Why the Bethe Ansatz Works: A Structural Explanation via Interaction Propagation

TL;DR

This paper explains why the Bethe Ansatz works or fails in quantum many-body systems by analyzing how interaction propagation behaves and identifying a structural boundary mechanism.

Contribution

It provides a representation-independent, structural explanation for the existence and failure of Bethe-type solvability based on interaction propagation behavior.

Findings

Bethe-type solvability occurs when interaction propagation terminates finitely.

Encountering structural boundaries leads to irreducible interaction data that obstruct solvability.

The success or failure of the Bethe Ansatz is governed by intrinsic boundary formation.

Abstract

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.