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Physica Scripta

Symmetry-based approach to ground-state properties of the one-dimensional negative-UBose–Hubbard trimer model

Feng Pan, Yin Wang, Yan Qiao, Xianzhe Bu, Jerry Paul Draayer

Published Dec 30, 2013• 4 citations
Publisher

Abstract

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.

DOI
10.1088/0031-8949/89/01/015402

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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.
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 consider a Bose–Hubbard (BH) trimer, i.e. an ultracold Bose gas populating three quantum states. The latter can be either different sites of a triple-well potential or three internal states of the atoms. The bosons can tunnel between different states with variable tunnelling strength between two of them. This will allow us to study; (i) different geometrical configurations, i.e. from a closed triangle to three aligned wells and (ii) a triangular configuration with a π -phase, i.e. by setting one of the tunnellings negative. By solving the corresponding three-site BH Hamiltonian we obtain the ground state of the system as a function of the trap topology. We characterize the different ground states by means of the coherence and entanglement properties. For small repulsive interactions, fragmented condensates are found for the π -phase case. These are found to be robust against small variations of the tunnelling in the small interaction regime. A low-energy effective many-body Hamiltonian restricted to the degenerate manifold provides a compelling description of the π -phase degeneration and explains the low-energy spectrum as excitations of discrete semifluxon states.
We study the effect of different open boundary conditions on the insulating ground states of the one-dimensional extended Bose-Hubbard model at and near unit filling. To this end, we employ the density-matrix renormalization-group method with system sizes up to 250 sites. To characterize the system, various order parameters and entanglement entropies are calculated. When opposite edge potentials are added to the two ends of the chain, the inversion symmetry is explicitly broken, and the regular bulk phases appear. On the other hand, simple open boundary conditions often exhibit nondegenerate ground states with a domain wall in the middle of the chain, which induces a sign flip of an order parameter. Such a domain wall can lead to an algebraic behavior of the off diagonals of the single-particle density matrix. We show that this algebraic behavior adds only a finite contribution to the entanglement entropy, which does not diverge as the system size increases. Therefore, it is not an indication of a superfluid phase. We confirm this picture by analytical calculations based on an effective Hamiltonian for a domain wall.
The superfluid-insulator transition in systems of lattice bosons is usually analyzed in the framework of the Bose-Hubbard model, and has been extensively studied by theory and simulations. Less attention has been paid to the remnants of the transition in truncated lattices, with or without periodic boundary conditions. Here we consider the hard-core limit of the extended Bose-Hubbard model on small square and triangular grids -- i.e., sections of the square and triangular lattices containing up to 13 sites. By mapping out the zero-temperature phase diagram through exact diagonalization, we find ground-state characteristics that are markedly different from those emerging in the thermodynamic limit, together with similarities. The dichotomy between superfluid-like and insulating-like behavior is then investigated in two-dimensional systems of a few interacting bosons in the continuum, subject to confining and optical-lattice potentials mimicking the 3×33\times 33×3 square grid. Using path-integral Monte Carlo simulations, we compute kinetic and potential energies, as well as superfluidity and exchange-cycle statistics, finding hints of Bose-Hubbard behavior even in systems of just five particles.
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.
We study the dissipative dynamics of a one-dimensional bosonic system described in terms of the bipartite Bose-Hubbard model with alternating gain and loss. This model exhibits the PT\mathcal{P}\mathcal{T}PT symmetry under some specific conditions and features a PT\mathcal{P}\mathcal{T}PT-symmetry phase transition. It is characterized by an order parameter corresponding to the population imbalance between even and odd sites,similar to the continuous phase transitions in the Hermitian realm. In the noninteracting limit, we solve the problem exactly and compute the parameter dependence of the order parameter. The interacting limit is addressed at the mean-field level, which allows us to construct the phase diagram for the model. We find that both the interaction and dissipation rates induce PT\mathcal{PT}PT-symmetry breaking. On the other hand, periodic modulation of the dissipative coupling in time stabilizes the PT\mathcal{PT}PT-symmetric regime. Our findings are corroborated numerically on a tight-binding chain with gain and loss.