Cluster-variation method applied in two-site approximation to the Hubbard model at high temperatures

In order to study the origin of metallic magnetism and the metal-insulator transition (Mott transition), the cluster variation method is applied in a two-site approximation to the Hubbard Hamiltonian ${H}_{\mathrm{Hubbard}}=\ensuremath{\Sigma}{i=1}^{\mathfrak{N}}\ensuremath{\Sigma}{j=1}^{\mathfrak{N}}\ensuremath{\Sigma}{\ensuremath{\sigma}}^{}{t}_{\mathrm{ij}}{c}_{i\ensuremath{\sigma}}^{\ifmmode\dagger\else\textdagger\fi{}}{c}_{j\ensuremath{\sigma}}+U\ensuremath{\Sigma}{i=1}^{\mathfrak{N}}{n}_{i\ensuremath{\uparrow}}{n}_{i\ensuremath{\downarrow}}$, where ${c}_{i\ensuremath{\sigma}}^{\ifmmode\dagger\else\textdagger\fi{}}$, ${c}_{i\ensuremath{\sigma}}$ are, respectively, the creation and destruction operators of an electron with spin $\ensuremath{\sigma}$ (\ensuremath{\uparrow} or \ensuremath{\downarrow}) on lattice site $i$, ${n}_{i\ensuremath{\sigma}}={c}_{i\ensuremath{\sigma}}^{\ifmmode\dagger\else\textdagger\fi{}}{c}_{i\ensuremath{\sigma}}$, $Ug0$ is the strength of the intrasite Coulomb repulsion between electrons having antiparallel spins, $\mathfrak{N}$ is the total number of sites, and ${t}_{\mathrm{ij}}$ is the "hopping" strength. With the restriction of only nearest-neighbor electron hoppings of strength $\ensuremath{-}t (tg0)$, and considering the half-filled-band case, both one- and three-dimensional (simple cubic lattice) results are calculated numerically. This is achieved by solving the basic equilibrium equations for the following two-site expectation values: ${〈{n}_{i\ensuremath{\uparrow}}{n}_{i\ensuremath{\downarrow}}〉}_{0}$, ${〈{n}_{i\ensuremath{\uparrow}}{n}_{j\ensuremath{\uparrow}}〉}_{0}$, ${〈{n}_{i\ensuremath{\uparrow}}{n}_{j\ensuremath{\downarrow}}〉}_{0}$, ${〈{n}_{i\ensuremath{\uparrow}}{n}_{i\ensuremath{\downarrow}}{n}_{j\ensuremath{\uparrow}}〉}_{0}$, ${〈{n}_{i\ensuremath{\uparrow}}{n}_{i\ensuremath{\downarrow}}{n}_{j\ensuremath{\uparrow}}{n}_{j\ensuremath{\downarrow}}〉}_{0}$, ${〈{c}_{i\ensuremath{\uparrow}}^{\ifmmode\dagger\else\textdagger\fi{}}{c}_{j\ensuremath{\uparrow}}〉}_{0}$, ${〈{c}_{i\ensuremath{\uparrow}}^{\ifmmode\dagger\else\textdagger\fi{}}{c}_{j\ensuremath{\uparrow}}{n}_{i\ensuremath{\downarrow}}〉}_{0}$, ${〈{c}_{i\ensuremath{\uparrow}}^{\ifmmode\dagger\else\textdagger\fi{}}{c}_{j\ensuremath{\uparrow}}{n}_{i\ensuremath{\downarrow}}{n}_{j\ensuremath{\downarrow}}〉}_{0}$, ${〈{c}_{i\ensuremath{\uparrow}}^{\ifmmode\dagger\else\textdagger\fi{}}{c}_{i\ensuremath{\downarrow}}^{\ifmmode\dagger\else\textdagger\fi{}}{c}_{j\ensuremath{\downarrow}}{c}_{j\ensuremath{\uparrow}}〉}_{0}$. These expectation values can in turn be used to compute various thermodynamic quantities, e.g., internal energy, entropy, specific heat, etc. For sufficiently large parametric value of $\frac{U}{t}$, a high-temperature maximum in the specific heat is resolved and is identified as an indication of a gradual metal-insulator transition. For the simple cubic structure, this high-temperature peak disappears, however, as one decreases the value of $\frac{U}{t}$ to values around or below 15. Also, correlation results strongly suggest that the three-dimensional half-filled-band Hubbard model admits an antiferromagnetic ground state.