Spin TQFTs and fermionic phases of matter

TL;DR

This work establishes a deep link between fermionic gapped phases and spin-TQFTs by showing that lattice constructions of fermionic SPTs inherently depend on a spin structure, with obstructions governed by the second Stiefel-Whitney class $[w_2]$. The Gu-Wen Grassmann integral is shown to provide a quadratic refinement of a bosonic pairing, encoding Koszul signs and higher cup products, and its gauge variation reveals a spin-structure–dependent 't Hooft anomaly that can be canceled by a spin-term, yielding a well-defined spin-TQFT partition function. The authors connect these lattice theories to spin cobordism, discuss state-sum constructions in low dimensions, and formulate a framework for fermionic anyon condensation that uses a kernel spin-TQFT $K_d$ to manage anomalies. They propose that spin structure is generally required for fermionic phases and outline potential Hamiltonian realizations, extended TFTs, and bosonization pathways in 2+1 dimensions. Overall, the paper provides a cohesive picture where fermionic phases emerge from and are constrained by spin topology, with concrete constructions in 2–4 dimensions and implications for mapping lattice models to spin-TQFTs and cobordism classifications.

Abstract

We study lattice constructions of gapped fermionic phases of matter. We show that the construction of fermionic Symmetry Protected Topological orders by Gu and Wen has a hidden dependence on a discrete spin structure on the Euclidean space-time. The spin structure is needed to resolve ambiguities which are otherwise present. An identical ambiguity is shown to arise in the fermionic analog of the string-net construction of 2D topological orders. We argue that the need for a spin structure is a general feature of lattice models with local fermionic degrees of freedom and is a lattice analog of the spin-statistics relation.

Figures (7)

• Figure 1: The barycentric subdivision of a triangle
• Figure 2: The two possible orientations of a triangle and ordering of vertices induced by a branching structure on a 2d triangulation
• Figure 3: Left: to each edge $e$ such that $\beta(e)=1$ we assign a $\theta_e$ Grassmann variable (black dot) and a $\bar{\theta}_e$ Grassmann variable (white dot). Each such edge contributes $d \theta_e d \bar{\theta}_e$ to the measure. Middle and right: as $\beta$ is a cochain, each triangle is associated to two Grassmann variables, which are ordered in the integrand according to the grey arrows.
• Figure 4: A collection of edges $e_i$ with $\beta(e_i)=1$, organized into a path in the dual graph to the triangulation. The Grassmann variables are encountered along the path in an order which differs by simple local permutations from the order they appear with in the measure and integrand of the Grassmann integral.
• Figure 5: As a path $C$ proceeds along the triangulation, it accumulates a sign when crossing each edge, and a sign when passing through each triangle, depending on if the Grassmann variables it encounters are set in canonical order in the integral, or not. In this figure we indicate the directions along which a path receives a $+$ sign. The opposite directions produce a $-$ sign.