Stacking Group Structure of Fermionic Symmetry-Protected Topological Phases

TL;DR

This work introduces a robust framework to determine how interacting fermionic SPT (FSPT) phases stack across dimensions 0+1 to 2+1D using fermionic local unitary transformations and decorated domain-wall pictures. By deriving explicit stacking rules for the data that classify FSPT states in each dimension, and employing the fermionic crystalline equivalence principle, the authors compute the stacking structures for 2D wallpaper-group protected FSPT states (including combinations with onsite time-reversal) and reveal a notable Z8 Majorana phase sector in 2+1D. The approach unifies unitary and anti-unitary symmetries, clarifies how Kitaev-chain and complex-fermion decorations transform under stacking, and provides computational tools (including graphical methods) to evaluate Majorana phases. These results extend the practical reach of FSPT classifications, offer a route to higher-dimensional generalizations, and connect crystalline and internal symmetry frameworks within a coherent FSLU-based methodology.

Abstract

In the past decade, there has been a systematic investigation of symmetry-protected topological (SPT) phases in interacting fermion systems. Specifically, by utilizing the concept of equivalence classes of finite-depth fermionic symmetric local unitary (FSLU) transformations and the fluctuating decorated symmetry domain wall picture, a large class of fixed-point wave functions have been constructed for fermionic SPT (FSPT) phases. Remarkably, this construction coincides with the Atiyah-Hirzebruch spectral sequence, enabling a complete classification of FSPT phases. However, unlike bosonic SPT phases, the stacking group structure in fermion systems proves to be much more intricate. The construction of fixed-point wave functions does not explicitly provide this information. In this paper, we employ FSLU transformations to investigate the stacking group structure of FSPT phases. Specifically, we demonstrate how to compute stacking FSPT data from the input FSPT data in each layer, considering both unitary and anti-unitary symmetry, up to 2+1 dimensions. As concrete examples, we explicitly compute the stacking group structure for crystalline FSPT phases in all 17 wallpaper groups and the mixture of wallpaper groups with onsite time-reversal symmetry using the fermionic crystalline equivalence principle. Importantly, our approach can be readily extended to higher dimensions, offering a versatile method for exploring the stacking group structure of FSPT phases.

Figures (18)

• Figure 1: The decorations in the complex fermions and Majorana chains are as follows: a black hollow circle represents no complex fermion decoration, indicated by $n_{1}( g_{0} ,g_{1}) =0$; a black solid circle represents one complex fermion decoration, indicated by $n_{1}( g_{0} ,g_{1}) =1$; a red solid circle represents a Majorana fermion; a red solid line with an arrow represents Majorana pairings; a red dashed line with an arrow represents the absence of Majorana pairing between the species labeled by $g_{i} \in G_{b}$ for the site $i$, while all other species $\sigma \neq g_{i}$ exhibit trivial pairing.
• Figure 2: The wavefunctions of BSPT: (a) the triangulation and group element labels of the two layers are both different; (b) the triangulation is the same, while the group element labels are different; (c) the triangulation and group element labels are both the same, which is the setup for the $\mathcal{P}$ move in Eq. (\ref{['eq4.23']}). The wavefunction in (c) is a subset of (b), and the wavefunction in (b) is a subset of (a).
• Figure 3: Four basic moves of FSLU transformation on Kitaev chains that are always Kasteleyn oriented.
• Figure 4: All possible FSLU transformations that reduce the two-layer system to one possibly non-trivial layer plus a trivial product state layer.
• Figure 5: Kitaev chain decoration convention.