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arXiv:cond-mat/0108314 · Journal of the Korean Physical Society

Bethe Ansatz Solutions and Excitation Gap of the Attractive Bose-Hubbard Model

Deok Sun Lee, Doo Chul Kim

Published Aug 20, 2001• 1 citation

Abstract

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.

DOI: 10.48550/arxiv.cond-mat/0108314

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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 study the ground state of the attractive one-dimensional Bose-Hubbard model, and in particular the nature of the crossover between the weak interaction and strong interaction regimes for finite system sizes. Indicator properties such as the gap between the ground and first excited energy levels, and the incremental ground-state wave function overlaps are used to locate different regimes. Using mean-field theory we predict that there are two distinct crossovers connected to spontaneous symmetry breaking of the ground state. The first crossover arises in an analysis valid for large

L

with finite

N

, where

L

is the number of lattice sites and

N

is the total particle number. An alternative approach valid for large

N

with finite

L

yields a second crossover. For small system sizes we numerically investigate the model and observe that there are signatures of both crossovers. We compare with exact results from Bethe ansatz methods in several limiting cases to explore the validity for these numerical and mean-field schemes. The results indicate that for finite attractive systems there are generically three ground-state phases of the model.

We present a simple approximation scheme for the solution of the integral equations resulting from the Bethe-ansatz diagonalization of the one-dimensional Fermi gas with \ensuremath{\delta}-function attraction. These equations arise for the Hubbard model with attractive interaction in the limit of weak coupling and low density. We obtain the ground-state energy as a function of coupling and density in very good agreement with numerical solutions, as well as a value for the parameter determining the gap in the magnetic excitation spectrum which strongly supports a conjecture of Larkin and Sak.

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 statistical properties of one-dimensional directed polymers in a short-range random potential by mapping the replicated problem to a many-body quantum boson system with attractive interactions. We find the full set of eigenvalues and eigenfunctions of the many-body system and perform the summation over the entire spectrum of excited states. The analytic continuation of the obtained exact expression for the replica partition function from integer to non-integer replica parameter N turns out to be ambiguous. Performing the analytic continuation simply by assuming that the parameter N can take arbitrary complex values, and going to the thermodynamic limit of the original directed polymer problem, we obtain the explicit universal expression for the probability distribution function of free energy fluctuations.

Ground-state properties of the attractive Hubbard model in one dimension are studied by means of both the exact Bethe-ansatz formalism and the self-consistent field (SCF) approach with renormalized chemical potential for general band fillings n and a wide range of coupling strength

U/t.

The energy, the concentration of double occupied sites, the kinetic energy and the chemical potential of the ground state are in a good numerical agreement with the exact results over a wide range of parameters

U/t

and n. The concentration of local pairs or double occupied sites in the Bethe-ansatz solution serves as a suitable parameter measuring the electron pairing correlations. The SCF theory provides a simple analytical relationship between the concentration of double occupied sites, the band filling and the BCS order parameter, valid for arbitrary

U/t

and n. The calculated energy gap, the BCS order parameter, the phase diagram and the compressibility are also discussed. The SCF theory in one dimension distinguishes the order parameter from the excitation gap and suggests a smooth crossover away from half-filling (at $n\ensuremath{\ne}1)$ from the BCS pairing to the Bose condensation regime under the variation of

U/t

and n.