This thesis aims at concluding the classification results for topological phases withsymmetry. First, we know that ;;trivial” (i.e., not topological) phases with symmetry canbe classified by Landau symmetry breaking theory. If the Hamiltonian of the system hassymmetry group G_H , the symmetry of the ground state, however, can be spontaneouslybroken and thus a smaller group G. In other words, different symmetry breaking patternsare classified by G ⊂ G_H.For topological phases, symmetry breaking is always a possibility. In this thesis, forsimplicity we assume that there is no symmetry breaking; equivalently we always workwith the symmetry group G of the ground states. We also restrict to the case that G isfinite and on-site.The classification of topological phases is far beyond symmetry breaking theory. Thereare two main exotic features in topological phases: (1) protected chiral, or non-chiralbut still gapless, edge states; (2) fractional, or (even more wild) anyonic, quasiparticleexcitations that can have non-integer internal degrees of freedom, fractional charges orspins and non-Abelian braiding statistics. In this thesis we achieved a full classification bystudying the properties of these exotic quasiparticle excitations.Firstly, we want to distinguish the exotic excitations with the ordinary ones. Herethe criteria is whether excitations can be created or annihilated by local operators. Theordinary 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 localoperators, for example a quasi-hole excitation with 1/3 charge in the ν = 1/3 Laughlinstate. Local operators can only create quasi-hole/quasi-electron pairs but never a singlequasi-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 fusioncategory, E = Rep(G) for boson systems or E = sRep(G^f) for fermion systems, consistingof the representations of the symmetry group and describing the local excitations withsymmetry.Thirdly, when we combine local excitations and topological excitations together, all theexcitations in the phase must form a consistent anyon model. This constitutes the secondlayer 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, thesubset of excitations that have trivial mutual statistics with all excitations (namely theMüger center) must coincide with the local excitations E. Thus, C is a non-degenerateunitary braided fusion category over E, or a UMTC/E.However, it turns out that only the information of excitations in the original phaseis 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 todynamical excitations, a.k.a., ;;gauge the symmetry”. We fully gauge the symmetry suchthat the gauged theory is a bosonic topological phase with no symmetry, described by aunitary 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 additionalexcitations coming from symmetry defects. It is a minimal modular extension of C. Mcaptures most information of the edge states and in particular fixes the chiral central chargeof the edge states modulo 8.We believe that the only thing missing is the E_8 state which has no bulk topologicalexcitations but non-trivial edge states with chiral central charge c = 8. So in additionwe add the central charge to complete the classification. Thus, topological phases withsymmetry are classified by (E ⊂ C ⊂ M,c).We want to emphasize that, the UBFCs E,C,M consist of large sets of data describingthe excitations, and large sets of consistent conditions between these data. The data andconditions are complete and rigid in the sense that the solutions are discrete and finite ata 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 andtabulate 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 froma 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 modularextensions of E = sRep(G^f). We also show that the UMTC/E C determines the topologicalphase with symmetry up to invertible ones.A special kind of anyon condensation is used in the study of stacking operations. Wethen study other kinds of anyon condensations. They allow us to group topological phasesinto equivalence classes and simplifies the classification. More importantly, anyon condensations reveal more relations between topological phases and correspond to certaintopological phase transitions.
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A Classification of (2+1)D Topological Phases with Symmetries
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