The double-semion model[132, 84, 157] is another topologically ordered state with two semions of statistics θ = ±π/2. In the notation of string-net states, the double-semion ground state can also be written as superposition of all closed string loops:
|ψDSi = X
X
where X represents a closed loop, and Nloops the number of loops. The above double-semion state can be described by a TNS with the following tensors T and Gm at g = 1 (see Fig.
3.5.2): T(αα0)(ββ0)(γγ0)(δδ0)= tαβγ0δ0δαβ0δβγδγ0δδδ0α0, t1000 = t1101 = −1, other tαβγ0δ0 = 1; Gm(αα0)(ββ0)= gααm0δαβδα0β0, g110= g011 = g, g000 = g011= 1, g100= g001 = g010= g010 = 0. (3.11)
Note that if we view α = β0, β = γ, γ0 = δ, and δ0 = α0 as indices that label “virtual qubits” in the squares, then the strings can be viewed as domain wall between the ”0” and ”1” states of the qubits. Also if we choose tαβγ0δ0 = 1, the above tensors will describe the
Z2 topologically ordered state discussed previously.
The Z2 gauge symmetry is generated by σx⊗ σx acting on each internal indices (αα0) followed by a transformation generated by ui
αα0, i = t, l, b, r acting on the links of the four orientations. Here ui αα0 must satisfy fαβγ0δ0 = ut βγ0ubαδ0ulβαurγ0δ0 (3.12) where f1000 = f0111 = f0010 = f1101 = −1, others fαβγ0δ0 = 1. (3.13) Furthermore ui
αα0 must also satisfy
gmαα0 = (utαα0)∗gmαα0(ubαα0)∗ gmαα0 = (ulαα0)∗gmαα0(urαα0)∗. (3.14) We find that ut = ub = 1 −1 1 1 , ur = ul= 1 1 −1 1 , (3.15)
See [120] for a general analysis of twisted gauge structures.
After the GSPTRG calculation, we find a phase transition at gc = 0.802. The S- and T -matrices for the nontrivial phase with g ∈ (0.802, 1] are given by Fig. 3.5.1, which agree with the modular matrices for the double semion model in string basis [158]. For the trivial phase near g = 0, the modular matrices become Tαβ = Sαβ = δα,0δβ,0.
3.7
Conclusion
We have developed a systematic approach, gauge-symmetry preserved tensor renormaliza- tion, to calculate modular matrices from a generic many-body wave function described by a tensor network. The modular matrices can be viewed as very robust ”topological order parameters” that only change at phase transitions. The tensor network approach gives rise to S and T matrices in a particular basis which is different from the standard quasiparti- cal basis.[124, 101, 103, 102, 106, 45, 108, 107, 55] The trivial phase will result in trivial modular matrices S = 1 and T = 1 (since there is no degeneracy on a torus), and the topological phase will give rise to nontrivial modular matrices, which contain topological informations, such as quasi-particles information, like statistic angle, fusion rule, quantum dimension, etc.
In particular, a general algorithm can be developed: the tensor network ansatz can be imposed with gauge symmetry G (or MPO symmetry, see below) in the beginning, and the corresponding update algorithm, which is used to find ground states, also preserves such a gauge symmetry. Therefore if the topological phase indeed has such a gauge theory description, the ansatz obviously is better than the normal tensor network ansatz. In Appendix B we perform such a benchmark computation using the Z2 phase of the Kitaev honeycomb model [56]. There we prepare an arbitrary tensor with Z2 symmetry, find the ground state (locally) numerically by gauge-symmetry preserved update and from there compute the modular matrices. A similar tensor network computation of Kitaev honeycomb model is developed in Ref. [159] where Z2 gauge structure is also imposed but expressed by Grassmann tensor network. The energy and nearest neighbor correlation are computed there.
After the completion and publication of the preprint of this chapter, the notion of (twisted) G-injectivity of [119,120] was generalized to the matrix product operator (MPO) case in [160] and it was shown that any string-net model is included with this generalization. The method developed in this chapter can thus similarly be generalized to any MPO symmetry and does not need any group structure (and thus not restricted to twisted discrete gauge theories).
The universal wave function overlap [1] (3.1) applies to any dimension and have already been investigated in exactly solvable models in 3+1D [161, 3, 162]. The method outlined in this chapter can similarly be generalized to higher dimensions to extract universal topo- logical information from generic gapped ground states.
Finally we note that although the universal wave function overlap [1] works for any topological order, the machinery developed in this chapter in 2+1D only works for non-
chiral topological order (gapped boundaries) as formulated here. This is only because the tensor network techniques used are best understood for non-chiral topological order, but a generalization for chiral topological order would be both interesting and important.
After the publication of the work presented in the previous and current chapter, lots of interesting work has been generated using these ideas. Some related to numerically detect- ing topological order and others related to 3+1D topological systems (see next chapter). In particular, I would like to mention [163]. There the ideas of this chapter was developed to study a much more realistic system that has been studied theoretically and experimentally for many years, namely the Heisenberg model on the Kagome lattice. It was shown that the modular matrices computed in this fashion, are a very sensitive probe of a gap above the ground-state. They find that the Heisenberg model is a gapped spin liquid with Z2 topological order and with a correlation length ξ ∼ 10 unit cells. This long correlation length explains the gapless behavior observed in many simulations on smaller systems.
Appendices
3.A
Appendix A: Robustness of modular matrices un-
der Z
2perturbations
In the phase diagram Fig. 3.5.1, it already demonstrates that T - and S-matrices are very robust characterization of topological order, which only depend on the phase. In order to address on this issue more explicitly, we will perturb Z2 topological state at g = 1, while the perturbation also respects internal Z2 gauge symmetry, i.e., the perturbation tensor T0 is written as:
Tαβγδ0α0β0γ0δ0 = r when α + β + γ + δ even (3.16)
where r is a uniform distributed random number ranging from [−1, 1] depending on α0, β0, γ0, δ0, α, β, γ, δ; and represents perturbation strength starting from zero. The initial tensor before RG will be T + T0.
As already shown in Ref. [24], Z2 topological phase is robust under tensor perturba- tions which respect the Z2 gauge symmetry, while fragile under perturbations breaking the Z2 gauge symmetry. Here we start from perturbed tensor T + T0 and calculate mod- ular matrices for different ’s, which will demonstrate the robustness of this topological characterization3.A.1.
Numerically it demonstrates that when 0 ≤ ≤ 0.35, T - and S-matrices are always eqn. 3.10. However, when > 0.35, the perturbations will possibly break the topological phase (and possibly not). In this case, T - and S-matrices have three possibilities as shown in the figure. Anyway, this calculation clearly demonstrates modular matrices are robust characterization of topological phase.
3.B
Appendix B: Gauge-Symmetry Preserved Update
For a typical tensor network algorithm, there are two main steps: updating local tensors to lower the energy to ground state energy and contracting all local tensors to compute physical quantities and norms. Here we only point out some details in gauge-symmetry preserved update algorithm, since the details in contraction have already been reviewed in the main text to some extent.
𝜖 𝜖 = 0.35
𝑍2Topological Phase
with Modular Matrices
Figure 3.A.1: Phase diagram under perturbation
We choose Kitaev honeycomb model as a benchmark. Kitaev honeycomb model is defined on the honeycomb lattice with spins on each site and different interactions along the three different links connected to each site
H = −Jx X x−links σxiσjx− Jy X y−links σiyσjy− Jz X z−links σizσjz. (B.1) Jγ are coupling constants along the γ−link. For simplicity we will assume they are all positive. For the coupling constants Jγ satisfying Jx+ Jy < Jz (or other permutations), a gapped phase will be acquired that indeed is a toric code phase by perturbation analysis [56].
We impose Z2 gauge symmetry on our tensor network ansatz. I.e., local tensors should satisfy
Tijkm = 0, if i + j + k odd. (B.2)
Other elements of tensors are random in the initial states before simple update. Gauge- symmetry preserved update differs from simple update only when we do SVD. Again, what we need to do in SVD approach is the following three steps: block diagonalization according to gauge symmetry, SVD in each block and rearrange the outcoming tensors back to the original form. Note that the gauge symmetry only acts on internal indices, so that block diagonalization only happens for internal indices. The procedure is also summarized in the figureB.1.
𝑇
1𝑇
2𝑒
−𝐻𝛾𝛿𝑡 Block Diagonalizaton𝐵
𝑜𝑜𝐵
𝑒𝑒𝐵
𝑒𝑒SVD
𝐵
𝑜𝑜 Recombine𝑇
1𝑇
2 (a) (b) (c) (d)Figure B.1: Illustration of gauge-symmetry preserved simple update. (a) shows that tenor T1 and T2 are contracted and acted with local imaginary evolution operator represented by the blue box. The legs with arrow are physical indices while legs without arrows are internal indices. (b) Block diagonalization according to internal indices. Bee and Boo represent the matrices with both legs even and odd. (c) Bee and Boo are SVD-ed. (d) The outcoming matrices are recombined into the original form as in figure (a).
We randomly pick up a few points in the gapped phase of Kitaev honeycomb model, use gauge-symmetry preserved update to obtain the ground states by Z2 symmetric ansatz
B.2. Modular matrices are calculated by the method explained in the main text, and the result is exactly the same matrices found in the main text:
S = 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 , T = 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 . (B.3)
Chapter 4
Universal Topological Data for
Gapped Quantum Liquids in Three
Dimensions and Fusion Algebra for
Non-Abelian String Excitations
This chapter was published in [3]
4.1
Introduction
For more than two decades exotic quantum states[16, 17, 18, 97, 98, 110, 111, 127, 128,
129, 130, 131] have attracted a lot attention from the condensed matter community. In particular gapped systems with non-trivial topological order,[99, 100, 124] which is a re- flection of long-range entanglement[24] of the ground state, have been studied intensely in 2 + 1 dimensions. Recently, people started to work on a general theory of topological order in higher than 2 + 1 dimensions.[84, 164, 1,165,166]
In a recent work Ref. [1], we conjectured that for a gapped system on a d-dimensional manifold M of volume V with the set of degenerate ground states {|ψαi}Nα=1 on M, we have the following overlaps
where ˆOA are transformations on the wave functions induced by the automorphisms A : M → M, α is a non-universal constant and MAis a universal matrix up to an overall U (1) phase. Here MA form a projective representation of the automorphism group AMG(M), which is robust against any local perturbations that do not close the bulk gap.[124, 101] In Ref. [1] we conjectured that such projective representations for different space manifold topologies fully characterize topological orders with finite ground state degeneracy in any dimension. Furthermore, we conjectured that projective representations of the mapping class groups MCG(M) = π0[AMG(M)] classify topological order with gapped boundaries.[124,
101] These quantities can be used as order parameters for topological order and detect transitions between different phases [2].
In this chapter we will study these universal quantities further in 3-dimensions for one of the most simple manifolds, the 3-torus M = T3. The mapping class group of the 3-torus is MCG(T3) = SL(3, Z). This group is generated by two elements of the form [123]
ˆ˜ S = 0 1 00 0 1 1 0 0 , T =ˆ˜ 1 0 01 1 0 0 0 1 . (4.2)
These matrices act on the unit vectors by ˆ˜S : ( ˆx, ˆy, ˆz) 7→ ( ˆz, ˆx, ˆy) and similarly ˆ˜T : ( ˆx, ˆy, ˆz) 7→ ( ˆx+ ˆy, ˆy, ˆz). Thus ˜S corresponds to a rotation, while ˜T is shear transformation in the xy-plane.
In this chapter, we will study the SL(3, Z) representations generated by a very sim- ple class of ZN models in detail and then consider models for any finite group G, which are 3-dimensional versions of Kitaevs quantum double models [69]. One can also gener- alize into twisted versions of these based on the group cohomology H4(G, U (1)) by direct generalization of Ref. [167] into 3+1D, which has been done for some simple groups in Ref. [166,168].
We will consider dimensional reduction of a 3D topological order C3D to 2D by making one direction of the 3D space into a small circle. In this limit, the 3D topologically ordered states C3D can be viewed as several 2D topological orders C2D
i , i = 1, 2, · · · which happen to have degenerate ground state energy. We denote such a dimensional reduction process as
C3D=M i
Ci2D. (4.3)
We can compute such a dimensional reduction using the representation of SL(3, Z) that we have calculated.
We consider SL(2, Z) ⊂ SL(3, Z) subgroup and the reduction of the SL(3, Z) represen- tation R3D to the SL(2, Z) representations R2D
i : R3D=M
i
R2Di . (4.4)
We will refer to this as branching rules for the SL(2, Z) subgroup. The SL(3, Z) represen- tation R3D describes the 3D topological order C3D and the SL(2, Z) representations R2Di describe the 2D topological orders C2D
i . The decomposition (4.4) gives us the dimensional reduction (4.3).
Let us use CG to denote the topological order described by the gauge theory with the finite gauge group G. Using the above result, we find that
CG3D = |G| M n=1
CG2D (4.5)
for Abelian G where |G| is the number of the group elements. For non-Abelian group G CG3D=M
C
CG2DC (4.6)
where LC sums over all different conjugacy classes C of G, and GC is a subgroup of G which commutes with an element in C. The results for G = ZN were mentioned in the paper [1] (chapter 2 of this thesis).
We also found that the reduction of SL(3, Z) representation, eqn. (4.4), encodes all the information about the three-string statistics discussed in Ref. [165] for Abelian groups. For non-Abelian groups, we will have a “non-Abelian” string braiding statistics and a non- trivial string fusion algebra. We also have a “non-Abelian” three-string braiding statistics and a non-trivial three-string fusion algebra. Within the dimension reduction picture, the 3D strings reduces to particles in 2D, and the (non-Abelian) statistics of the particles encode the (non-Abelian) statistics of the strings.