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Progress of Theoretical Physics

Low-Temperature Specific-Heat of One-Dimensional Hubbard Model

Minoru Takahashi

Published Jul 1, 1974• 79 citations
Publisher

Abstract

Low-temperature specific heat per site (C) of one-dimensional Hubbard model is investigated by the method of non-linear integral equations. For the half-filled case we show lim H →0 lim T →0C / T = πI0( π/ 2 U )/( 6 I1(π/ 2 U ) ), where T is temperature, H is magnetic field, U is the coupling constant, and I0 and I1 are modified Bessel functions. Although this equation yields lim T, H →0C / T = π/ 6 in the limit U →0+, the true value of lim T, H →0C / T at U = 0 is π/ 3. This means that lim T, H →0C / T is a discontinuous function of U at U = 0. This discontinuity disappears when the band is not half filled.

DOI
10.1143/ptp.52.103

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Low-temperature specific heat per site (C) of one-dimensional Hubbard model is investigated by the method of non-linear integral equations. For the half-filled case we show lim H →0 lim T →0C / T = πI0( π/ 2 U )/( 6 I1(π/ 2 U ) ), where T is temperature, H is magnetic field, U is the coupling constant, and I0 and I1 are modified Bessel functions. Although this equation yields lim T, H →0C / T = π/ 6 in the limit U →0+, the true value of lim T, H →0C / T at U = 0 is π/ 3. This means that lim T, H →0C / T is a discontinuous function of U at U = 0. This discontinuity disappears when the band is not half filled.
The Feynman path-integral quantum Monte Carlo (PI QMC) approach has been employed to study the specific heat C of the one-dimensional (1D) half-filled Hubbard chain as a function of temperature. For not too strong many-body interactions C exhibits a single maximum only. With increasing correlation strength the C curve shows an intermediate minimum separating a sharp low-temperature maximum and a broad high-temperature profile. The first peak is caused by spin excitations and the high-temperature values of C are determined by charge degrees of freedom. The contribution of both quantities to the net specific heat curve has been quantified in the present theoretical setup. The specific heat data are interpreted with the aid of calculated charge fluctuations, probabilities Pi(n) to find n = 2, 1, 0 electrons at the respective lattice sites, a spin density wave (SDW) order parameter and the electronic chemical potential μ.
We investigate the grand potential of the one-dimensional Hubbard model in the high-temperature limit, calculating the coefficients of the high-temperature expansion $(\ensuremath{\beta}$-expansion) of this function up to order ${\ensuremath{\beta}}^{4}$ by an alternative method. The results derived are analytical and do not involve any perturbation expansion in the hopping constant, being valid for arbitrary density of electrons in the one-dimensional model. In the half-filled case, we compare our analytical results for the specific heat and the magnetic susceptibility, in the high-temperature limit, with the ones obtained by Beni et al. [Phys. Rev. B 8, 3329 (1973)] and Takahashi's integral equations, showing that the latter result does not take into account the complete energy spectrum of the one-dimensional Hubbard model. The exact integral solution by J\"uttner et al. [Nucl. Phys. B 522, 471 (1998)] is applied to the determination of the range of validity of our expansion in $\ensuremath{\beta}$ in the half-filled case, for several different values of U.
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.
We have calculated the spin correlation length of the half-filled one-dimensional Hubbard model down to the temperature of T =0.01×(transfer integral) by the Bethe-Ansatz method for the quantum transfer matrix. Its temperature dependence shows the crossover from (2 log π T ) -1 at high temperatures to a T -1 with the log corrections at low temperatures, the low- T behavior being consistent with the prediction by the conformal field theory. When the Coulomb repulsion U is small, we have found at low temperatures another crossover in the proportional constant a from the value for the free electron system to that for the antiferromagnetic Heisenberg model. This crossover is due to the disappearance of charge fluctuation. A similar crossover has been observed for the linear coefficient of specific heat, C / T .
We have calculated the spin correlation length of the half-filled one-dimensional Hubbard model down to the temperature of T =0.01×(transfer integral) by the Bethe-Ansatz method for the quantum transfer matrix. Its temperature dependence shows the crossover from (2 log π T ) -1 at high temperatures to a T -1 with the log corrections at low temperatures, the low- T behavior being consistent with the prediction by the conformal field theory. When the Coulomb repulsion U is small, we have found at low temperatures another crossover in the proportional constant a from the value for the free electron system to that for the antiferromagnetic Heisenberg model. This crossover is due to the disappearance of charge fluctuation. A similar crossover has been observed for the linear coefficient of specific heat, C / T .