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arXiv:2509.01989 · Journal of Statistical Mechanics Theory and Experiment

Bethe Ansatz solution for a model of global-range interacting bosons on the square lattice

Jon Links

Published Feb 1, 2026• 1 citation
arXivHTMLPDF

Abstract

Abstract Quantum systems on a one-dimensional lattice are ubiquitous in the study of models exactly-solved by Bethe Ansatz techniques. Here it is shown that including global-range interaction opens scope for Bethe Ansatz solutions that are not constrained to one-dimensional quantum systems. A bosonic model on a square lattice is defined, and the exact Bethe Ansatz solution is provided for open, cylindrical, and toroidal boundary conditions. Generalising the result for an integrable defect leads to a Bethe Ansatz solution that is not expressible in an exact, closed-form manner.

DOI
10.1088/1742-5468/ae40ee

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Abstract Quantum systems on a one-dimensional lattice are ubiquitous in the study of models exactly-solved by Bethe Ansatz techniques. Here it is shown that including global-range interaction opens scope for Bethe Ansatz solutions that are not constrained to one-dimensional quantum systems. A bosonic model on a square lattice is defined, and the exact Bethe Ansatz solution is provided for open, cylindrical, and toroidal boundary conditions. Generalising the result for an integrable defect leads to a Bethe Ansatz solution that is not expressible in an exact, closed-form manner.
The Bethe Ansatz provides exact solutions for certain interacting quantum many-body systems, yet its success is confined to narrow regimes and breaks down abruptly outside them. Despite extensive developments in integrable systems, a structural explanation of this phenomenon has remained elusive. In this paper we give a representation-independent account of both the existence and the failure of Bethe-type exact solvability. We identify a single governing mechanism: the behaviour of interaction propagation. For systems in which propagation terminates after finite depth without encountering structural boundaries, global interaction data factor through finitely many local components, forcing Bethe-type solvability. Conversely, once a structural boundary is encountered, irreducible interaction data arise that obstruct such finite factorization and preclude Bethe-type solutions. This yields a sharp structural dichotomy. Within this regime, exact solvability is not an analytic accident but a rigidity phenomenon, while its failure is governed by intrinsic boundary formation. In this way, the Bethe Ansatz is understood as a consequence of constrained interaction propagation rather than as a model-specific construction.
Abstract The Lieb–Liniger model describes one-dimensional bosons with contact interactions. This many-body system admits an exact solution in terms of the Bethe ansatz. Some of the exact and perturbative results for this model are reviewed. Particular attention is devoted to the explicit evaluation, in terms of the interaction parameter, of physical quantities that can be formally exactly extracted from the Bethe ansatz solution. Another goal of this review is to stress exact relations between various quantities. The technical developments are explained in detail. The most relevant experimental realisations of the studied problems are eventually discussed. This review also contains several new results such as the study of convergence of the ground-state energy series at strong interactions, the excitation spectrum at high energies, and the evaluation of the boundary energy.