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Journal of Statistical Mechanics Theory and Experiment

A Bethe ansatz study of the ground state energy for the repulsive Bose–Hubbard dimer

Jon Links, Shao-you Zhao A

Published Mar 6, 2009• 12 citations
Publisher

Abstract

We employ an exact Bethe ansatz solution to study the repulsive Bose–Hubbard dimer model. Using a continuum approximation we solve for the ground state root density and in turn compute the ground state energy per particle. The formula so obtained is accurate for a finite-sized system in the weak coupling limit. Surprisingly, the formula diverges from the numerically calculated value as the number of particles increases. This indicates that the formula fails to predict the ground state energy per particle in the thermodynamic limit.

DOI
10.1088/1742-5468/2009/03/p03013

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We employ an exact Bethe ansatz solution to study the repulsive Bose–Hubbard dimer model. Using a continuum approximation we solve for the ground state root density and in turn compute the ground state energy per particle. The formula so obtained is accurate for a finite-sized system in the weak coupling limit. Surprisingly, the formula diverges from the numerically calculated value as the number of particles increases. This indicates that the formula fails to predict the ground state energy per particle in the thermodynamic limit.
The ground-state energy and energy gap to the first excited state are calculated for the attractive Hubbard model in one dimension using both the Bethe ansatz equations and the variational BCS wave function. Comparisons are provided as a function of coupling strength and electron density. While the ground-state energies are always in very good agreement, the BCS energy gap is sometimes incorrect by an order of magnitude, particularly at half-filling. Finite-size effects are also briefly discussed for cases where an exact solution in the thermodynamic limit is not possible. In general, the BCS result for the energy gap is poor compared to the exact result.
The attractive two-site Bose-Hubbard model is studied within the framework of the analytical solution obtained by the application of the quantum inverse scattering method. The structure of the ground and excited states is analyzed in terms of solutions of Bethe equations, and an approximate solution for the Bethe roots is given. This yields approximate formulas for the ground-state energy and for the first excited-state energy. The obtained formulas work with remarkable precision for a wide range of parameters of the model, and are confirmed numerically. An expansion of the Bethe state vectors into a Fock space is also provided for evaluation of expectation values, although this does not have accuracy similar to that of the energies.
We study the ground-state properties and excitation spectrum of the Lieb-Liniger model, i.e. the one-dimensional Bose gas with repulsive contact interactions. We solve the Bethe-Ansatz equations in the thermodynamic limit by using an analytic method based on a series expansion on orthogonal polynomials developed in and push the expansion to an unprecedented order. By a careful analysis of the mathematical structure of the series expansion, we make a conjecture for the analytic exact result at zero temperature and show that the partially resummed expressions thereby obtained compete with accurate numerical calculations. This allows us to evaluate the density of quasi-momenta, the ground-state energy, the local two-body correlation function and Tan’s contact. Then, we study the two branches of the excitation spectrum. Using a general analysis of their properties and symmetries, we obtain novel analytical expressions at arbitrary interaction strength which are found to be extremely accurate in a wide range of intermediate to strong interactions.
The energy gap between the ground state and the first excited state of the one-dimensional attractive Bose-Hubbard Hamiltonian is investigated in connection with directed polymers in random media. The excitation gap Δis obtained by exact diagonalization of the Hamiltonian in the two- and three-particle sectors and also by an exact Bethe Ansatz solution in the two-particle sector. The dynamic exponent z is found to be 2. However, in the intermediate range of the size L where UL~O(1), U being the attractive interaction, the effective dynamic exponent shows an anomalous peak reaching high values of 2.4 and 2.7 for the two- and the three-particle sectors, respectively. The anomalous behavior is related to a change in the sign of the first excited-state energy. In the two-particle sector, we use the Bethe Ansatz solution to obtain the effective dynamic exponent as a function of the scaling variable UL/π. The continuum version, the attractive delta-function Bose-gas Hamiltonian, is integrable by the Bethe Ansatz with suitable quantum numbers, the distributions of which are not known in general. Quantum numbers are proposed for the first excited state and are confirmed numerically for an arbitrary number of particles.
We revisit the one-dimensional attractive Hubbard model by using the Bethe-ansatz-based density-functional theory and density-matrix renormalization method. The ground-state properties of this model are discussed in details for different fillings and different confining conditions in weak-to-intermediate coupling regime. We investigate the ground-state energy, energy gap, and pair-binding energy and compare them with those calculated from the canonical Bardeen-Cooper-Schrieffer approximation. We find that the Bethe-ansatz-based density-functional theory is computationally easy and yields an accurate description of the ground-state properties for weak-to-intermediate interaction strength, different fillings, and confinements. In order to characterize the quantum phase transition in the presence of a harmonic confinement, we calculate the thermodynamic stiffness, the density-functional fidelity, and fidelity susceptibility, respectively. It is shown that with the increase in the number of particles or attractive interaction strength, the system can be driven from the Luther-Emery-type phase to the composite phase of Luther-Emery-type in the wings and insulatinglike in the center.
Exactly solvable models provide a unique method, via qualitative changes in the distribution of the ground-state roots of the Bethe ansatz equations, to identify quantum phase transitions. Here we expand on this approach, in a quantitative manner, for two models of Bose–Einstein condensates. The first model deals with the interconversion of bosonic atoms and molecules. The second is the two-site Bose–Hubbard model, widely used to describe tunneling phenomena in Bose–Einstein condensates. For these systems we calculate the ground-state root density. This facilitates the determination of analytic forms for the ground-state energy, and associated correlation functions through the Hellmann–Feynman theorem. These calculations provide a clear identification of the quantum phase transition in each model. For the first model we obtain an expression for the molecular fraction expectation value. For the two-site Bose–Hubbard model we find that there is a simple characterization of condensate fragmentation.