Fermion decoration construction of symmetry protected trivial orders for fermion systems with any symmetries $G_f$ and in any dimensions

TL;DR

This work develops a general, constructive framework to realize fermionic symmetry-protected trivial (SPT) orders in any dimension for arbitrary fermion symmetries $G_f=Z_2^f\rtimes G_b$, extending Gu–Wen group super-cohomology to generic extensions. It combines higher-dimensional bosonization with fermion decoration to produce exactly soluble bosonized path integrals labeled by higher-group data, enabling systematic construction and classification of fermionic SPT states, including time-reversal and space-time symmetry enrichments. The paper provides explicit models and invariants, demonstrates equivalence relations among label data, and shows an Abelian group structure under stacking, with detailed examples across 1+1D, 2+1D, and 3+1D for various symmetry groups (e.g., $Z_2\times Z_2^f$, $Z_4^f$, $Z_2^f\times Z_2^T$, $SU_2^f$, and U(1)-related groups). It connects the decorations to higher-group cocycles, spin/Pin structures, and cobordism classifications, and discusses how results align with or extend known cobordism and spin-cobordism calculations, including interacting topological insulators. The framework offers a powerful, geometrical lens for predicting and cataloging interacting fermionic SPT phases beyond prior formalisms, with implications for understanding symmetry-protected phases in strongly correlated fermionic systems.

Abstract

We use higher dimensional bosonization and fermion decoration to construct exactly soluble interacting fermion models to realize fermionic symmetry protected trivial (SPT) orders (which are also known as symmetry protected topological orders) in any dimensions and for generic fermion symmetries $G_f$, which can be a non-trivial $Z_2^f$ extension (where $Z_2^f$ is the fermion-number-parity symmetry). This generalizes the previous results from group superconhomology of Gu and Wen (arXiv:1201.2648), where $G_f$ is assumed to be a trivial $Z_2^f$ extension. We find that the SPT phases from fermion decoration construction can be described in a compact way using higher groups.

Figures (6)

• Figure 1: (Color online) The vertices $i$ and $j$ mark the regions with order parameter $g_i$ and $g_j$. The link labeled by $(ij)$ connects the vertices $i$ and $j$.
• Figure 2: (Color online) Three space-time $\cM_1$, $\cM_2$, and $\cM_1\sqcup -\cM_2$, plus their extensions. $\cN_1$, $\cN_2$, and $\cN$. The ratio of the action amplitudes (which are pure $U(1)$ phases) for space-time (a) and (b): $Z(a)/Z(b)=Z(a)Z^*(b)$ is given by the action amplitude for space-time (c): $Z(a)Z^*(b)=Z(c)$.
• Figure 3: (Color online) The boundary of $I\times \cN^{d+2}$ is given by $\cN^{d+2} \sqcup I\times \cM^{d+1} \sqcup \cN^{d+2}$. The boundary of $I\times \cM^{d+1}$ is given by $\cM^{d+1} \sqcup \cM^{d+1}$.
• Figure 4: (Color online) Two branched simplices with opposite orientations. (a) A branched simplex with positive orientation and (b) a branched simplex with negative orientation.
• Figure 5: (Color online) A 1-cochain $a$ has a value $1$ on the red links: $a_{ik}=a_{jk}= 1$ and a value $0$ on other links: $a_{ij}=a_{kl}=0$. $\dd a$ is non-zero on the shaded triangles: $(\dd a)_{jkl} = a_{jk} + a_{kl} - a_{jl}$. For such 1-cohain, we also have $a\smile a=0$. So when viewed as a $\Z_2$-valued cochain, $\beta_2 a \neq a\smile a$ mod 2.