Integrability of Multispecies Long-Range Swap Models with Species-Dependent Interpolation

Abstract: 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 $i$ is assigned a parameter $μ_i$ 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 $μ_i \in {0,1}$ for arbitrary species compositions. In the continuous parameter regime $μ_i \in (0,1)$, 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.