Algebra of free fermions: Classifying spaces, Hamiltonians, and computation
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.
Versions
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BibTeX
@misc{yuan2026algebra,
title = {{Algebra of free fermions: Classifying spaces, Hamiltonians, and computation}},
author = {Tian Yuan and Yang Qi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.11655},
eprint = {2605.11655},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.11655},
}
RIS
TY - UNPB
TI - Algebra of free fermions: Classifying spaces, Hamiltonians, and computation
AU - Tian Yuan
AU - Yang Qi
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2605.11655
UR - https://arxiv.org/abs/2605.11655
AB - 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 ℤ_2-graded algebra A_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 ℤ_2-graded Wedderburn-Artin decomposition of A_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.
ER -