Why the Bethe Ansatz Works: A Structural Explanation via Interaction Propagation
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866917400132190208 |
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| author | Gildea, Joe |
| author_facet | Gildea, Joe |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_09844 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Why the Bethe Ansatz Works: A Structural Explanation via Interaction Propagation Gildea, Joe Rings and Algebras Mathematical Physics Primary 81Q05, Secondary 82B23, 81R50, 16T25 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. |
| title | Why the Bethe Ansatz Works: A Structural Explanation via Interaction Propagation |
| topic | Rings and Algebras Mathematical Physics Primary 81Q05, Secondary 82B23, 81R50, 16T25 |
| url | https://arxiv.org/abs/2604.09844 |