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Physical review. B, Solid state

Thermodynamic and correlation functions of half-filled-band Hubbard dimer, dimerized, and uniform Hubbard chains

Charles C. Chen

Published Aug 15, 1977• 7 citations
Publisher

Abstract

The thermodynamic and two-site correlation functions for the half-filled-band isolated Hubbard dimer, dimerized, and uniform Hubbard chains for different values of the Coulomb energy $U$ are evaluated in a unified way in the pair approximation of the cluster-variation method. The energy, specific heat, and correlation functions as functions of temperature for different ratios of $\frac{{t}^{\ensuremath{'}}}{t}$ ($t$, ${t}^{\ensuremath{'}}$ are intradimer and interdimer transfer integrals, respectively) and $\frac{U}{t}$ are obtained. Our solutions for the Hubbard dimer are exact and agree with those obtained from the partition function. Solutions for the dimerized and uniform Hubbard chains correspond to the Bethe approximations in which all two-site correlations are taken into account. It is observed that the chemical potential for the half-filled dimerized Hubbard Hamiltonian is exactly $\frac{U}{2}$ at all temperatures, independent of the transfer integrals $t$ or ${t}^{\ensuremath{'}}$.

DOI
10.1103/physrevb.16.1312

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The thermodynamic and two-site correlation functions for the half-filled-band isolated Hubbard dimer, dimerized, and uniform Hubbard chains for different values of the Coulomb energy UUU are evaluated in a unified way in the pair approximation of the cluster-variation method. The energy, specific heat, and correlation functions as functions of temperature for different ratios of t\ensuremath′t\frac{{t}^{\ensuremath{'}}}{t}tt\ensuremath′​ (ttt, t\ensuremath′{t}^{\ensuremath{'}}t\ensuremath′ are intradimer and interdimer transfer integrals, respectively) and Ut\frac{U}{t}tU​ are obtained. Our solutions for the Hubbard dimer are exact and agree with those obtained from the partition function. Solutions for the dimerized and uniform Hubbard chains correspond to the Bethe approximations in which all two-site correlations are taken into account. It is observed that the chemical potential for the half-filled dimerized Hubbard Hamiltonian is exactly U2\frac{U}{2}2U​ at all temperatures, independent of the transfer integrals ttt or t\ensuremath′{t}^{\ensuremath{'}}t\ensuremath′.
The thermodynamic and correlation functions for one- and three- dimensional systems described by the half-filled band Hubbard Hamiltonians are evaluated in the pair-approximation of cluster variation method. The energy, specific heat, magnetization, susceptibility and various equilibrium correlation functions are computed as functions of temperature as well as the external magnetic field for different values of coulomb energies. Our solutions correspond to Bethe approximation in which the relevant pair-correlation functions are included in our formulation. It is observed that the chemical potential for the half-filled band is exactly half of the coulomb energy at all temperatures, irrespective of the value of the applied magnetic field.
The dimerized-chain Hubbard model is characterized by a Coulomb energy UUU and two alternating transfer constants, an intradimer constant ttt, and a smaller interdimer constant ${t}^{\ensuremath{'}}$. We have used thermodynamic perturbation theory to treat the half-filled-band case, taking the interdimer transfer as the perturbation term. The unperturbed problem is a lattice of isolated dimers. We give the energy, specific heat, and magnetic susceptibility as functions of temperature for the unperturbed case and for the first nonvanishing order in the perturbation. The results are consistent with the U=0U=0U=0 case, which is solved exactly. The interdimer transfer ${t}^{\ensuremath{'}}$ has the effect of distorting the peaks in the specific heat and reducing the susceptibility; this is shown graphically.
Abstract The thermodynamics of the uniform and dimerized half‐filled Hubbard chains is investigated within an approximation which takes into account the correlations in a given cluster of sites and which is valid for all model parameters and for all temperatures. The dependence of the specific heat on temperature changes from a one‐peak to a two‐peak structure with increasing interaction strength. The influence of dimerization is discussed. The local magnetic moment is found to decrease with increasing temperature in the high‐temperature region. It is suggested that this qualitative behaviour is more general and independent on dimension and band filling.
In order to investigate the thermal properties (specific heat, magnetic susceptibility, entropy, internal energy, and some correlation functions) of the one-dimensional half-filled-band Hubbard model, we have studied linear chains and rings containing two to six atoms, by performing machine calculations. Supplementing the low-temperature behavior obtained from the exact solution for the infinite chain by Lieb and Wu, our results should be suggestive of the properties of the infinite system throughout the entire temperature domain. It is shown that when the ratio of the correlation energy UUU to the total width $\ensuremath{\Delta}$ of the band of single-particle excitations is larger than 1, the specific heat has two peaks. The high-temperature peak arises from the gradual metal-insulator transition (or the gradual formation of local moments), while the low-temperature peak is associated with the antiferromagnetic short-range ordering. When $\frac{U}{\ensuremath{\Delta}}$ becomes small, the two peaks merge into one. This picture is consistent with all other thermal properties, including correlation functions. The high-temperature properties are compared with the results predicted by Hubbard's approximate theory based on the truncation of the equations of motion of the Green's functions.
In order to determine finite-temperature properties of the infinite one-dimensional Hubbard model for the half-filled band, finite systems such as chains with 2 to 5 atoms and rings with 4 and 6 atoms are studied with the application of the grand canonical ensemble. The obtained temperature dependence of the specific heat, the entropy, the magnetic susceptibility and some correlation functions clearly shows a rearrangement of the electronic state of the system with the increase of the strength of the Coulomb interaction relative to the transfer integral. In the light of our new results the functional integral approach recently proposed by Kimball and Schrieffer as well as Hubbard's approximate theory is examined.