The Algebra of Free Fermions: Classifying Spaces, Hamiltonians, and Computation

TL;DR

This paper develops a unified algebraic framework for classifying free fermion systems with symmetries, connecting topological phases, K-theory, and computational tools like GAP.

Contribution

It introduces a $G$-$ ext{Omega}$-spectrum structure for free fermions and provides a computational method and software for their classification and Hamiltonian construction.

Findings

Established the connection between free fermion systems and $G$-$ ext{Omega}$-spectra.

Developed a computational method for $ ext{Z}_2$-graded Wedderburn-Artin decomposition.

Created a GAP package to automate classification and Hamiltonian construction.

Abstract

Research on topological phases of matter is a core field in modern condensed matter physics. Free fermion systems, such as topological insulators and superconductors, have been studied using the "Tenfold Way" and K-theory. Building on Kitaev's idea of $\Omega$-spectrum and classifying space, as well as Freed-Moore's K-theory, this work demonstrates that free fermionic systems form a genuine $G$-$\Omega$-spectrum and clarifies its connection to several distinct classification schemes appearing in the physical literature. By introducing the $\mathbb{Z}_2$-graded algebra $A_{\mathrm{sym}}^V$, the classification problem for systems with general symmetries, including antilinear symmetries, antisymmetries, projective representations, and point group symmetries, is turned into an extension problem in representation theory. To solve this, a computational method for the $\mathbb{Z}_2$-graded Wedderburn-Artin decomposition of $A_{\mathrm{sym}}^V$ is developed. This decomposition not only yields a classification but also enables the explicit construction of the corresponding Dirac Hamiltonian. Furthermore, a GAP programming package has been developed to automate these calculations.