Symmetry-protected topological orders for interacting fermions: Fermionic topological nonlinearσmodels and a special group supercohomology theory

Abstract

Symmetry-protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry G, which can all be smoothly connected to the trivial product states if we break the symmetry. It has been shown that a large class of interacting bosonic SPT phases can be systematically described by group cohomology theory. In this paper, we introduce a (special) group supercohomology theory which is a generalization of the standard group cohomology theory. We show that a large class of short-range interacting fermionic SPT phases can be described by the group supercohomology theory. Using the data of supercocycles, we can obtain the ideal ground state wave function for the corresponding fermionic SPT phase. We can also obtain the bulk Hamiltonian that realizes the SPT phase, as well as the anomalous (i.e., non-onsite) symmetry for the boundary effective Hamiltonian. The anomalous symmetry on the boundary implies that the symmetric boundary must be gapless for (1+1)-dimensional [(1+1)D] boundary, and must be gapless or topologically ordered beyond (1+1)D. As an application of this general result, we construct a new SPT phase in three dimensions, for interacting fermionic superconductors with coplanar spin order (which have T 2 = 1 time-reversal Z T 2 and fermion-number-parity Z f 2 symmetries described by a full symmetry group Z T 2 × Z f 2 ). Such a fermionic SPT state can neither be realized by free fermions nor by interacting bosons (formed by fermion pairs), and thus are not included in the K-theory classification for free fermions or group cohomology description for interacting bosons. We also construct three interacting fermionic SPT phases in two dimensions (2D) with a full symmetry group Z 2 × Z f 2 . Those 2D fermionic SPT phases all have central-charge c = 1 gapless edge excitations, if the symmetry is not broken.

Classification of Interacting Topological Floquet Phases in One Dimension

“…In the absence of interactions, periodic driving raises the new possibility of obtaining topologically protected edge modes with quasienergy π, [5,6] in addition to those with zero quasienergy that are familiar from nondriven equilibrium systems [31]. As for the equilibrium SPTs, we find that interactions generally tend to reduce the set of nontrivial phases when the noninteracting classification contains integer topological invariants [32][33][34][35][36][37][38]. In the Floquet context, this reduction arises from a nontrivial interplay of the zero-and π-quasienergy modes.…”

Loop Braiding Statistics and Interacting Fermionic Symmetry-Protected Topological Phases in Three Dimensions

“…A systematic construction of fermionic SPT phases has been proposed in Ref. [18], although no explicit examples in 3D for unitary symmetries (except the bosonic ones) were given. In addition, it is not clear what kind of physical properties characterize the constructed FSPT states in Ref.…”

Axion topological field theory of topological superconductors

“…The effective action describes both topological trivial and nontrivial superconductors. Although not explicit in the effective action, the topological invariant N is determined by the Chern numbers C 1i and the ground state value of θ i , which is thus determined for a given effective action (24).…”

“…However, in reality all electron systems are interacting, so that the physically interesting TSM are those which are robust even with interaction. In general it is difficult to directly study the topological classification of gapped interacting Hamiltonians or their ground states, except in some special cases such as in one dimension [16][17][18][19] , and in some special classes of models in higher dimensions [20][21][22][23][24] . A general approach to characterize interacting TSM's is by writing down the possible topological response theories, which describe some observable physical properties of the system, and contain a"topological order parameter" that is required to be quantized by general principles.…”