Charge and magnetic states for the Hubbard model on a three-site cluster

Abstract

The Hubbard model for small size clusters is investigated. Using direct diagonalization of the small cluster Hamiltonian, electronic, magnetic and spin susceptibilities are calculated and their behavior at low temperature is investigated. The temperature behavior of susceptibilities for two- and three-site clusters is investigated for both center and edge of the Brillouin zone. If at the given concentration the ground state is polarized, the susceptibilities diverge following the Curie law and the system is transformed into the magnetically ordered state at T = 0. Compounds with transition- and rare-earth metals (oxides, suldes and others) possess unique physical properties and attract great attention in the elds of practical applications and general-theoretical study. Re- garding the magnetic properties of such compounds, they are ferromagnetic, ferrimagnetic, and antiferromagnetic. The peculiarity of some of them is a possibility of mag- netic transitions which are followed by a change of the magnetic-order type under the external inuence. Ac- cording to the electrical properties, transition- and rare- earth metal compounds may be divided into four groups: insulators, metals, compounds which possess a possibility of a metalinsulator transition (caused by external fac- tors) with a simultaneous change of the magnetic prop- erties and magnetic-disordered compounds in which a metal-insulator transition takes place too. But at the same time, the mechanisms of exchange interactions at the metalinsulator transition in such compounds have been studied insucien tly. This is connected with strong electron correlation. In this respect there is a need for the investigation of physics of such phenomena as ferromag- netism, antiferromagnetism, etc. by using the Hubbard model. The Hubbard model (1 3) was originally proposed for the description of correlations in narrow-band materi- als on a three-dimensional lattice. It takes into account the main system characteristics, namely electron hopping and Coulomb interaction. Its two-dimensional version is often considered as the minimal model for describing the copper oxide planes in high-Tc superconductors (4, 5). The one-dimensional Hubbard model has an exact solu- tion in terms of the Bethe ansatz (6). It displays Luttinger liquid and Mott insulator phases and has received much attention (7). In innite dimensions other exact solutions for the Hubbard model were obtained within Dynamical Mean Field Theory (8). In other dimensions, for which there are no exact so- lutions, a great variety of approximate treatments have been proposed in order to accommodate a suitable theo- retical framework. In this connection, many eorts were exerted to obtain exact ground states of few electrons for the Hubbard model on a nite size cluster (see, Ref. (9) and references therein) from which the ground state ener- gy for the low-dimensional cases can be estimated using a concept of dimensional scaling (10). The latest period has provided new motivations for further investigations into the systems containing a small number of particles conned in a device or unit as, for ex- ample, in the case of quantum dots, quantum well struc- tures, mesoscopic systems, experimental entanglement, etc., for which the knowledge of the ground state does not suce. The consideration of the excited states is a much more complicated problem. It has been studied in detail only for the two-site (11 13) and four-site (14) clus- ters. In Ref. (15) linear chains and rings containing two to six atoms were studied numerically and it was found that with the increase of the cluster size the thermodynamic properties of the model at half-lling approach its one- dimensional limit whereas magnetic susceptibility shows a clear even-odd eect in the very low temperature re- gion.