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Hierarchical algebraic structure and recursive solution of the Bose-Hubbard model

Feng Pan1,2,3,*, Lianrong Dai1,†, and Jerry P. Draayer2

  • *Contact author: daipan@dlut.edu.cn
  • †Contact author: dailianrong@zjhu.edu.cn
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Phys. Rev. Research 8, 033318 – Published 15 September, 2026

DOI: https://doi.org/10.1103/psg9-5rv3

Abstract

We uncover a previously unrecognized hierarchical algebraic structure in the Bose-Hubbard model that enables a systematic construction of its spectrum without relying on integrability. We develop a recursive framework in which, within each fixed particle-number sector, the eigenproblem of a p-site system is reduced to a closed set of algebraic relations determined entirely by the eigenpairs and single-particle transfer amplitudes of the (p−1)-site subsystem. This reduction yields two complementary sets of Bethe-type equations whose solutions exhaust the spectrum. The underlying structure originates from the locality of the Hamiltonian, which constrains the coupling between subsystems and gives rise to an emergent sparse, near-tridiagonal representation of the many-body problem. As an explicit demonstration, we construct the complete spectrum of the three-site Bose-Hubbard model from that of the two-site system for particle numbers n≤3, obtaining exact agreement with direct diagonalization. The resulting formulation substantially reduces computational cost and reveals a form of hierarchical solvability in a nonintegrable quantum many-body system. More broadly, our results suggest that recursive algebraic constructions may provide a general framework for analyzing lattice models beyond the conventional integrable paradigm.

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Phys. Rev. Research 8, 033318– Published 15 September, 2026
  • Received 30 March 2026
  • Accepted 1 September 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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