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arXiv:1505.03364 · Acta Physica Polonica A

Some Exact Results for the Zero-Bandwidth Extended Hubbard Model with Intersite Charge and Magnetic Interactions

Konrad Jerzy Kapcia, Waldemar Kłobus, S. Robaszkiewicz

Published Feb 1, 2015• 5 citations
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Abstract

The extended Hubbard model in the zero-bandwidth limit is studied.The eective Hamiltonian consists of (i) on-site U interaction, (ii) intersite densitydensity interaction W , and (iii) Ising-like magnetic exchange interaction J between the nearest-neighbors.We present rigorous (and analytical) results obtained within the transfer-matrix method for 1D chain in two particular cases: (a) W = 0 and n = 1; (b) U → +∞ and n = 1/2 (W = 0, J = 0).We obtain the exact formulae for the partition functions which enables to calculate thermodynamic properties such as entropy, specic heat (c), and double occupancy per site.In both cases the system exhibits an interesting temperature dependence of c involving a characteristic two-peak structure.There are no phase transitions at nite temperatures and the only transitions occur in the ground state.

DOI
10.12693/aphyspola.127.284

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The extended Hubbard model in the zero-bandwidth limit is studied.The eective Hamiltonian consists of (i) on-site U interaction, (ii) intersite densitydensity interaction W , and (iii) Ising-like magnetic exchange interaction J between the nearest-neighbors.We present rigorous (and analytical) results obtained within the transfer-matrix method for 1D chain in two particular cases: (a) W = 0 and n = 1; (b) U → +∞ and n = 1/2 (W = 0, J = 0).We obtain the exact formulae for the partition functions which enables to calculate thermodynamic properties such as entropy, specic heat (c), and double occupancy per site.In both cases the system exhibits an interesting temperature dependence of c involving a characteristic two-peak structure.There are no phase transitions at nite temperatures and the only transitions occur in the ground state.
In this paper the two-dimensional extended Hubbard model with intersite magnetic Ising-like interaction in the atomic limit is analyzed by means of the classical Monte Carlo method in the grand canonical ensemble.Such an eective simple model could describe behavior of insulating (anti)ferromagnets.In the model considered the Coulomb interaction (U ) is on-site and the magnetic interactions in z-direction (J > 0, antiferromagnetic) are restricted to nearest-neighbors.Simulations of the model have been performed on a square lattice consisting of N = L × L = 400 sites (L = 20) in order to obtain the full phase diagram for U/(4J) = 1.Results obtained for on-site repulsion (U > 0) show that, apart from homogeneous non-ordered (NO) and ordered magnetic (antiferromagnetic, AF) phases, there is also a region of phase separation (PS: AF/NO) occurrence.We present a phase diagram as well as some thermodynamic properties of the model for the case of U/(4J) = 1 (and arbitrary chemical potential and arbitrary electron concentration).The AFNO transition can be second-order as well as rst-order and the tricritical point occurs on the diagram.
A one-dimensional two-band Hubbard model describing CuO chains is considered in the limit of zero bandwidth. Onsite and nearest-neighbor Coulomb interactions are included in the description. Various density correlation functions are calculated exactly at arbitrary concentration of holes. For increasing values of the intersite Coulomb interaction V, the system is driven towards both a charge-transfer instability and a phase separation instability when doped away from half filling. In the case of one hole added to the half-filled system, it is shown that a hole binds to a biexciton. Three-body bound states recently proposed to produce the marginal Fermi-liquid spectrum also occur naturally in the model, but a finite amount of doping away from half filling is necessary for them to form to an appreciable extent in the thermodynamic limit.
We present the exact solution of the one-dimensional extended Hubbard model in the atomic limit within the Green's function and equations of motion formalism. We provide a comprehensive and systematic analysis of the model by considering all the relevant response and correlation functions as well as thermodynamic quantities in the whole parameters space. At zero temperature we identify four phases in the plane (U,n) ( U is the on-site potential and n is the filling) and relative phase transitions as well as different types of charge ordering. These features are endorsed by investigating at T=0 the chemical potential and pertinent local correlators, the particle and double occupancy correlation functions, the entropy, and by studying the behavior in the limit T-->0 of the charge and spin susceptibilities. A detailed study of the thermodynamic quantities is also presented at finite temperature. This study evidences that a finite-range order persists for a wide range of the temperature, as shown by the behavior of the correlation functions and by the two-peak structure exhibited by the charge susceptibility and by the entropy. Moreover, the equations of motion formalism, together with the use of composite operators, allows us to exactly determine the set of elementary excitations. As a result, the density of states can be determined and a detailed analysis of the specific heat allows for identifying the excitations and for ascribing its two-peak structure to a redistribution of the charge density.
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The eigenvalue problem of the two-site and three-site extended Hubbard Hamiltonian, which contains the intraatomic repulsion UUU, the nearest-neighbor repulsion VVV, the isotropic exchange JJJ, the anisotropic exchange ${J}^{\ensuremath{'}}$, the hopping term ttt, the semihopping term SSS, and the double hopping term DDD, has been solved exactly and the grand partition function is obtained. The magnetic and thermodynamic quantities such as the magnetic susceptibility and the specific heat are calculated for arbitrary electron density. Several features believed to have wider validity have been deduced from this cluster model.
The recently introduced concept of exact mean-field theories is applied to the extended simplified Hubbard model, which is a one-band extended Hubbard model (including nearest-neighbor interaction) where only the \ensuremath{\downarrow} electrons can hop. An exact mean-field Hamiltonian is derived for the model on a Bethe lattice in the limit of a large coordination number (Z\ensuremath{\rightarrow}\ensuremath{\infty}). The mean-field Hamiltonian is used to calculate the Green functions and the free energy at half-filling. This exact solution is analyzed analytically at small and large values of the on-site (U) and nearest-neighbor (V) interaction strengths. One finds that a phase transition occurs at sufficiently low temperatures for all U,V>0. The low-temperature phase is a charge-density wave for V>1/2U and a spin-density wave for V1/2U. Special attention is paid to (i) the critical temperature as a function of U and V, (ii) the order parameter as a function of temperature, and (iii) the density of states. The density of states shows that the low-temperature phase is an insulator for all U,V provided that V/U\ensuremath{\ne}1/2, while the high-temperature phase is a paramagnetic metal for all V if U2, and a paramagnetic insulator if U>2. Many detailed results are given.