The Hubbard model extended by nearest‐neighbor Coulomb and exchange interaction on a cubic cluster – rigorous and exact results

Abstract

Key words extended Hubbard model, nearest-neighbor interaction, cubic cluster PACS 85. 80.+n, 73.22.-f, 71.27.+aThe Hubbard model on a cube was revisited and extended by both nearest-neighbor (nn) Coulomb correlation W and nearest-neighbor Heisenberg exchange J. The complete eigensystem was computed exactly for all electron occupancies and all model parameters ranging from minus infinity to plus infinity. For two electrons on the cluster the eigensystem is given in analytical form. For six electrons and infinite on-site correlation U we determinded the groundstate and the groundstate energy of the pure Hubbard model analytically. For fixed electron numbers we found a multitude of ground state level crossings. in dependence on the various model parameters. Furthermore the groundstates of the pure Hubbard model in dependence on a magnetic field h coupled to the spins are shown for the complete U − h plane. The critical magnetic field, where the zero spin groundstate breaks down is given for four and six electrons. Suprisingly we found parameter regions, where the ground state spin does not depend monotonously on J in the extended model. For the cubic cluster gas, i.e. an ensemble of clusters coupled to an electron bath, we calculated the density n(µ, T, h) and the thermodynamical density of states from the grand potential. The ground states and the various spin-spin correlation functions are studied for both attractive and repulsive values of the three interaction constants. We determined the various anomalous degeneration lines, where n(µ, T = 0, h = 0) steps more than one, since in this parameter regions exotic phenomena as phase separation are to expect in extended models. For the cases where these lines end in triple points, i.e. groundstates of three different occupation numbers are degenerated, we give the related parameter values. Regarding the influence of the nn-exchange and the nn-Coulomb correlation onto the anomalous degeneration we find both lifting and inducing of degeneracies in dependence on the parameter values.

Hubbard pair cluster in the external fields. Studies of the magnetic properties

“…For instance, the chemical potential µ for x = 1, found in [64], amounts to µ = U/2 and is the same as for the infinite system [15,52]. Another general property is the electron-hole symmetry with respect to the half-filling case (x = 1) [21,52], which results here in the symmetric form of Figs. 7 and 12.…”

“…3) has 15 Accepted manuscript. The final version was published in: Physica A: Statistical Mechanics and its Applications 499, 395-406 (2018), DOI:10.1016/j.physa.2018.02.017 also been reported both for the infinite systems [52], and for the finite (8 × 8) 2D lattice [10], as well as for the cubic cluster [21].…”

“…However, the spontaneous magnetic ordering, possible for infinite systems, has not been found here for the paircluster, which remains paramagnetic when the external fields are absent. Regarding the influence of the external field, it should be mentioned that the critical magnetic field, when the zero-spin ground state breaks down, has also been found for the cubic cluster [21], where the exact and rigorous calculations have been performed for the extended Hubbard model.…”

Hubbard pair cluster in the external fields. Studies of the chemical potential

“…The influence of temperature, Hubbard on-site Coulombic energy U and electron concentration on the chemical potential is investigated and illustrated in figures. In particular, a discontinuous behaviour of the chemical potential (or electron concentration) in the ground state is discussed.Quantum Monte Carlo (QMC) methods [18,42] and Dynamical Cluster Approximation (DCA) [20] deserve particular attention.The possible extensions of the domain where the exact solutions can be found for the model include the zero-dimensional, cluster systems (see for example [22,23]) as well as clusters embedded in the environment of localized spins [24][25][26][27]. It is worthy to mention that the studies of such geometrically confined, cluster systems are important from the point of view of nanophysics and nanotechnology.…”

“…It is worthy to mention that the studies of such geometrically confined, cluster systems are important from the point of view of nanophysics and nanotechnology. In particular, it has been shown in [22,23] that for some small number of electrons the analytic solutions can be obtained in addition to the numerical calculations. The magnetic field has been taken into account in [22], however, the influence of the external electric field on the cluster properties has not been studied there.Motivated by such a possibility, we undertake the exact study of the Hubbard simplest cluster, namely the pair (or dimer), embedded simultaneously in the external magnetic and electric fields.…”