The Algebra of Free Fermions: Classifying Spaces, Hamiltonians, and Computation
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910211426484224 |
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| author | Yuan, Tian Qi, Yang |
| author_facet | Yuan, Tian Qi, Yang |
| contents | 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 $\mathbb{Z}_2$-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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_11655 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Algebra of Free Fermions: Classifying Spaces, Hamiltonians, and Computation Yuan, Tian Qi, Yang Mesoscale and Nanoscale Physics Mathematical Physics 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 $\mathbb{Z}_2$-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. |
| title | The Algebra of Free Fermions: Classifying Spaces, Hamiltonians, and Computation |
| topic | Mesoscale and Nanoscale Physics Mathematical Physics |
| url | https://arxiv.org/abs/2605.11655 |