Completeness of the coordinate Bethe ansatz for the periodic ASEP with tunable interaction (the general-U Bethe equations of Nakerst–Prosen–Haque)

The Markov generator of the periodic asymmetric simple exclusion process (ASEP) can be written, via a Jordan–Wigner transformation, as a non-Hermitian spinless-fermion model $H_U = H_0 + U\mathcal{I}$ with a tunable nearest-neighbor density-density interaction: $U=0$ is a free-fermion point and $U=1$ recovers the standard ASEP (under the source's convention $p+q=1$). Nakerst, Prosen and Haque (2024) derived coordinate-Bethe-ansatz equations for arbitrary $U$ and found numerically, in small systems, that every eigenvalue is a sum of Bethe roots; they explicitly state that a formal completeness proof for these equations is lacking. The problem is to prove or refute completeness of this Bethe ansatz — that its solutions, counted with multiplicity, account for all $\binom{L}{N}$ eigenvalues — in particular in the interacting regime $U \neq 1$, which is covered neither by the completeness results for periodic TASEP nor by the completeness claimed for the standard periodic ASEP at generic hopping rates.

Background

The asymmetric simple exclusion process (ASEP) is a paradigmatic stochastic model of one-dimensional transport: particles occupy a lattice of $L$ sites with at most one particle per site, and hop to the right or left neighboring site with rates $p$ and $q$ if that site is empty. The configuration probabilities $P(t)$ obey a master equation $\frac{d}{dt}P = HP(t)$, where $H$ is the Markov generator (a matrix with non-negative off-diagonal entries and zero column sums); the spectrum of $H$ encodes all relaxation rates of the process. On a ring (periodic boundary conditions) with $N$ particles, the particle number is conserved, so $H$ acts on a finite space of dimension $\binom{L}{N}$. Gwa and Spohn (1992) constructed the coordinate Bethe-ansatz solution of the periodic ASEP generator — the coordinate Bethe ansatz being the standard integrability technique in which eigenfunctions are written as superpositions of plane waves whose quasimomenta (Bethe roots) satisfy coupled algebraic equations fixed by two-particle scattering — and used it to extract the dynamical exponent of the Kardar–Parisi–Zhang (KPZ) universality class of one-dimensional driven-diffusive systems; their derivation does not address whether the Bethe solutions exhaust the spectrum.

Nakerst, Prosen and Haque (2024) recast the periodic ASEP generator through a Jordan–Wigner transformation (a standard mapping of spin chains to fermions) as a non-Hermitian interacting spinless-fermion model $H_U = H_0 + U\mathcal{I}$: $H_0$ is a quadratic hopping term with right/left rates $p,q$ and a boundary twist $(-1)^{N+1}$, while $\mathcal{I} = \sum_j n_j n_{j+1} - N$ (with $n_j = c_j^\dagger c_j$ the particle-number operator and site $L+1$ identified with site $1$) is a quartic density-density interaction. The interaction strength $U$ interpolates between free fermions ($U=0$, where the spectrum is obtained exactly by single-particle diagonalization) and the standard ASEP ($U=1$); the paper adopts the convention $p+q=1$ unless specified otherwise, under which $H_1$ is the Markov generator of the ASEP (for general rates the Markov generator is instead $H_0 + (p+q)\mathcal{I}$). They then derived, apparently for the first time, coordinate-Bethe-ansatz equations for arbitrary $U$: the Bethe roots $z_1,\dots,z_N$ satisfy recurrent algebraic equations, and each solution yields an eigenvalue as a sum of single-root contributions.

The Bethe ansatz is called complete for this model if the solutions of these equations, counted with multiplicity, account for all $\binom{L}{N}$ eigenvalues of $H_U$ (equivalently — wherever $H_U$ is diagonalizable — the Bethe vectors span the state space). Completeness is what licenses any Bethe-ansatz-based spectral statement, but for interacting models it is a genuinely hard question. Rigorous completeness results exist for other Bethe-ansatz-solvable systems — e.g., the nonlinear Schrödinger model (Dorlas 1993) and the algebro-geometric program of Langlands and Saint-Aubin (1995). For exclusion processes, Golinelli and Mallick (2005) mapped the TASEP (totally asymmetric, $q=0$) Bethe roots numerically and uncovered spectral degeneracies of the Markov matrix whenever $\gcd(L,N)>1$, showing that completeness comes with nontrivial degeneracy bookkeeping; the periodic-TASEP case was settled only in 2025 by Iwao and Motegi via an algebro-geometric counting argument on a Riemann surface (the one-complex-dimensional algebraic curve parametrizing the Bethe roots), and completeness of the standard periodic ASEP at generic hopping rates is claimed in an unpublished preprint of Brattain, Do and Saenz. None of these covers the interacting family $U \neq 1$. Nakerst, Prosen and Haque verified numerically in small systems that all eigenvalues conform to their general-$U$ Bethe ansatz, and state that a formal completeness proof is lacking.

Problem Statement

Prove, or refute by an explicit finite counterexample, that the coordinate Bethe ansatz is complete for the periodic ASEP with tunable interaction, as formulated by Nakerst–Prosen–Haque. For the generator $H_U = H_0 + U\mathcal{I}$ on a ring of $L$ sites with $N$ particles and hopping rates $p$ (right) and $q$ (left), determine whether the $N$-tuples $(z_1,\dots,z_N)$ solving the Bethe equations

$z_j^L \;=\; \prod_{k \neq j}^{N}\left(-\frac{q + p\, z_j z_k - U z_j}{q + p\, z_j z_k - U z_k}\right), \qquad j = 1,\dots,N,$

with associated eigenvalues $E = \sum_{j=1}^{N}\left(p\, z_j + q\, z_j^{-1} - U\right)$, account for all $\binom{L}{N}$ eigenvalues of $H_U$ counted with multiplicity — for arbitrary interaction strength $U$ (where $U=0$ is the free-fermion point and $U=1$ the standard ASEP at $p+q=1$), in particular in the interacting regime $U \neq 1$, and for arbitrary hopping rates $p,q$.

References

1. Goran Nakerst, Tomaž Prosen, Masud Haque, 'Spectral boundary of the asymmetric simple exclusion process: Free fermions, Bethe ansatz, and random matrix theory', Physical Review E 110, 014110 (2024). DOI: 10.1103/PhysRevE.110.014110; arXiv:2402.00662. https://doi.org/10.1103/PhysRevE.110.014110
2. Eric Brattain, Norman Do, Axel Saenz, 'The completeness of the Bethe ansatz for the periodic ASEP', arXiv:1511.03762 (v1 2015, v2 2017, unpublished preprint). https://arxiv.org/abs/1511.03762
3. Shinsuke Iwao, Kohei Motegi, 'Bethe roots for periodic TASEP and algebraic curve', arXiv:2504.19690 (2025). https://arxiv.org/abs/2504.19690
4. Anastasiia Trofimova, Lu Xu, 'Scaling Properties of Current Fluctuations in Periodic TASEP', arXiv:2507.17750 (2025). https://arxiv.org/abs/2507.17750
5. T. C. Dorlas, 'Orthogonality and completeness of the Bethe ansatz eigenstates of the nonlinear Schrödinger model', Communications in Mathematical Physics 154, 347–376 (1993). DOI: 10.1007/BF02097001. https://doi.org/10.1007/BF02097001
6. Robert P. Langlands, Yvan Saint-Aubin, 'Algebro-geometric aspects of the Bethe equations', in Strings and Symmetries, Lecture Notes in Physics Vol. 447, pp. 40–53 (Springer, Berlin, 1995). DOI: 10.1007/3-540-59163-X_254. https://doi.org/10.1007/3-540-59163-X_254
7. Leh-Hun Gwa, Herbert Spohn, 'Bethe solution for the dynamical-scaling exponent of the noisy Burgers equation', Physical Review A 46, 844–854 (1992). DOI: 10.1103/PhysRevA.46.844. https://doi.org/10.1103/PhysRevA.46.844
8. O. Golinelli, K. Mallick, 'Spectral Degeneracies in the Totally Asymmetric Exclusion Process', Journal of Statistical Physics 120, 779–798 (2005). DOI: 10.1007/s10955-005-6972-7. https://doi.org/10.1007/s10955-005-6972-7
9. Sylvain Prolhac, 'Riemann surfaces for KPZ with periodic boundaries', SciPost Physics 8, 008 (2020). DOI: 10.21468/SciPostPhys.8.1.008; arXiv:1908.08907. https://doi.org/10.21468/SciPostPhys.8.1.008
10. Yunfeng Jiang, Yang Zhang, 'Algebraic geometry and Bethe ansatz. Part I. The quotient ring for BAE', Journal of High Energy Physics 2018, 87 (2018). DOI: 10.1007/JHEP03(2018)087; arXiv:1710.04693. https://doi.org/10.1007/JHEP03(2018)087