Construction and classification of symmetry protected topological phases in interacting fermion systems

TL;DR

This paper develops a comprehensive framework for classifying symmetry-protected topological phases in interacting fermion systems by building fixed-point wave functions via a layered decoration scheme and equivalence classes of fermionic SLU (FSLU) transformations. Central to the approach is the use of domain-wall decoration with Kitaev chains, complex fermions, and, in higher dimensions, p+ip superconductors, organized by the symmetry data encoded in ω_2 and s_1 that define the central extension G_f=Z_2^f×_{ω_2}G_b. Consistency is enforced through a hierarchy of twisted cocycle equations (d n_d = O_{d+1}) and obstruction functions O_{d+1}, with trivialization subgroups Γ^i capturing states realizable as boundary ASPT states in one lower dimension. The framework reproduces known classifications from cobordism theory, yields commuting-projector Hamiltonians for fixed-point FSPT states, and exposes how boundary theories encode obstructions and trivializations, including in 2D and 3D where Kitaev chains and p+ip layers interplay. The results offer a path to generalizations to spatial and Lie group symmetries and highlight the deep ties between SPT physics, cobordism, and fermionic algebraic structures.

Abstract

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 $G_f=\mathbb{Z}_2^f \times_{ω_2} G_b$ which is a central extension of bosonic symmetry group $G_b$ (may contain time reversal symmetry) by the fermion parity symmetry group $\mathbb{Z}_2^f = \{1,P_f\}$. Our construction is based on the concept of equivalence class of finite depth fermionic symmetric local unitary (FSLU) transformations and decorating symmetry domain wall picture, subjected to certain obstructions. We will also discuss the systematical construction and classification of boundary anomalous SPT (ASPT) 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/space group symmetry as well as continuum Lie group symmetry.

Figures (18)

• Figure 1: The self consistent equation on a big patch. Mathematically it is known as the 3-cocycle equation.
• Figure 2: Kasteleyn orientations of the resolved dual lattice. For a given triangulation of the 2D spacial spin manifold (shown by black links), we can construct a resolved dual lattice (shown by red links). The Majorana fermion pairings should respect the red link arrows in the figures.
• Figure 3: Standard super pentagon equation. The dual trivalent graph of the triangulation is the usual string diagram pentagon equation for tensor category. Algebraically, this standard super pentagon condition corresponds to Eq. (\ref{['2D:dF']}). Since the group element label of the first vertex is $e\in G_b$, all the fermionic $F$ moves are standard except $\,\!^{\bar{0}1}F_{1234}$. Note that we used a simpler notation $F_{ijkl}=F(e,\bar{i}j,\bar{j}k,\bar{k}l)$ in the figure. Blue arrows indicate that the Majorana fermion pairing directions may be changed compared to the red arrow Kasteleyn orientations.
• Figure 4: FSLU transforms a 2D "FSPT" state to a fermion product state. There are two FSLU transformations $(a)\xrightarrow{U_1}(b)\xrightarrow{U_2}(c)$ to trivialize the initial 2D FSPT state. (a) Majorana fermions are in vacuum pairs (green arrows). (b) There is exactly one Kitaev's Majorana chain inside the triangle (red links). (c) There is one Kitaev's Majorana chain around each vertex (inside the gray arc). And we can shrink it to the vertex and redefine the basis state $|g_i\rangle$ for the vertex.
• Figure 5: FSLU transformations for "FSPT" state on 2D surface with boundary. The degrees of freedom inside the gray circle in (c) are combined to be the new basis state $|g_\ast\rangle'$. There is a 1D ASPT state along the boundary of the 2D bulk shown in (c).