Characterization of Mott-insulating and superfluid phases in the one-dimensional Bose--Hubbard model

TL;DR

This paper investigates the static and dynamical properties of the one-dimensional Bose--Hubbard model in both Mott-insulating and superfluid phases using multiple advanced computational methods, providing detailed insights and applicability ranges.

Contribution

It compares strong-coupling, VCA, and DMRG methods for the Bose--Hubbard model, establishing their accuracy and limits in describing phase properties and excitations.

Findings

Strong-coupling and VCA methods accurately reproduce DMRG results within certain parameter ranges.

The transition points and central charge are determined from entanglement entropy analysis.

Dynamical structure factors reveal resonances and phonon modes consistent with theoretical predictions.

Abstract

We use strong-coupling perturbation theory, the variational cluster approach (VCA), and the dynamical density-matrix renormalization group (DDMRG) method to investigate static and dynamical properties of the one-dimensional Bose--Hubbard model in both the Mott-insulating and superfluid phases. From the von Neumann entanglement entropy we determine the central charge and the transition points for the first two Mott lobes. Our DMRG results for the ground-state energy, momentum distribution function, boson correlation function decay, Mott gap, and single particle-spectral function are reproduced very well by the strong-coupling expansion to fifth order, and by VCA with clusters up to 12 sites as long as the ratio between the hopping amplitude and on-site repulsion, t/U, is smaller than 0.15 and 0.25, respectively. In addition, in the superfluid phase VCA captures well the ground-state energy and the sound velocity of the linear phonon modes. This comparison provides an authoritative estimate for the range of applicability of these methods. In strong-coupling theory for the Mott phase, the dynamical structure factor is obtained from the solution of an effective single-particle problem with an attractive potential. The resulting resonances show up as double-peak structure close to the Brillouin zone boundary. These high-energy features also appear in the superfluid phase which is characterized by a pronounced phonon mode at small momenta and energies, as predicted by Bogoliubov and field theory. In one dimension, there are no traces of an amplitude mode in the dynamical single-particle and two-particle correlation functions.