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Bethe ansatz solution to integrable bosonic cube networks

Lachlan Bennett, Phillip S. Isaac, Jon Links

TL;DR

This work develops Bethe ansatz solutions for two interacting bosonic networks restricted to a cube graph, introducing two extended Bose-Hubbard Hamiltonians. By applying two canonical transformations, the cube is mapped to tractable subgraphs (double squares and a square with dimers), enabling a systematic SU(2) symmetry analysis and identification of eight commuting conserved quantities that guarantee integrability. The authors construct explicit lowest-weight and recursive bases, derive Bethe equations for each model, and provide closed-form expressions for eigenvectors and energies in terms of rapidities. The results deepen the understanding of exact solvability in bosonic networks on graphs and have potential relevance for quantum control and engineered bosonic devices on complex geometries.

Abstract

We study two extended Bose-Hubbard-type Hamiltonians representing bosonic networks restricted to the graph of a cube. For both Hamiltonians, we demonstrate that Bethe ansatz methods of solution can be employed after applying a canonical transformation of operators. We provide the resulting Bethe ansatz equations, and corresponding formulae for states and energies of both Hamiltonians.

Bethe ansatz solution to integrable bosonic cube networks

TL;DR

This work develops Bethe ansatz solutions for two interacting bosonic networks restricted to a cube graph, introducing two extended Bose-Hubbard Hamiltonians. By applying two canonical transformations, the cube is mapped to tractable subgraphs (double squares and a square with dimers), enabling a systematic SU(2) symmetry analysis and identification of eight commuting conserved quantities that guarantee integrability. The authors construct explicit lowest-weight and recursive bases, derive Bethe equations for each model, and provide closed-form expressions for eigenvectors and energies in terms of rapidities. The results deepen the understanding of exact solvability in bosonic networks on graphs and have potential relevance for quantum control and engineered bosonic devices on complex geometries.

Abstract

We study two extended Bose-Hubbard-type Hamiltonians representing bosonic networks restricted to the graph of a cube. For both Hamiltonians, we demonstrate that Bethe ansatz methods of solution can be employed after applying a canonical transformation of operators. We provide the resulting Bethe ansatz equations, and corresponding formulae for states and energies of both Hamiltonians.
Paper Structure (24 sections, 98 equations, 3 figures)

This paper contains 24 sections, 98 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Bosonic cube models
  3. Canonical transformations and integrability
  4. Canonical transformation i
  5. Canonical transformation ii
  6. Symmetries
  7. The double square model
  8. The double dimer and square model
  9. Double square model Bethe ansatz solution
  10. Lowest weight states
  11. Proof that $f_1 |\psi_m^{\,\ell}\rangle = 0$
  12. Proof that $f_\alpha |\psi_m^{\,\ell}\rangle = 0$
  13. Recursive basis
  14. Counting modules
  15. Total Hilbert space
  16. ...and 9 more sections

Figures (3)

  • Figure 1: Labelling of the $Q_{3}$ cube graph. The bosonic networks (\ref{['cubeHam 1']}) and (\ref{['cubeHam 2']}) are defined on this graph: the bosons can occupy each vertex site and can tunnel to an adjacent vertex joined by an edge.
  • Figure 2: Transformation i applied to the cube $Q_{3}$. Vertices shown in red indicate sites in the original $a$-basis, while vertices in light blue mark the transformed $b$-basis. In this new basis the tunnelling network decomposes into two disjoint squares, matching the connectivity of Hamiltonian (\ref{['new cubeHam1']}). The corresponding Bethe ansatz solution is presented in Section \ref{['Section 4']}.
  • Figure 3: Transformation ii of the cube. Again, red vertices correspond to the original $a$-basis; vertices in dark blue label the transformed $\widetilde{b}$-basis. The resulting graph decomposes into one square and two dimers, in agreement with the structure of Hamiltonian (\ref{['new cubeHam2']}). The corresponding Bethe ansatz solution is presented in Section \ref{['Section 5']}.