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Journal of Statistical Mechanics Theory and Experiment

Integrability of multispecies long-range swap models with species-dependent interpolation

Eunghyun Lee

Published Jul 1, 2026• 0 citations
Publisher

Abstract

Abstract We introduce a class of multispecies exclusion processes in which each species follows its own intra-species interaction mechanism. In contrast to previously studied integrable multispecies models—such as the multispecies TASEP, where the intra-species interaction rule is identical across all species—our framework allows the intra-species rule to vary across species, while inter-species interactions obey a long-range swap rule. We prove that Bethe ansatz integrability is preserved in the binary regime, where each species follows either TASEP-type or drop-push-type dynamics, by establishing the Yang–Baxter structure of the associated scattering matrices. Furthermore, we identify nontrivial families of species compositions through a continuous interpolation between these dynamics that remain integrable, and extend the model to include bidirectional motion. These results show that Bethe ansatz integrability extends to a broad class of multispecies systems with species-dependent interaction mechanisms, providing a framework for deriving exact transition probabilities and analyzing nonequilibrium dynamics.

DOI
10.1088/1742-5468/ae80b5

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Abstract We introduce a class of multispecies exclusion processes in which each species follows its own intra-species interaction mechanism. In contrast to previously studied integrable multispecies models—such as the multispecies TASEP, where the intra-species interaction rule is identical across all species—our framework allows the intra-species rule to vary across species, while inter-species interactions obey a long-range swap rule. We prove that Bethe ansatz integrability is preserved in the binary regime, where each species follows either TASEP-type or drop-push-type dynamics, by establishing the Yang–Baxter structure of the associated scattering matrices. Furthermore, we identify nontrivial families of species compositions through a continuous interpolation between these dynamics that remain integrable, and extend the model to include bidirectional motion. These results show that Bethe ansatz integrability extends to a broad class of multispecies systems with species-dependent interaction mechanisms, providing a framework for deriving exact transition probabilities and analyzing nonequilibrium dynamics.
We introduce a class of multispecies exclusion processes with long-range swap interactions, incorporating species-dependent interpolation between TASEP-type and drop--push-type dynamics: each species iii is assigned a parameter μiμ_iμi​ governing same-species interactions, resulting in a heterogeneous system in which different species follow distinct microscopic interaction mechanisms. In contrast to previously studied integrable multispecies models, where species dependence typically enters through jump rates, the present framework allows the interaction mechanism itself to depend on the species. Our main result establishes integrability of the model in the binary parameter regime μi∈{0,1}μ_i \in \{0,1\}μi​∈{0,1} for arbitrary species compositions. In the continuous parameter regime μi∈(0,1)μ_i \in (0,1)μi​∈(0,1), we identify several nontrivial classes of species compositions for which integrability is preserved. We further extend the model to include bidirectional motion, going beyond totally asymmetric dynamics. Using the coordinate Bethe ansatz, we prove two-particle reducibility and derive the associated scattering matrix, which is shown to satisfy the Yang--Baxter equation. The resulting scattering matrix exhibits genuinely species-dependent diagonal entries.
We introduce the multispecies totally asymmetric simple exclusion process (mTASEP) with long-range swap, a new interacting particle system combining the backward-push rule with the forward-jump rule. Although governed by local dynamics, the model induces effective long-range particle exchanges. We establish its integrability by proving two-particle reducibility and showing that the associated scattering matrix satisfies the Yang-Baxter equation. In addition, we derive explicit contour integral formulas for transition probabilities. These results position the long-range swap model as a novel exactly solvable multispecies process, characterized by distinctive algebraic features and opening new directions for further study in integrable probability and statistical mechanics.
Abstract We work towards the classification of all one-dimensional exclusion processes with two species of particles that can be solved by a nested coordinate Bethe ansatz (BA). Using the Yang–Baxter equations, we obtain conditions on the model parameters that ensure that the underlying system is integrable. Three classes of integrable models are thus found. Of these, two classes are well known in literature, but the third has not been studied until recently, and never in the context of the BA. The Bethe equations are derived for the latter model as well as for the associated dynamics encoding the large deviation of the currents.
We consider a multispecies generalization of the totally asymmetric simple exclusion process (TASEP) with the simple hopping rule: for the α and βth-class particles (α < β), the transition αβ → βα occurs with a rate independent from the values α and β. Ferrari and Martin (2007 Ann. Prob. 35 807) obtained the stationary state of this model thanks to a combinatorial algorithm, which was subsequently interpreted as a matrix product representation by Evans et al (2009 J. Stat. Phys. 135 217). This 'matrix ansatz' shows that the stationary state of the multispecies TASEP with N classes of particles ( N -TASEP) can be constructed algebraically by the action of an operator on the ( N − 1)-TASEP stationary state. Besides, Arita et al ( 2009 J. Phys. A. Math Theor. 45 345002 ) analyzed the spectral structure of the Markov matrix: they showed that the set of eigenvalues of the N -TASEP contains those of the ( N − 1)-TASEP and that the various spectral inclusions can be encoded in a hierarchical set-theoretic structure known as the Hasse diagram. Inspired by these works, we define nontrivial operators that allow us to construct eigenvectors of the N -TASEP by lifting the eigenvectors of the ( N − 1)-TASEP. This goal is achieved by generalizing the matrix product representation and the Ferrari–Martin algorithm. In particular, we show that the matrix ansatz is not only a convenient tool to write the stationary state but in fact intertwines Markov matrices of different values of N .