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 ${t}^{\ensuremath{'}}$.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 ${t}^{\ensuremath{'}}$.

Physical review. B, Solid stateAug 15, 19777 citations

Two-dimensional Hubbard model: Numerical simulation study

J. E. Hirsch

We have studied the two-dimensional Hubbard model on a square lattice with nearest-neighbor hopping. We first discuss the properties of the model within the mean-field approximation: Because of the form of the band structure, some peculiar features are found. We then discuss the simulation algorithm used and compare simulation results with exact results for 6-site chains to test the reliability of the approach. We present results for thermodynamic properties and correlation functions for lattices up to 8\ifmmode\times\else\texttimes\fi{}8 in spatial size. The system is found to be an antiferromagnetic insulator for all values of the coupling constant at zero temperature in the half-filled-band case, but the long-range order is much smaller than predicted by mean-field theory. We perform a finite-size-scaling analysis to determine the character of the transition at zero coupling. For non-half-filled-band cases, our results suggest that the system is always paramagnetic, in contradiction with Hartree-Fock predictions. The system does not show tendency to ferromagnetism nor triplet superconductivity in the parameter range studied. We also discuss some properties of the attractive Hubbard model in the half-filled-band case.We have studied the two-dimensional Hubbard model on a square lattice with nearest-neighbor hopping. We first discuss the properties of the model within the mean-field approximation: Because of the form of the band structure, some peculiar features are found. We then discuss the simulation algorithm used and compare simulation results with exact results for 6-site chains to test the reliability of the approach. We present results for thermodynamic properties and correlation functions for lattices up to 8\ifmmode\times\else\texttimes\fi{}8 in spatial size. The system is found to be an antiferromagnetic insulator for all values of the coupling constant at zero temperature in the half-filled-band case, but the long-range order is much smaller than predicted by mean-field theory. We perform a finite-size-scaling analysis to determine the character of the transition at zero coupling. For non-half-filled-band cases, our results suggest that the system is always paramagnetic, in contradiction with Hartree-Fock predictions. The system does not show tendency to ferromagnetism nor triplet superconductivity in the parameter range studied. We also discuss some properties of the attractive Hubbard model in the half-filled-band case.

Physical review. B, Condensed matterApr 1, 19851,082 citations

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 ${t}^{\ensuremath{'}}$.