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Journal of Physics A Mathematical and General

Simple exact solutions of one-dimensional finite-chain hard-core Bose-Hubbard and Fermi-Hubbard models

Feng Pan, Jerry Paul Draayer

Published Nov 29, 2000• 9 citations
Publisher

Abstract

Exact algebraic solutions of a one-dimensional finite-chain hard-core Bose-Hubbard model with nearest-neighbour interactions, which is related to a state-dependent nuclear pairing interaction with a nearest-state interaction approximation, are derived based on a simple algebraic approach. Further extensions to arbitrary spin cases and the corresponding Fermi-Hubbard model are also presented.

DOI
10.1088/0305-4470/33/49/305

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Exact algebraic solutions of a one-dimensional finite-chain hard-core Bose-Hubbard model with nearest-neighbour interactions, which is related to a state-dependent nuclear pairing interaction with a nearest-state interaction approximation, are derived based on a simple algebraic approach. Further extensions to arbitrary spin cases and the corresponding Fermi-Hubbard model are also presented.
A unified algebraic approach to both the hard-core Fermi– and Bose–Hubbard models is extended to both the finite- and infinite-site with periodic condition cases. Excitation energies and the corresponding wavefunctions of both the models with nearest neighbor hopping are exactly derived by using a new and simple algebraic method. It is found that spectra of both the models are determined simply by eigenvalue problem of N × N hopping matrix, where N is the number of sites for finite system or the period of sites for infinite system.
We determine the zero-temperature phase diagram of the hard-core Bose-Hubbard model on a square lattice by mean-field theory supplemented by a linear spin-wave analysis. Due to the interplay between nearest- and next-nearest-neighbor interaction and cubic anisotropy several supersolid phases with checkerboard, stripe-domain, or intermediate symmetry are stabilized. The phase diagrams show three different topologies depending on the relative strength of nearest- and next-nearest-neighbor interaction. We also find a rich variety of new quantum critical behavior and multicritical points and discuss the corresponding effective actions and universality classes.
The phase transitions at finite temperatures in the systems described by the Bose-Fermi-Hubbard model are investigated in this work in the framework of the selfconsistent random phase approximation. The case of the hard-core bosons is considered and pseudospin formalism is used. The density-density correlator is calculated in the random phase approximation and the possibilities of transitions from superfluid to supersolid phases are investigated. It is shown that the transitions between uniform and charge ordered phases can be of the second or the first order, depending on the system parameters.
The one-dimensional Holstein model of spinless fermions interacting with dispersionless phonons is solved by using a Bethe ansatz in analogue to that for the one-dimensional spinless Fermi–Hubbard model. Excitation energies and the corresponding wavefunctions of the model are determined by a set of partial differential equations. It is shown that the model is, at least, quasi-exactly solvable for the two-site case, when the phonon frequency, the electron–phonon coupling strength and the hopping integral satisfy certain relations. As examples, some quasi-exact solutions of the model for the two-site case are derived.