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Journal of Statistical Mechanics Theory and Experiment

An exact solution of the homogenous trimer Bose-Hubbard model

Feng Pan, Aoxue Li, Yingxin Wu, Jerry Paul Draayer

Published Mar 1, 2023• 2 citations
Publisher

Abstract

Abstract It is shown that the homogenous three-site (trimer) Bose–Hubbard model with a periodic boundary condition can be solved exactly by using an extended Bethe ansatz based on the solution of the dimer model and the site-permutation symmetry. A solution of the model within the S 3 symmetric or non-symmetric subspace is presented. Coupled differential equations of a series of extended Heine–Stieltjes polynomials, the zeros of which are related to the solution of the model, are derived. Numerical examples of the solution of the model with N ⩽ 4 bosons, including the related Heine–Stieltjes polynomials and the Van Vleck polynomials, are presented, which serve to validate the procedure and illustrate the completeness of the solutions they render.

DOI
10.1088/1742-5468/acb7ec

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Abstract It is shown that the extended one-dimensional dimer Bose–Hubbard model with multi-body interactions can be solved exactly by using the algebraic Bethe ansatz mainly due to the site-permutation S 2 symmetry. The solution for the model with up to three-particle hopping and three-body on-site interaction is explicitly shown. As an example of the application, lower part of the excitation energy levels and the ground-state entanglement measure of the standard Bose–Hubbard Hamiltonian with the attractive two-body on-site interaction plus the three-body on-site interaction for 100 bosons with variation of the control parameter are calculated by using the exact solution. It is shown that the attractive three-body on-site interaction reinforces the critical point entanglement of the system, which may be helpful for design of an optical lattice for ultracold atoms or a tuneable superconducting quantum interference device with maximal entanglement.
Abstract The Bose–Hubbard model is a paradigm for the study of strongly correlated bosonic systems. We study the two-site Bose–Hubbard model with generic integrable open boundaries specified by the most general non-diagonal reflecting matrices. Besides the inhomogeneous parameters, the model itself has three free boundary parameters, which break the U(1)-symmetry, in other words, break the particle number conservation. The Hamiltonian H under these circumstances is constructed. With the help of the off-diagonal Bethe Ansatz method, we successfully obtain the corresponding Bethe Ansatz equations as well as the eigenvalues.
It is shown that the ground state of the one-dimensional negative-U three-site Bose-Hubbard (trimer) model with periodic (boundary) condition or for a closed chain can be solved almost exactly within a small symmetric subspace with a few boson-trines through an analysis of the percentage of the boson-trines in the ground state. It is also shown that there is a sudden increase of the boson-trine content of the ground state near the critical point of the quantum phase transition, which may be used as an effective order parameter to study the phase transition of the model. The ground-state energy per particle and the entanglement measure with the variation of the control parameter are also analyzed to show their transitional behavior. © 2014 The Royal Swedish Academy of Sciences.
The Bose-Hubbard dimer Hamiltonian is a simple yet effective model for describing tunneling phenomena of Bose-Einstein condensates.One of the significant mathematical properties of the model is that it can be exactly solved by Bethe ansatz methods.Here we review the known exact solutions, highlighting the contributions of V.B. Kuznetsov to this field.Two of the exact solutions arise in the context of the Quantum Inverse Scattering Method, while the third solution uses a differential operator realisation of the su(2) Lie algebra.
It is shown that the two component Fermi or Bose many-body Hamiltonian, such as the two-orbit fermion pairing and two-site Bose-Hubbard model with arbitrary finite higher order terms can always be solved exactly by using Bethe ansatz vector construction based on the permutation of two components of bosons or fermions involved. As examples of the solution, the extended one-dimensional dimer Bose -Hubbard model with multi-body interactions and the mean-fifield plus orbit-dependent non-separable pairing model with two non-degenerate j-orbits are demonstrated with the eigenstates and the eigen-energy and the related Bethe ansatz equations. It is shown that the main feature of the solutions lies in the fact that the Bethe ansatz vectors can be expressed in terms of binomials of the boson or fermion operators times the related symmetric functions. As the consequence, two-component quantum many-body systems , such as the extended Lipkin-Meshkov-Glick model with higher-order interactions, can be solved in a similar way.