Spectrum in multi-species asymmetric simple exclusion process on a ring

The spectrum of Hamiltonian (Markov matrix) of a multi-species asymmetric simple exclusion process on a ring is studied. The dynamical exponent concerning the relaxation time is found to coincide with the one-species case. It implies that the system belongs to the Kardar-Parisi-Zhang or Edwards-Wilkinson universality classes depending on whether the hopping rate is asymmetric or symmetric, respectively. Our derivation exploits a poset structure of the particle sectors, leading to a new spectral duality and inclusion relations. The Bethe ansatz integrability is also demonstrated.

💡 Research Summary

This paper investigates the full spectrum of the Markov generator (Hamiltonian) of a multi‑species asymmetric simple exclusion process (ASEP) defined on a periodic lattice. The authors first formulate the model: L sites form a ring, each site can be occupied by at most one particle, and there are N distinct particle species. Particles of species k hop to the right with rate p_k and to the left with rate q_k, provided the target site is empty. The stochastic dynamics are encoded in a Markov matrix H whose off‑diagonal entries are the transition rates and whose diagonal entries ensure probability conservation.

A central conceptual tool introduced is the notion of “sectors” (or particle configurations) grouped by the numbers M_k of each species and their spatial ordering. The set of all sectors is equipped with a partial‑order structure (poset) defined by inclusion: sector A is below sector B if the particle arrangement of A can be obtained by deleting particles from B. By block‑diagonalising H according to this poset, the authors prove an inclusion relation for eigenvalues: the spectrum of a lower sector is a subset of the spectrum of any higher sector that contains it.

From this hierarchical organization they uncover a new “spectral duality”. For any sector, the complement sector (obtained by swapping occupied and empty sites while preserving species counts) has a spectrum that is shifted by a simple integer amount depending on the total bias (p‑q) and the system size. This duality follows from a symmetry of the Bethe equations and dramatically reduces the amount of independent spectral data that must be computed.

Integrability is demonstrated by constructing Bethe‑Ansatz equations for each sector. Introducing rapidities λ_j, the eigenvalues are expressed as E = Σ_j ε(λ_j) with ε(λ) the single‑particle dispersion. The rapidities satisfy coupled algebraic equations of the form

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