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This paper establishes a rigorous mathematical framework for classifying one-dimensional integrable fermions by defining massless regimes through the simultaneous satisfaction of the Yang-Baxter and Shastry's decorated equations. The work identifies two distinct mechanisms for generating massive fermions—breaking time-reversal symmetry via external fields or introducing symmetric interactions—and provides schematic procedures to determine the corresponding R-matrices for models like the XY chain and Hubbard model. These findings offer a generalized algebraic structure that bridges free-fermion algebras with broader classes of exactly solvable quantum models.
Integrable Massless and Massive Fermions
arXiv:2603.11172v2 Announce Type: replace-cross Abstract: One-dimensional integrable fermions can be classified into massless and massive regimes, and the $R$-operator for the latter can be constructed from that of the former. Here, I define integrable massless fermions by the simultaneous satisfaction of the Yang-Baxter equation (YBE) and Shastry's decorated YBE (DYBE) by the $R$-matrix. This notion is strictly more general than Maassarani's `free-fermion algebra', yet more restrictive than the notion of free fermions in exactly solvable quantum models or in integrable two-dimensional classical vertex models dual to quantum spin chains. Within this framework, there emerge two archetypal mechanisms for opening a spectral gap and generating massive fermions: (i) breaking time-reversal symmetry by coupling to external field, and (ii) introducing time-reversal symmetric interactions. These paradigms are realized, respectively, in the XY chain in a longitudinal field and in the Hubbard model, both of which possess non-relativistic, bivariate $R$-matrices. Integrability conditions on local Hamiltonians for both massless and massive fermions are identified, and schematic procedures for uniquely determining their $R$-matrices are proposed.
Abstract:One-dimensional integrable fermions can be classified into massless and massive regimes, and the $R$-operator for the latter can be constructed from that of the former. Here, I define integrable massless fermions by the simultaneous satisfaction of the Yang-Baxter equation (YBE) and Shastry's decorated YBE (DYBE) by the $R$-matrix. This notion is strictly more general than Maassarani's `free-fermion algebra', yet more restrictive than the notion of free fermions in exactly solvable quantum models or in integrable two-dimensional classical vertex models dual to quantum spin chains. Within this framework, there emerge two archetypal mechanisms for opening a spectral gap and generating massive fermions: (i) breaking time-reversal symmetry by coupling to external field, and (ii) introducing time-reversal symmetric interactions. These paradigms are realized, respectively, in the XY chain in a longitudinal field and in the Hubbard model, both of which possess non-relativistic, bivariate $R$-matrices. Integrability conditions on local Hamiltonians for both massless and massive fermions are identified, and schematic procedures for uniquely determining their $R$-matrices are proposed.
Exactly Solvable and Integrable Systems (nlin.SI); Statistical Mechanics (cond-mat.stat-mech); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)
arXiv:2603.11172 [nlin.SI]
(or arXiv:2603.11172v2 [nlin.SI] for this version)
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