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

Abstract

The chemical potential of the two-site Hubbard cluster (pair) embedded in the external electric and magnetic fields is studied by exact diagonalization of the Hamiltonian. The formalism of the grand canonical ensemble is adopted. 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. 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. We note that such system has already attracted some attention [28][29][30][31][32][33][34][35]. Also the effect of the electric field on the properties of the Hubbard model was studied [36,37]. However, the simultaneous influence of both external electric and magnetic field on the Hubbard dimer was not discussed. It can be mentioned that the usefulness of such model can be related to hydrogen molecule [38] or several layered organic strongly correlated compounds [39]. We perform analytic diagonalization of the pair Hamiltonian and proceed using the formalism of the grand canonical ensemble, where the mean number of electrons in the system can vary and results from thermodynamic equilibrium conditions. This enables to obtain the grand thermodynamic potential as well as the average values of relevant operators. However, the statistical-thermodynamic calculations are possible provided the chemical potential is known. Therefore, as the first stage, we found it particularly important to calculate accurately the chemical potential for such an open system, interacting with the environment.In this paper we concentrate exclusively on the comprehensive calculations of the chemical potential for the Hubbard pair, especially we study its behaviour in the external magnetic and electric fields, in a wide range of temperatures. In particular, for low temperature range the quantum changes of the chemical potential are investigated. The influence of Hubbard energy U on the chemical potential...

Hubbard pair cluster with elastic interactions. Studies of thermal expansion, magnetostriction and electrostriction

“…It is worth noticing that by including the new term, Wn a n b , the complexity of diagonalization procedure will not increase in comparison to that presented in Ref. [41]. We note that n a n b is a diagonal 16 × 16 matrix and corresponds to 16 possible states.…”

“…For such extended Hamiltonian, the diagonalization procedure can be performed exactly using the method outlined in Ref. [41].…”

“…Therefore, the aim of the present paper is extension of the recent studies, basing on the formalism developed in Ref. [41], by including elastic interactions in the description of the Hubbard dimer. Such interactions are especially important since they are responsible for stability of the dimer structure and allow to account for the energy of thermal vibrations.…”

“…Among such systems, a two-atomic cluster (dimer) plays a special role, being the smallest nanosystem where the Hubbard model can be adopted, and the exact analytical solutions can be found. As a consequence, numerous aspects of physics of Hubbard dimers focused considerable attention in the literature [30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49]. These facts serve as a sound motivation for further comprehensive study of the two-atomic Hubbard system aimed at its full and exact thermodynamic description.…”

“…In line with this trend, recently the Hubbard pair cluster embedded simultaneously in the external magnetic and electric fields has been studied by the exact analytical methods in the framework of the grand canonical ensemble. In this approach the system is open, so that it is able to exchange the electrons with its environment [41][42][43]. The statistical and thermodynamic properties, both magnetic and electric ones, have been extensively studied there.…”