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Physical review. B, Condensed matter

Bethe ansatz for the one-dimensional boson Hubbard model

Werner Krauth

Published Nov 1, 1991• 55 citations
Publisher

Abstract

The Bethe ansatz (BA) for the one-dimensional boson Hubbard model is studied. Even though the model cannot be solved exactly by this method, it is argued that it gives an excellent approximation for densities \ensuremath{\rho}\ensuremath{\le}1. This claim is substantiated by a detailed study of the internal consistency of the BA solution, and by comparison with quantum Monte Carlo simulations.

DOI
10.1103/physrevb.44.9772

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The Bethe ansatz (BA) for the one-dimensional boson Hubbard model is studied. Even though the model cannot be solved exactly by this method, it is argued that it gives an excellent approximation for densities \ensuremath{\rho}\ensuremath{\le}1. This claim is substantiated by a detailed study of the internal consistency of the BA solution, and by comparison with quantum Monte Carlo simulations.
We study the Mott insulating phase of the one-dimensional Hubbard model using a local-density approximation (LDA) that is based on the Bethe Ansatz (BA).Unlike conventional functionals, the BA-LDA has an explicit derivative discontinuity.We demonstrate that, as a consequence of this discontinuity, the BA-LDA yields the correct Mott gap, independently of the strength of the correlations.A convenient analytical formula for the Mott gap in the thermodynamic limit is also derived.We find that in one-dimensional quantum systems the contribution of the discontinuity to the full gap is more important than that of the band-structure gap, and discuss some consequences this finding has for electronic-structure calculations.
The exact solution of a quantum Bethe lattice model in the thermodynamic limit amounts to solve a functional self-consistent equation. In this paper we obtain this equation for the Bose-Hubbard model on the Bethe lattice, under two equivalent forms. The first one, based on a coherent-state path integral, leads in the large connectivity limit to the mean-field treatment of Fisher et al. [Phys. Rev. B 40, 546 (1989)] at the leading order, and to the bosonic dynamical mean field theory as a first correction, as recently derived by Byczuk and Vollhardt [Phys. Rev. B 77, 235106 (2008)]. We obtain an alternative form of the equation using the occupation number representation, which can be easily solved with an arbitrary numerical precision, for any finite connectivity. We thus compute the transition line between the superfluid and Mott insulator phases of the model, along with thermodynamic observables and the space and imaginary-time dependence of correlation functions. The finite connectivity of the Bethe lattice induces a richer physical content with respect to its infinitely connected counterpart: a notion of distance between sites of the lattice is preserved, and the bosons are still weakly mobile in the Mott insulator phase. The Bethe lattice construction can be viewed as an approximation to the finite-dimensional version of the model. We show indeed a quantitatively reasonable agreement between our predictions and the results of Quantum Monte Carlo simulations in two and three dimensions.