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Physical review. B, Condensed matter

Magnetic phase diagram of the half-filled Hubbard model for a simple cubic lattice

Y. Kakehashi, Hideo Hasegawa

Published Sep 1, 1987• 23 citations
Publisher

Abstract

We have calculated the $U\ensuremath{-}T$ phase diagram of the half-filled simple-cubic Hubbard model ($U$: interaction, $T$: temperature) by making use of the two interpolation theories, the single-site spin-fluctuation (SSF) theory proposed by Hubbard and Hasegawa and the Gutzwiller-type variational approach (VA) by Kakehashi and Fulde. The results are compared with those of the recent Monte Carlo (MC) calculations made by Hirsch for a three-dimensional 4\ifmmode\times\else\texttimes\fi{}4\ifmmode\times\else\texttimes\fi{}4 lattice. The N\'eel temperatures ${T}_{N}$ calculated with the MC method are about 70% higher than those calculated with the mean-field-type theories of the SSF and VA though the $U$ dependence of the amplitudes of the local moments and charge fluctuations agrees well with our results. This suggests an overestimation of ${T}_{N}$ up to 100% in Hirsch's calculations, whose conceivable origins are discussed. We present also the result of the internal energy as a function of $T$ and $U$ calculated using the SSF and VA.

DOI
10.1103/physrevb.36.4066

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We have calculated the U\ensuremath−TU\ensuremath{-}TU\ensuremath−T phase diagram of the half-filled simple-cubic Hubbard model (UUU: interaction, TTT: temperature) by making use of the two interpolation theories, the single-site spin-fluctuation (SSF) theory proposed by Hubbard and Hasegawa and the Gutzwiller-type variational approach (VA) by Kakehashi and Fulde. The results are compared with those of the recent Monte Carlo (MC) calculations made by Hirsch for a three-dimensional 4\ifmmode\times\else\texttimes\fi{}4\ifmmode\times\else\texttimes\fi{}4 lattice. The N'eel temperatures TN{T}_{N}TN​ calculated with the MC method are about 70% higher than those calculated with the mean-field-type theories of the SSF and VA though the UUU dependence of the amplitudes of the local moments and charge fluctuations agrees well with our results. This suggests an overestimation of TN{T}_{N}TN​ up to 100% in Hirsch's calculations, whose conceivable origins are discussed. We present also the result of the internal energy as a function of TTT and UUU calculated using the SSF and VA.
Magnetic and thermodynamical properties of the half-filled Hubbard model on the simple-cubic lattice have been investigated by using the two analytical theories for band magnetism: the single-site spin fluctuation (SSF) theory proposed independently by Hubbard and Hasegawa and the Gutzwiller-type variational approach (VA) developed by Kakehashi and Fulde. We have calculated, as a function of the temperature T and the electron-electron interaction U, the sublattice magnetization, amplitude of local moments, susceptibilities, energy, and entropy, from which the U-T phase diagram is obtained. Our ground-state results, particularly of the VA, are in good agreement with those obtained in the variational Monte Carlo calculations performed by Yokoyama and Shiba. Our finite-temperature calculations are compared with those in the recent Monte Carlo (MC) simulations made by Hirsch. It is shown that, as far as local quantities such as the amplitude of local moments and the Curie constants are concerned, our results are consistent with the MC results. It is realized, however, that the antiferromagnetic correlation is much overestimated in a small 4\ifmmode\times\else\texttimes\fi{}4\ifmmode\times\else\texttimes\fi{}4 cluster in Hirsch's MC simulation, yielding N\'eel temperatures, TN{T}_{N}TN​, about 70% higher than those in the mean-field-type theories of the SSF and VA approaches.
Magnetic properties of the half-filled Hubbard model on three-dimensional bipartite lattices have been studied by using the linked-cluster series expansion method. The series for the free energy and the staggered susceptibility have been calculated as a function of \ensuremath{\mathrm{B}}t for three-dimensional bipartite lattices for an arbitrary value of the Coulomb interaction U. We obtain a transition from a paramagnetic to an antiferromagnetic U/t-TN{\mathrm{T}}_{\mathrm{N}}TN​ /t phase diagram from the divergence of the staggered susceptibility series. Comparisons with Monte Carlo simulation, Onsager reaction field result of an effective spin Hamiltonian, and large-U Heisenberg high-temperature solutions are made.
In this work we obtain the magnetic and thermodynamic properties of the classical Hubbard model. This one is easier to handle than the original Hubbard model, and it maintains fundamental aspects like charge and particle-hole symmetry. We find the relationship between this and the Ising model. For square and simple cubic lattices, the thermodynamic properties and phase diagrams were calculated by Monte Carlo method. We show the behavior of internal energy, specific heat, and magnetic susceptibility as a function of the temperature. For the considered lattices, the ground state is anti-ferromagnetic. The Néel temperatures were found as a function of the electronic interaction in the case of half-filled band. In the limit of vanishing of diverging interaction, our results agree with the Ising model.
As a continuation of a previous paper [J. Phys. Soc. Jpn. 56 (1987) 1490], the variational Monte-Carlo method is extended to include the antiferromagnetic long-range order. The theory is based on the Gutzwiller-type correlation factor and its effect is exactly taken into account by the Monte-Carlo procedure. An application is made to the half-filled-band case of one-dimensional lattice (50 sites), two-dimensional square lattice (up to 20×20 sites) and three-dimensional simple cubic lattice (6×6×6 sites). The result is qualitatively different from previous studies relying on the random-phase-type "Gutzwiller approximation." The variational energy for two and three dimensions is favorably compared with Hirsch's quantum Monte-Carlo data.
The antiferromagnetic Ne\'el temperature TN{T}_{N}TN​ is calculated as a function of the Coulomb interaction U for a half-filled 3D Hubbard model. Numerical results are given for the specific heat and staggered magnetic suspectibility on periodic lattices ranging from 43{4}^{3}43--103{10}^{3}103 sites, and used to determine TN{T}_{N}TN​. The size dependence of the results suggests that the transition is weakly first order at small to intermediate values of U.
In this work we obtain the magnetic and thermodynamic properties of the classical Hubbard model. This one is easier to handle than the original Hubbard model, and it maintains fundamental aspects like charge and particle-hole symmetry. We find the relationship between this and the Ising model. For square and simple cubic lattices, the thermodynamic properties and phase diagrams were calculated by Monte Carlo method. We show the behavior of internal energy, specific heat, and magnetic susceptibility as a function of the temperature. For the considered lattices, the ground state is anti-ferromagnetic. The Néel temperatures were found as a function of the electronic interaction in the case of half-filled band. In the limit of vanishing of diverging interaction, our results agree with the Ising model.