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physica status solidi (b)

Specific Heat of the Half-Filled One-Dimensional Hubbard Model: Feynman Path-Integral Simulations

J. Schulte, M.C. B�hm

Published Jan 1, 1997• 6 citations
Publisher

Abstract

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.

DOI
10.1002/1521-3951(199701)199:1<59::aid-pssb59>3.0.co;2-r

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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.
The specific heat C of the one-dimensional half-filled Hubbard model is studied by Feynman path-integral quantum Monte Carlo simulations as a function of temperature (T). The strength of the electronic correlations is modified by a variable intramolecular two-electron repulsion U. For weak correlations, C shows a single maximum. For intermediate correlations C has two maxima in the T interval studied. For strong correlations only one maximum and one minimum in C are found in the T interval considered. The influence of spin and electronic excitations on the T dependence of C are discussed. The C data are correlated with calculated probabilities of single occupancy, electronic charge fluctuations, and the electronic chemical potential. \textcopyright{} 1996 The American Physical Society.
Quantum Monte Carlo results for the specific heat c of the two-dimensional Hubbard model are presented. At half filling it was observed that c\ensuremath{\sim}T2{\mathrm{T}}^{2}T2 at very low-temperatures. Two distinct features were also identified: a low-temperature peak related to the spin degrees of freedom and a higher-temperature broad peak related to the charge degrees of freedom. Away from half filling the spin-induced feature slowly disappears as a function of hole doping while the charge feature moves to lower temperature. A comparison with experimental results for the high-temperature cuprates is discussed.
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 .
Feynman path-integral quantum Monte Carlo (QMC) simulations and an analytic many-body approach are used to study the ground state properties of one-dimensional (1D) chains in the theoretical framework of model Hamiltonians of the Hubbard type. The QMC algorithm is employed to derive position-space quantities, while band structure properties are evaluated by combining QMC data with expressions derived in momentum (k) space. Bridging link between both representations is the quasi-chemical approximation (QCA). Electronic charge fluctuations <(Δn 2 i )> and the fluctuations of the magnetic local moments <(Δs 2 i )> are studied as a function of the on-site density <n i > and correlation strength, which is given by the ratio between two-electron interaction and kinetic hopping. Caused by the non-analytic behaviour of the chemical potential μ = ∂E/∂<n i > (with E denoting the electronic energy), strict 1D systems with an on-site density <n i > of 1·0 do not exhibit the properties of a conductor for any non-zero interaction beyond the mean-field approximation. The QMC simulations lead to straightforward access to the probabilities P i (n) of finding n = 0, 1, 2 electrons at the ith lattice site. The P i (n) elements allow to calculate the enhancement factors on the electron spin susceptibility χ, effective electronic mass m* and Knight shift κ. m* is enhanced by a bandwidth renormalization factor D -1 0, κ by an element ηK mapping the additional localization of the correlated electrons in the presence of an external magnetic field B and χ by the product D -1 0 ηK. Available experimental data are discussed in the light of the present theoretical findings.
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.