In this paper, we present an exactly solvable model for two dimensional topological superconductor with helical Majorana edge modes protected by time reversal symmetry. Our construction is based on the idea of decorated domain walls and makes use of the Kasteleyn orientation on a two dimensional lattice, which was used for the construction of the symmetry protected fermion phase with Z2 symmetry in Ref. 1 and 2. By decorating the time reversal domain walls with spinful Majorana chains, we are able to construct a commuting projector Hamiltonian with zero correlation length ground state wave function that realizes a strongly interacting version of the two dimensional topological superconductor. From our construction, it can be seen that the T 2 = −1 transformation rule for the fermions is crucial for the existence of such a nontrivial phase; with T 2 = 1, our construction does not work.
While two-dimensional symmetry-enriched topological phases (SETs) have been studied intensively and systematically, three-dimensional ones are still open issues. We propose an algorithmic approach of imposing global symmetry Gs on gauge theories (denoted by GT) with gauge group Gg. The resulting symmetric gauge theories are dubbed "symmetry-enriched gauge theories" (SEG), which may be served as low-energy effective theories of three-dimensional symmetric topological quantum spin liquids. We focus on SEGs with gauge group Gg = ZN 1 × ZN 2 × · · · and on-site unitary symmetry group Gs = ZK 1 × ZK 2 × · · · or Gs = U(1) × ZK 1 × · · · . Each SEG(Gg, Gs) is described in the path integral formalism associated with certain symmetry assignment. From the path-integral expression, we propose how to physically diagnose the ground state properties (i.e., SET orders) of SEGs in experiments of charge-loop braidings (patterns of symmetry fractionalization) and the mixed multi-loop braidings among deconfined loop excitations and confined symmetry fluxes. From these symmetry-enriched properties, one can obtain the map from SEGs to SETs. By giving full dynamics to background gauge fields, SEGs may be eventually promoted to a set of new gauge theories (denoted by GT * ). Based on their gauge groups, GT * s may be further regrouped into different classes each of which is labeled by a gauge group G * g . Finally, a web of gauge theories involving GT, SEG, SET and GT * is achieved. We demonstrate the above symmetry-enrichment physics and the web of gauge theories through many concrete examples.
We classify symmetry fractionalization and anomalies in a (3+1)d U(1) gauge theory enriched by a global symmetry group G. We find that, in general, a symmetry-enrichment pattern is specified by four pieces of data: ρ, a map from G to the duality symmetry group of this U(1) gauge theory which physically encodes how the symmetry permutes the fractional excitations, ν ∈ H 2 ρ [G, U T (1)], the symmetry actions on the electric charge, p ∈ H 1 [G, Z T ], indication of certain domain wall decoration with bosonic integer quantum Hall (BIQH) states, and a torsor n over H 3 ρ [G, Z], the symmetry actions on the magnetic monopole. However, certain choices of (ρ, ν, p, n) are not physically realizable, i.e., they are anomalous. We find that there are two levels of anomalies. The first level of anomalies obstruct the fractional excitations being deconfined, thus are referred to as the deconfinement anomaly. States with these anomalies can be realized on the boundary of a (4+1)d long-range entangled state. If a state does not suffer from a deconfinement anomaly, there can still be the second level of anomaly, the more familiar 't Hooft anomaly, which forbids certain types of symmetry fractionalization patterns to be implemented in an on-site fashion. States with these anomalies can be realized on the boundary of a (4+1)d short-range entangled state. We apply these results to some interesting physical examples.
Strong interactions can give rise to new fermionic symmetry protected topological phases which have no analogs in free fermion systems. As an example, we have systematically studied a spinless fermion model with U (1) charge conservation and time reversal symmetry on a three-leg ladder using density-matrix renormalization group. In the non-interacting limit, there are no topological phases. Turning on interactions, we found two gapped phases. One is trivial and is adiabatically connected to a band insulator, while another one is a nontrivial symmetry protected topological phase resulting from strong interactions.Introduction. Gapped quantum states can be classified by their quantum entanglement 1 . The long-range entangled states carry intrinsic topological orders 2-4 , while the short-range entangled states are trivial and can be adiabatically connected to direct product states (or slater determinant states for fermionic systems). If the system has some symmetries, there will be more phases.For instance, short-range entangled states without symmetry breaking can belong to di↵erent phases. Besides the trivial symmetric phases, there may exist symmetry protected topological (SPT) phases 5,6 which have nontrivial edge excitations. A typical example of bosonic interacting SPT phase is the S = 1 Haldane phase 7,8 , which is protected by spin rotational symmetry, or time reversal symmetry, or spatial inversion symmetry. Bosonic SPT phases with symmetry G in d-dimension can be classified by the (d + 1)th group cohomology H d+1 (G, U (1)) 9 . SPT phases also exist in free fermion systems. Topological insulators 10-14 are well known SPT phases protected by U (1) charge conservation and time reversal symmetry. Free fermion systems with di↵erent symmetry groups in di↵erent dimensions can be classified using homotopy theory, with 10 di↵erent classes of topological phases [15][16][17][18][19] . SPT phases may also exist in the presence of strong fermion-fermion interactions. However, the classification of the interacting fermionic SPT phases are usually di↵erent from those of free fermions. In 1D, interaction fermionic SPT phases can be classified by projective representations of the symmetry group 20,21 . In higher dimensions, the classification is more di cult and is partially described by the super-cohomology of the symmetry group 22 . Some examples of 2D interacting fermionic SPT phases are studied recently [23][24][25] . An interesting question is what is the relation between the classification of SPT phases for the interacting and non-interaction systems. For bosonic systems (including spin systems), there are no nontrivial SPT phases without interaction. So all nontrivial Bosonic SPT phases are induced from interactions. In contrast, situations are quite di↵erent for fermionic systems. Naively speaking, strong interactions will reduce the number of SPT
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