A Classification of (2+1)D Topological Phases with Symmetries

Abstract

This thesis aims at concluding the classification results for topological phases with symmetry. First, we know that “trivial” (i.e., not topological) phases with symmetry can be classified by Landau symmetry breaking theory. If the Hamiltonian of the system has symmetry group G_H , the symmetry of the ground state, however, can be spontaneously broken and thus a smaller group G. In other words, different symmetry breaking patterns are classified by G ⊂ G_H.

For topological phases, symmetry breaking is always a possibility. In this thesis, for simplicity we assume that there is no symmetry breaking; equivalently we always work with the symmetry group G of the ground states. We also restrict to the case that G is finite and on-site.

The classification of topological phases is far beyond symmetry breaking theory. There are two main exotic features in topological phases: (1) protected chiral, or non-chiral but still gapless, edge states; (2) fractional, or (even more wild) anyonic, quasiparticle excitations that can have non-integer internal degrees of freedom, fractional charges or spins and non-Abelian braiding statistics. In this thesis we achieved a full classification by studying the properties of these exotic quasiparticle excitations.

Firstly, we want to distinguish the exotic excitations with the ordinary ones. Here the criteria is whether excitations can be created or annihilated by local operators. The ordinary ones can be created by local operators, such as a spin flip in the Ising model, and will be referred to as local excitations. The exotic ones can not be created by local operators, for example a quasi-hole excitation with 1/3 charge in the ν = 1/3 Laughlin state. Local operators can only create quasi-hole/quasi-electron pairs but never a single quasi-hole. They will be referred to as topological excitations.

Secondly, we know that local excitations always carries the representations of the sym- metry group G. This constitutes the first layer of our classification, a symmetric fusion category, E = Rep(G) for boson systems or E = sRep(G^f) for fermion systems, consisting of the representations of the symmetry group and describing the local excitations with symmetry.

Thirdly, when we combine local excitations and topological excitations together, all the excitations in the phase must form a consistent anyon model. This constitutes the second layer of our classification, a unitary braided fusion category C describing all the quasiparticle excitations in the bulk. It is clear that E ⊂ C. Due to braiding non-degeneracy, the subset of excitations that have trivial mutual statistics with all excitations (namely the Müger center) must coincide with the local excitations E. Thus, C is a non-degenerate unitary braided fusion category over E, or a UMTC/E.

However, it turns out that only the information of excitations in the original phase is not enough. Most importantly, we miss the information of the protected edge states. To fix this weak point, we consider the extrinsic symmetry defects, and promote them to dynamical excitations, a.k.a., “gauge the symmetry”. We fully gauge the symmetry such that the gauged theory is a bosonic topological phase with no symmetry, described by a unitary modular tensor category M, which constitutes the third layer of our classification. It is clear that M contains all excitations in the original phase, C ⊂ M, plus additional excitations coming from symmetry defects. It is a minimal modular extension of C. M captures most information of the edge states and in particular fixes the chiral central charge of the edge states modulo 8.

We believe that the only thing missing is the E_8 state which has no bulk topological excitations but non-trivial edge states with chiral central charge c = 8. So in addition we add the central charge to complete the classification. Thus, topological phases with symmetry are classified by (E ⊂ C ⊂ M,c).

We want to emphasize that, the UBFCs E,C,M consist of large sets of data describing the excitations, and large sets of consistent conditions between these data. The data and conditions are complete and rigid in the sense that the solutions are discrete and finite at a fixed rank.

As a first application, we use a subset of data (gauge-invariant physical observables) and conditions between them to numerically search for possible topological orders and tabulate them. We also study the stacking of topological phases with symmetry based on such classification. We recovered the known classification H^3(G,U(1)) for bosonic SPT phases from a different perspective, via the stacking of modular extensions of E = Rep(G). Moreover, we predict the classification of invertible fermionic phases with symmetry, by the modular extensions of E = sRep(G^f). We also show that the UMTC/E C determines the topological phase with symmetry up to invertible ones.

A special kind of anyon condensation is used in the study of stacking operations. We then study other kinds of anyon condensations. They allow us to group topological phases into equivalence classes and simplifies the classification. More importantly, anyon condensations reveal more relations between topological phases and correspond to certain topological phase transitions.