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Exactly Solvable Bose and Fermi many-body Hamiltonian with higher order terms based on the S2 symmetry

Feng Pan, L. R. Dai

Published Aug 3, 2021• 0 citations
OpenAlex

Abstract

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.

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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.
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.
A solvable extended Hamiltonian that includes multi-pair interac- tions among s- and d-bosons up to infinite order within the framework of the interacting boson model is proposed to gain a better description of E(5) model results for finite-N systems. Similarly, an extended pairing Hamiltonian to de- scribe pairing interactions among valence nucleon monopole pairs up to infinite order in a spherical mean-field can also be built, which includes the extended pairing interaction model within a deformed mean-field theory as a special case. These mean-field plus pairing models are all constructed based on the local ~ E2 algebraic structure.
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.
We present a family of many-body models which are exactly solvable analytically. The models are an extended n-body interaction Lipkin-Meshkov-Glick model which considers spin-flip terms which are associated with the interaction of an external classical field which coherently manipulates the state of the system in order to, for example, process quantum information. The models also describe a two-mode Bose-Einstein condensate with a Josephson-type interaction which includes n-particle elastic and inelastic collisions. One of the models corresponds to the canonical two-mode Bose-Einstein Hamitonian plus a term which we argue must be considered in the description of the two-mode condensate. Intriguingly, this extra term allows for an exact and analytical solution of the two-particle collision two-mode BEC problem. Our results open up an arena to study many-body system properties analytically.