𝑔 2 ∈𝐼𝐺𝐺
Associativity ⇒ 𝜒( 𝑔 1 , 𝑔 2 ) 𝜒( 𝑔 1 𝑔 2 , 𝑔 3 )= 𝑔 1 𝜒 𝑔 2 , 𝑔 3 𝜒 𝑔 1 , 𝑔 2 𝑔 3 .
Can we tune this 𝜒( 𝑔 1 , 𝑔 2 ) away?
It turns out that
𝑊 𝑔 1 →𝜒 𝑔 1 𝑊 𝑔 1 , 𝑊 𝑔 2 →𝜒 𝑔 2 𝑊 𝑔 2 , 𝑊 𝑔 1 𝑔 2 →𝜒 𝑔 1 𝑔 2 𝑊 𝑔 1 𝑔 2
⇒𝜒( 𝑔 1 , 𝑔 2 )→ 𝜒 𝑔 1 , 𝑔 2 ∙[ 𝜒 𝑔 𝑔 1 𝜒 𝑔 2 𝜒 𝑔 1 𝑔 2 ] 𝜒
𝜒 −1 𝜒
𝜒 −1
𝜒 −1 𝜒
𝜒 −1 𝜒
IGG:

22. SPT phases: 𝐻 𝑑+1 𝑆𝐺,𝑈 1 (d=1)𝑔 =
Symmetry Condition:
𝑊 𝑔 (𝑎,𝑙)
𝑊 𝑔 (𝑎,𝑟)
⟹ 𝑊 𝑔 1 𝑔 1 𝑊 𝑔 2 𝑔 2 =𝜒( 𝑔 1 , 𝑔 2 ) 𝑊 𝑔 1 𝑔 2 𝑔 1 𝑔 2 , 𝜒 𝑔 1 , 𝑔 2 ∈𝐼𝐺𝐺
Associativity ⇒ 𝜒( 𝑔 1 , 𝑔 2 ) 𝜒( 𝑔 1 𝑔 2 , 𝑔 3 )= 𝑔 1 𝜒 𝑔 2 , 𝑔 3 𝜒 𝑔 1 , 𝑔 2 𝑔 (1)
It turns out that
𝑊 𝑔 1 →𝜒 𝑔 1 𝑊 𝑔 1 , 𝑊 𝑔 2 →𝜒 𝑔 2 𝑊 𝑔 2 , 𝑊 𝑔 1 𝑔 2 →𝜒 𝑔 1 𝑔 2 𝑊 𝑔 1 𝑔 2
⇒𝜒( 𝑔 1 , 𝑔 2 )→ 𝜒 𝑔 1 , 𝑔 2 ∙[ 𝜒 𝑔 𝑔 1 𝜒 𝑔 2 𝜒 𝑔 1 𝑔 2 ] (2)
Mathematically, equivalence classes of group functions satisfying Eq.(1) up to Eq.(2)
are nothing but 𝐻 2 𝑆𝐺,𝑈 1 , which classifies topologically distinct 𝑊 𝑔 transformation rules.

23. Plan
Next I will go to d=2.
Instead of discussion SPT classifications/constructions related to 𝐻 𝑑+1 𝑆𝐺,𝑈 1 , let me firstly discuss an example for SET phases, which turns out to be conceptually simpler.

24. Plan Spin liquids on the kagome lattice SPT phases: 𝐻 𝑑+1 (𝑆𝐺,𝑈 1 )
Anyon condensation mechanism:
“conventional” SET phases  SPT phases

25. SET example: The Heisenberg model on the kagome lattice
For systems on the kagome lattice with spin per site, what is the quantum phase of the following Hamiltonian?
𝐻=𝐽 𝑖𝑗 𝑆 𝑖 ∙ 𝑆 𝑗
Symmetries: on-site spin rotation, time reversal,
lattice translation, rotation, reflection
Sachdev, Marston, Senthil, Singh, Evenbly, Vidal, Ran, Hermele, Wen, Lee, Wang, Vishwanath, Iqbal, Becca, Sorella, Poilblanc, White, Huse, Depenbrock, McCulloch, Schollwock, Jiang, Balents, Mei, Xiang, He, Zaletel, Oshikawa, Pollmann… and many more
DMRG  spin liquid phase (SET for both onsite and lattice symmetries)?
Which spin liquid???

26. Implementing spin rotation symmetry
𝑈 𝜃 𝑛
𝑊 𝜃 𝑛
𝑊 𝜃 𝑛 =
𝑊 𝜃 𝑛
𝑊 𝜃 𝑛
𝑊 𝜃 𝑛 : representation of SU(2) symmetry  Local tensors are spin singlets
1 2
0⊕ 1 2
1 2
To see how we obtain the consistent equation for 𝑊 𝑔 . We consider a spin-1/2 system on square lattice. We want to construct states which are spin-singlet, and preserves translation symmetry.
The symmetry condition indicates that the every tensor is a spin singlet.

27. Invariant Gauge Group (𝑰𝑮𝑮)
2𝜋 spin rotation: =
(−) 𝐽 𝐽
𝐽 = − −1 | 0 ⟩ |↑ ⟩ |↓ ⟩
𝐽 2 =𝐼, 𝑍 2 matrix 𝐼𝐺𝐺:
a pure gauge transformation
leaves a single tensor invariant
relate to 𝑍 2 gauge theory (Swingle, Wen, Poilblanc, Schuch, Pérez-García,Cirac ….)
Minimal required 𝑍 2 𝐼𝐺𝐺 in spin-1/2 kagome system ~ no trivial symmetric phase (consistent with the Hastings-Oshikawa-Lieb-Schultz-Mattis theorem)
There is always “trivial 𝐼𝐺𝐺”— leg dependent phase factors
To construct SET classes, we should first figure out the way to represent topological order in tensor networks. We introduce a concept 𝐼𝐺𝐺 which will be defined below.
Consider PEPS defined on square lattice. 𝐽 is some nontrivial matrix, and 𝐽 2 =𝐼. …
Before interpret the physical meaning of this 𝑍 2 𝐼𝐺𝐺, let me give you an example, where the 𝑍 2 𝐼𝐺𝐺 naturally emerges.

28. Physical interpretation of 𝑰𝑮𝑮
What is the physical meaning of the 𝑍 2 matrix 𝐼𝐺𝐺?
𝑍 2 𝐼𝐺𝐺 ~ 𝑍 2 gauge theory (toric code); 𝐽 ~ flux line
Four-fold GSD on torus =
𝐺𝑆 1
: 𝑱 action
𝐺𝑆 2

29. Physical interpretation of 𝑰𝑮𝑮
Topological excitations: =
(−) 𝑱 𝑱 1
𝑒 (spinon) =
(+) 𝑱 𝑱
𝑚 (vison) 𝑚
𝑓 = 𝑒×𝑚
deconfined  spin liquid
confined  ordered phase (VBS, magnetic order)

30. 𝜒: leg dependent 𝑈 1 , 𝜂=𝐼 𝑜𝑟 𝐽
Tensor equations: interplay between symmetries and 𝑰𝑮𝑮 =
𝑊 𝑇 1
𝑇 (𝑥,𝑦)
𝑇 (𝑥−1,𝑦)
𝑊 𝑇 2
𝑇 (𝑥,𝑦−1)
So, we conclude, the local transformation rule of tensors satisfy group multiplication rule up to an 𝐼𝐺𝐺 element!
translation form a 𝑍×𝑍 group, defined by 𝑇 1 𝑇 2 = 𝑇 2 𝑇 1
𝑊 𝑇 1 𝑇 1 𝑊 𝑇 2 𝑇 2 =𝜒⋅𝜂⋅ 𝑊 𝑇 2 𝑇 2 𝑊 𝑇 1 𝑇 1
𝜒: leg dependent 𝑈 1 , 𝜂=𝐼 𝑜𝑟 𝐽

31. Symmetry fractionalization from tensor equations
𝑊 𝑇 1 𝑇 1 𝑊 𝑇 2 𝑇 2 =𝜂⋅ 𝑊 𝑇 2 𝑇 2 𝑊 𝑇 1 𝑇 1
𝜂=𝐼, trivial SET
𝜂=𝐽, 𝑒 carries fractional “translational” quantum number
𝑇 2 −1 𝑇 1 −1 𝑇 2 𝑇 1 →𝐽 𝐽 𝐽 = − 𝑒 𝐽

32. 𝑊 𝑔 1 𝑔 1 𝑊 𝑔 2 𝑔 2 =𝜒( 𝑔 1 , 𝑔 2 )𝜂 𝑔 1 , 𝑔 2 𝑊 𝑔 1 𝑔 2 𝑔 1 𝑔 2
symmetries
of the model
Identify 𝐼𝐺𝐺
List tensor equations
𝑊 𝑔 1 𝑔 1 𝑊 𝑔 2 𝑔 2 =𝜒( 𝑔 1 , 𝑔 2 )𝜂 𝑔 1 , 𝑔 2 𝑊 𝑔 1 𝑔 2 𝑔 1 𝑔 2
solve 𝑊 𝑔 by fixing gauge
gauge inequivalent 𝑊 𝑔 (classes of symmetric tensor networks)
𝑊 𝑔 𝑔∘ 𝑇 𝑎 = 𝑇 𝑎
constraint sub-Hilbert space for every class (generic wavefunctions)
Now let us summarize
tensor numerics
determine phase diagram

33. Kagome Heisenberg model
Topological invariants of tensor symmetry transformation rules:
𝜒 𝜎 , 𝜒 𝑇
“weak SPT” index, 2D AKLT like physics
𝜂 12 , 𝜂 𝑐6 , 𝜂 𝜎
label symmetry fractionalization of spinon-𝑒 in the 𝑍 2 QSL member phase.
𝑑=2
For 𝐷=6, can realize four classes
Unconstraint: 𝐷𝐼𝑀 𝑡𝑜𝑡 = 𝑑 𝐷 6 ≈𝟐𝟔𝟎𝟎
Constraint: 𝐷𝐼𝑀 𝑐𝑜𝑛𝑠𝑡𝑟𝑎𝑖𝑛𝑡 =𝟏𝟗
𝐷=6 (0⊕ 1 2 ⊕1)

34. Energy densities for optimal state of four classes
𝐷 𝑐𝑢𝑡 ~ virtual states kept when performing tensor contraction
We develop a tensor algorithm, which is suitable for the symmetric PEPS. It took us one year to get this picture…
For 𝐷=7, 8×8×3 lattice size, 𝐸~−𝟎.𝟒𝟑𝟔𝟔(𝟑)𝑱, comparable to (slightly higher than) DMRG report.
Two zero-flux classes have nearly degenerate energy
Which class?

35. Competing spin liquids?
𝐸(Zero-Flux I) ≈ 𝐸(Zero-Flux 2)
Sachdev’s Q1=Q2 state
Q1=-Q2 state
Two possibilities:
Their energy densities are different. But our numerics is not accurate enough to distinguish them.
They indeed share degenerate energy. Any physical reason?
𝑈(1) Dirac spin liquid?
(Hastings, Ran, Hermele, Lee, Wen, Iqbal, Becca, Poilblanc…)
Higgs II
Higgs I
zero-flux I
zero-flux II
(Lu, Ran, Lee)
Future work:
Long-range behavior?
Excitation spectrum?
More advanced optimization method

36. Plan Spin liquids on the kagome lattice SPT phases: 𝐻 𝑑+1 (𝑆𝐺,𝑈 1 )
Anyon condensation mechanism:
“conventional” SET phases  SPT phases

37. Bosonic cohomological SPT phases
In our framework, SPT: 𝐻 𝑑+1 (𝑆𝐺,𝑈 1 ) (not complete)
𝑆𝐺: on-site and lattice symmetries (onsite (Chen, Liu, Gu, Wen), lattice (Chen, Hermele, Fu, Qi, Furusaki,Cheng…))
𝑇 and 𝑃(mirror) ~ “anti-unitary”
Example:
1d (Yesterday), 𝐻 2 (𝑍 2 𝑇 ,𝑈(1))= 𝐻 2 (𝑍 2 𝑃 ,𝑈(1))= 𝑍 2 , the “Haldane phase”
2d, 𝑃 & 𝑇  𝐻 3 𝑍 2 𝑇 × 𝑍 2 𝑃 ,𝑈 1 = 𝑍 2 2
Generic wavefunctions (constrained tensor Hilbert space) for every class
A by-product: a general connection between SET and SPT phases in 2D.
𝑺=𝟏 Haldane chain
spin- 1 2
Protected by 𝑆𝑂 3
Our tensor construction also indicates a useful connection between SET and SPT phases. This byproduct can be understood beyond tensor language. So let us come to this byproduct first

38. SPT phases: 𝐻 𝑑+1 𝑆𝐺,𝑈 1 (d=2)
Plaquette IGG:
For every plaquette, there is
at least a phase variable ∈𝐼𝐺𝐺.
It turns out that

39. SPT phases: 𝐻 𝑑+1 𝑆𝐺,𝑈 1 (d=2)
Plaquette IGG:
For every plaquette, there is
at least a phase variable ∈𝐼𝐺𝐺.
It turns out that, to describe strong SPT
phases, plaquette-𝐼𝐺𝐺 need to contain
nontrivial matrix transformations.
It turns out that

40. SPT phases: 𝐻 𝑑+1 𝑆𝐺,𝑈 1 (d=2)
Plaquette IGG:
In order to describe cohomological
SPT phases, we assume:
global matrix 𝐼𝐺𝐺 decomposition:
𝐽= 𝑝 𝜆 𝑝 𝜆
It turns out that, to describe strong SPT
phases, plaquette-𝐼𝐺𝐺 need to contain
nontrivial matrix transformations.
It turns out that

41. SPT phases: 𝐻 𝑑+1 𝑆𝐺,𝑈 1 (d=2)
In order to describe cohomological
SPT phases, we assume:
Physical interpretation:
If 𝐽 cannot be decomposed, then the
tensor-network would be topologically ordered.
global matrix 𝐼𝐺𝐺 decomposition:
𝐽= 𝑝 𝜆 𝑝 𝐽 𝜆 𝐽 ≠
It turns out that 𝐽

42. SPT phases: 𝐻 𝑑+1 𝑆𝐺,𝑈 1 (d=2)
In order to describe cohomological
SPT phases, we assume:
Physical interpretation:
If 𝐽 can be decomposed, then:
global matrix 𝐼𝐺𝐺 decomposition:
𝐽= 𝑝 𝜆 𝑝 𝐽 𝜆 𝐽 =
It turns out that 𝐽
i.e. the would-be topological order is confined
by the 𝐽-string condensation.

43. SPT phases: 𝐻 𝑑+1 𝑆𝐺,𝑈 1 (d=2)
In order to describe cohomological
SPT phases, we assume:
This decomposition has an
overall U(1) phase ambiguity:
global matrix 𝐼𝐺𝐺 decomposition:
𝐽= 𝑝 𝜆 𝑝 = 𝑝 𝜒∙𝜆 𝑝 𝜒
𝜒 ∗ 𝜒
𝜒 ∗ 𝜆 𝜒
𝜒 ∗ 𝜒
𝜒 ∗
It turns out that

44. SPT phases: 𝐻 𝑑+1 𝑆𝐺,𝑈 1 (d=2)
In order to describe cohomological
SPT phases, we assume:
This decomposition has an
overall U(1) phase ambiguity:
global matrix 𝐼𝐺𝐺 decomposition:
𝐽= 𝑝 𝜆 𝑝 = 𝑝 𝜒∙𝜆 𝑝 𝜒
𝜒 ∗ 𝜒
𝜒 ∗ 𝜆 𝜒
𝜒 ∗ 𝜒
𝜒 ∗
It turns out that
It turns out that the 𝜒 ambiguity in this decomposition directly leads to
topological invariants of tensor symmetry transformation rules: 3-cocycles

45. SPT phases: 𝐻 𝑑+1 𝑆𝐺,𝑈 1 (d=2)
𝑊 𝑔 1 𝑔 1 𝑊 𝑔 2 𝑔 2 =𝜂 𝑔 1 , 𝑔 2 𝑊 𝑔 1 𝑔 2 𝑔 1 𝑔 2
𝜂 𝑔 1 , 𝑔 2 𝜂 𝑔 1 𝑔 2 , 𝑔 3 = 𝑊 𝑔 1 𝑔 1 𝜂 𝑔 2 , 𝑔 3 𝜂 ( 𝑔 1 , 𝑔 2 𝑔 3 )

46. SPT phases: 𝐻 𝑑+1 𝑆𝐺,𝑈 1 (d=2)
𝑊 𝑔 1 𝑔 1 𝑊 𝑔 2 𝑔 2 =𝜂 𝑔 1 , 𝑔 2 𝑊 𝑔 1 𝑔 2 𝑔 1 𝑔 2
𝜂 𝑔 1 , 𝑔 2 𝜂 𝑔 1 𝑔 2 , 𝑔 3 = 𝑊 𝑔 1 𝑔 1 𝜂 𝑔 2 , 𝑔 3 𝜂 ( 𝑔 1 , 𝑔 2 𝑔 3 )
“Global” 𝐼𝐺𝐺 𝜂 𝑔 1 , 𝑔 2 can be decomposed to plaquette 𝐼𝐺𝐺:
𝜂= 𝑝 𝜆 𝑝

47. SPT phases: 𝐻 𝑑+1 𝑆𝐺,𝑈 1 (d=2)
𝑊 𝑔 1 𝑔 1 𝑊 𝑔 2 𝑔 2 =𝜂 𝑔 1 , 𝑔 2 𝑊 𝑔 1 𝑔 2 𝑔 1 𝑔 2
𝜂 𝑔 1 , 𝑔 2 𝜂 𝑔 1 𝑔 2 , 𝑔 3 = 𝑊 𝑔 1 𝑔 1 𝜂 𝑔 2 , 𝑔 3 𝜂 ( 𝑔 1 , 𝑔 2 𝑔 3 )
“Global” 𝐼𝐺𝐺 𝜂 𝑔 1 , 𝑔 2 can be decomposed to plaquette 𝐼𝐺𝐺:
𝜂= 𝑝 𝜆 𝑝
𝜆 𝑝 𝑔 1 , 𝑔 2 𝜆 𝑝 𝑔 1 𝑔 2 , 𝑔 3 =
𝜒 𝑔 1 , 𝑔 2 , 𝑔 𝑊 𝑔 1 𝑔 1 𝜆 𝑝 𝑔 2 , 𝑔 3 𝜆 𝑝 ( 𝑔 1 , 𝑔 2 𝑔 3 )
3-cocycle

48. SPT phases: 𝐻 𝑑+1 𝑆𝐺,𝑈 1 (d=2)
𝑊 𝑔 1 𝑔 1 𝑊 𝑔 2 𝑔 2 =𝜂 𝑔 1 , 𝑔 2 𝑊 𝑔 1 𝑔 2 𝑔 1 𝑔 2
𝜂 𝑔 1 , 𝑔 2 𝜂 𝑔 1 𝑔 2 , 𝑔 3 = 𝑊 𝑔 1 𝑔 1 𝜂 𝑔 2 , 𝑔 3 𝜂 ( 𝑔 1 , 𝑔 2 𝑔 3 )
“Global” 𝐼𝐺𝐺 𝜂 𝑔 1 , 𝑔 2 can be decomposed to plaquette 𝐼𝐺𝐺:
𝜂= 𝑝 𝜆 𝑝
𝜆 𝑝 𝑔 1 , 𝑔 2 𝜆 𝑝 𝑔 1 𝑔 2 , 𝑔 3 =
𝜒 𝑔 1 , 𝑔 2 , 𝑔 𝑊 𝑔 1 𝑔 1 𝜆 𝑝 𝑔 2 , 𝑔 3 𝜆 𝑝 ( 𝑔 1 , 𝑔 2 𝑔 3 ) P
3-cocycle 𝜒
𝜒 ∗ 𝜒
𝜒 ∗
Just like time-reversal,
mirror reflections send χ→ 𝜒 −1 (anti-unitary),
while rotations/translations send χ→𝜒 (unitary) 𝜒
𝜒 ∗ 𝜒
𝜒 ∗

49. SPT phases: 𝐻 𝑑+1 𝑆𝐺,𝑈 1 (d=2)
Examples:
𝐻 3 𝑍 2 ,𝑈 1 = 𝑍 2
Constructing the nontrivial SPT with D=4 PEPS (square lattice)
Constrained tensor sub-Hilbert space 15 variational parameters
𝐻 3 (𝑍 2 𝑇 × 𝑍 2 𝑃 ,𝑈(1))= 𝑍 2 × 𝑍 2
Constructing three nontrivial SPT with D=6 PEPS (square lattice)
Constrained sub-Hilbert spaces 79/79/87 variational parameters

50. SPT phases: 𝐻 𝑑+1 𝑆𝐺,𝑈 1 (d=3)
Similar construction can be generalized to 3D. (need cubic IGG& plaquette IGG).
After complicated algebra, one can show that the topological invariants of tensor symmetry transformation rules are given by four cocycles.
And time-reversal and mirror reflections should be treated as anti-unitary.

51. Plan Spin liquids on the kagome lattice SPT phases: 𝐻 𝑑+1 (𝑆𝐺,𝑈 1 )
Anyon condensation mechanism:
“conventional” SET phases  SPT phases

52. General criteria for anyon condensation
Condense certain fluxes
an SET phase
an SPT phase
Gauge group: 𝑍 𝑁 1 × 𝑍 𝑁 2 ×… & symmetry group: 𝑆𝐺
e-particles feature nontrivial symmetry fractionalization
m-particles have trivial fractionalization, but can carry usual quantum numbers
Ω 𝑔 1 ⋅ Ω 𝑔 2 =𝝀 𝒈 𝟏 , 𝒈 𝟐 ⋅ Ω 𝑔 1 𝑔 Ω 𝑔 ~ symmetry defect, 𝜆 ~ certain 𝑚-particle
(Barkeshli, Bonderson, Cheng, Wang, Hermele, Chen,Fidkowski…)

53. General criteria for anyon condensation
Condense certain fluxes
an SET phase
an SPT phase
Gauge group: 𝑍 𝑁 1 × 𝑍 𝑁 2 ×… & symmetry group: 𝑆𝐺
e-particles feature nontrivial symmetry fractionalization
m-particles have trivial fractionalization, but can carry usual quantum numbers
Ω 𝑔 1 ⋅ Ω 𝑔 2 =𝝀 𝒈 𝟏 , 𝒈 𝟐 ⋅ Ω 𝑔 1 𝑔 Ω 𝑔 ~ symmetry defect, 𝜆 ~ certain 𝑚 particle
Condensing m-particles without breaking symmetry, which requires:
Condensed 𝑚’s carry 1D symmetry irrep: 𝝌 𝒎 𝒈
𝜒 𝑚 (𝑔)⋅ 𝜒 𝑚 ′ (𝑔)= 𝜒 𝑚 𝑚 ′ (𝑔)
After condensing those 𝑚’s, we get an SPT phase
𝜔 𝑔 1 , 𝑔 2 , 𝑔 3 ≡ 𝜒 𝜆 𝑔 2 , 𝑔 𝑔 1 , [𝜔]∈ 𝐻 3 (𝑆𝐺,𝑈(1))

54. Possible model realizations?
Quantum dimer models on non-bipartite lattice can host 𝑍 2 toric code topological order. (Rokhsar,Kivelson,Sondhi,Moessner)
These models can be mapped to hard-core boson models (or XXZ models), with a U(1) symmetry, and spinons carry half-charge. (Balents, Fisher, Girvin, Isakov, Kim…)
Tuning parameters, these models (e.g.: kagome 1/3-filled hard-core boson model) can go from the 𝑍 2 spin liquid into Valence Bond Solid phases (VBS) via vison condensation. (Pollmann et.al. )
One could add interactions breaking the U(1) down to Ising. If the the condensed vison is Ising odd, then, the resulting VBS phase is a nontrivial Ising-SPT phase. (SPT-VBS)
(In tensor-based algorithms, the quantum number carried by low energy vison near condensation can be measured.)

55. Summary SET SPT SPT partially classified by 𝐻 𝑑+1 [𝑆𝐺,𝑈 1 ]
32 gapped 𝑍 2 spin liquid
Nearly degenerate energy densities for two classes
SPT partially classified by 𝐻 𝑑+1 [𝑆𝐺,𝑈 1 ]
𝑆𝐺: on-site & spatial symmetries
𝑇 and 𝑃 antiunitary
anyon condensation
SET
SPT

56. Discussion/future directions
Previously in 2D PEPS, MPO-invariance was used to characterize onsite SPT in PEPS(Williamson et.al.), connection with our formulation?
Classifying and simulating fermion phases?
Combining state of the art numerical techniques with this analytical construction (Vanderstraeten, Verstraete, Corboz…)
More accurate energy density, correlators, …
Excitation spectrum?
Possible realization of SPT?
Numerical simulation for SPT tensor wavefunctions
Condensing visons carrying nontrivial quantum number in spin liquid phases  SPT-VBS phase?
Thank You!