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arXiv:1610.04605 · Acta Physica Polonica A

Exact Diagonalization Study of an Extended Hubbard Model for a Cubic Cluster at Quarter Filling

Karol Szałowski, T. Balcerzak, M. Jaščur, A. Bobák, Milan ˇ Zukovič

Published Apr 1, 2017• 4 citations
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Abstract

In the paper the thermodynamics of a cubic cluster with 8 sites at quarter filling is characterized by means of exact diagonalization technique. Particular emphasis is put on the behaviour of such response functions as specific heat and magnetic susceptibility. The system is modelled with extended Hubbard model which includes electron hopping between both first and second nearest neighbours as well as coulombic interactions, both on-site and between nearest-neighbour sites. The importance of hopping between second nearest neighbours and coulombic interactions between nearest neighbours for the temperature dependences of thermodynamic response functions is analysed. In particular, the predictions of Schottky model are compared with the calculations based on the full energy spectrum.

DOI
10.12693/aphyspola.131.1012

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In the paper the thermodynamics of a cubic cluster with 8 sites at quarter filling is characterized by means of exact diagonalization technique. Particular emphasis is put on the behaviour of such response functions as specific heat and magnetic susceptibility. The system is modelled with extended Hubbard model which includes electron hopping between both first and second nearest neighbours as well as coulombic interactions, both on-site and between nearest-neighbour sites. The importance of hopping between second nearest neighbours and coulombic interactions between nearest neighbours for the temperature dependences of thermodynamic response functions is analysed. In particular, the predictions of Schottky model are compared with the calculations based on the full energy spectrum.
We study the thermodynamics of a zero-dimensional, cubic cluster described with a Hubbard Hamiltonian, focusing our interest on the magnetic properties. The range in which the studied cluster is paramagnetic is considered. The results are obtained by means of exact numerical diagonalization. Such thermodynamic quantities as entropy, specific heat, magnetic susceptibility, spin-spin correlations and double occupancy are discussed. Particular emphasis is put on the behaviour of local maxima of specific heat and susceptibility, which are analysed in terms of Schottky anomalies.
The ground-state and thermodynamic properties of a simple Hubbard Hamiltonian are studied for an eight-site cubic cluster as functions of the number of electrons and the strength of the electron interaction. The ground-state energy and spin are determined, and the excitation spectrum is computed. Thermodynamic properties including the specific heat, the magnetic susceptibility, and spin correlation functions are calculated. Computations are performed both by explicit diagonalization of the Hamiltonian and by Monte Carlo simulation.
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 fermi linearization scheme for a two-site cluster (dimer) is applied to the extended Hubbard model, including Coulomb interaction between nearest neighbour sites. The partition function and the order parameter, as well as the average double-occupation and local magnetic moment are evaluated. The theory is checked by extended numerical analysis, performed here in the homogeneous case, when pairing is zero, and deriving the explicit temperature and filling dependence of the above quantities. An interesting feature that emerges from such calculations is that the mean-field parameter characteristic of fermi-linearization is capable of describing two different phases, one of which is not present in the conventional Hubbard model. The order parameters corresponding to the two phases are explicitly derived. The novel phase can be ascribed to the n.n. Coulomb interaction, the role of whose strength V is analysed.
By use of the conservation laws a four-site Hubbard model coupled to a particle bath within an external magnetic field in z-direction was diagonalized. The analytical dependence of both the eigenvalues and the eigenstates on the interaction strength, the chemical potential and magnetic field was calculated. It is demonstrated that the low temperature behaviour is determined by a delicate interplay between many-particle states differing in electron number and spin if the electron density is away from half-filling. The grand partition sum is calculated and the specific heat, the susceptibility as well as various correlation functions and spectral functions are given in dependence of the interaction strength, the electron occupation and the applied magnetic field. For both the grand canonical and the canonical ensemble the high-temperature crossing points of the specific heat are calculated. Whereas in the weak correlation regime the universal value calculated by second order perturbation theory for several Hubbard systems being in the thermodynamic limit is confirmed, these crossing points vanish for intermediate to strong correlation.
With the aim of contributing to a better understanding of the finite-temperature properties of low-dimensional strongly correlated systems we study a one-dimensional half-filled-band extended Hubbard model with a term of exchange interaction between neighboring sites. We analyze the temperature dependence of the specific heat, internal energy, entropy, number of doubly occupied sites, magnetic susceptibility, spin-spin correlation functions, and also investigate a detailed diagram of magnetic correlations. We use the method of small-cluster exact diagonalization calculations with the application of the grand canonical ensemble and of extrapolation techniques to the infinite chain. Our results show that the examined model exhibits a rich structure of magnetic properties due to the competition between the onsite Coulombian repulsion and the exchange interaction.