The classification and construction of symmetry-protected topological (SPT) phases in interacting boson and fermion systems have become a fascinating theoretical direction in recent years. It has been shown that (generalized) group cohomology theory or cobordism theory gives rise to a complete classification of SPT phases in interacting boson or spin systems. The construction and classification of SPT phases in interacting fermion systems are much more complicated, especially in three dimensions. In this work, we revisit this problem based on an equivalence class of fermionic symmetric local unitary transformations. We construct very general fixed-point SPT wave functions for interacting fermion systems. We naturally reproduce the partial classifications given by special group supercohomology theory, and we show that with an additional \tilde{B}H^2(G_b,\mathbb{Z}_2) structure [the so-called obstruction-free subgroup of H^2(G_b,\mathbb{Z}_2)], a complete classification of SPT phases for three-dimensional interacting fermion systems with a total symmetry group G_f = G_b × \mathbb{Z}^f_2 can be obtained for unitary symmetry group G_b. We also discuss the procedure for deriving a general group supercohomology theory in arbitrary dimensions.

The classification and lattice model construction of symmetry-protected topological (SPT) phases in interacting fermion systems are very interesting but challenging. In this paper, we give a systematic fixed-point wave function construction of fermionic SPT (FSPT) states for generic fermionic symmetry group G_f = \mathbb{Z}^f_2 ×_{ω_2} G_b which is a central extension of bosonic symmetry group Gb (may contain time-reversal symmetry) by the fermion parity symmetry group \mathbb{Z}^f_2 = {1,P_f}. Our construction is based on the concept of an equivalence class of finite-depth fermionic symmetric local unitary transformations and decorating symmetry domain wall picture, subjected to certain obstructions. We also discuss the systematical construction and classification of boundary anomalous SPT states which leads to a trivialization of the corresponding bulk FSPT states. Thus, we conjecture that the obstruction-free and trivialization-free constructions naturally lead to a classification of FSPT phases. Each fixed-point wave function admits an exactly solvable commuting-projector Hamiltonian. We believe that our classification scheme can be generalized to point and space group symmetry as well as continuum Lie group symmetry.

In the past decade, tremendous efforts have been made towards understanding fermionic symmetry protected topological (FSPT) phases in interacting systems. Nevertheless, for systems with continuum symmetry, e.g., electronic insulators, it is still unclear how to construct an exactly solvable model with a finite dimensional Hilbert space in general. In this paper, we give a lattice model construction and classification for 2D interacting electronic insulators. Based on the physical picture of U(1)_f-charge decorations, we illustrate the key idea by considering the well known 2D interacting topological insulator. Then we generalize our construction to an arbitrary 2D interacting electronic insulator with symmetry Gf=U(1)_f ⋊_{ρ_1,ω_2} G, where U(1)_f is the charge conservation symmetry and ρ_1,ω_2 are additional data which fully characterize the group structure of G_f. Finally we study more examples, including the full interacting classification of 2D crystalline topological insulators.

Topological qauntum field theory(TQFT) is a very powerful theoretical tool to study topological phases and phase transitions. In 2+1D, it is well known that the Chern-Simons theory captures all the universal topological data of topological phases, e.g., quasi-particle braiding statistics, chiral central charge and even provides us a deep insight for the nature of topological phase transitions. Recently, topological phases of quantum matter are also intensively studied in 3+1D and it has been shown that loop like excitation obeys the so-called three-loop-braiding statistics. In this paper, we will try to establish a TQFT framework to understand the quantum statistics of particle and loop like excitation in 3+1D. We will focus on Abelian topological phases for simplicity, however, the general framework developed here is not limited to Abelian topological phases.

Fractional statistics is one of the most intriguing features of topological phases in 2D. In particular, the so-called non-Abelian statistics plays a crucial role towards realizing universal topological quantum computation. Recently, the study of topological phases has been extended to 3D and it has been proposed that loop-like extensive objects can also carry fractional statistics. In this work, we systematically study the so-called three-loop braiding statistics for loop-like excitations for 3D fermionic topological phases. Most surprisingly, we discovered new types of non-Abelian three-loop braiding statistics that can only be realized in fermionic systems (or equivalently bosonic systems with fermionic particles). The simplest example of such non-Abelian braiding statistics can be realized in interacting fermionic systems with a gauge group Z2×Z8 or Z4×Z4, and the physical origin of non-Abelian statistics can be viewed as attaching an open Majorana chain onto a pair of linked loops, which will naturally reduce to the well known Ising non-Abelian statistics via the standard dimension reduction scheme. Moreover, due to the correspondence between gauge theories with fermionic particles and classifying fermionic symmetry-protected topological (FSPT) phases with unitary symmetries, our study also give rise to an alternative way to classify FSPT phases with unitary symmetries. We further compare the classification results for FSPT phases with arbitrary Abelian total symmetry G^f and find systematical agreement with previous studies using other methods. We believe that the proposed framework of understanding three-loop braiding statistics (including both Abelian and non-Abelian cases) in interacting fermion systems applies for generic fermonic topological phases in 3D.