The entire energy spectrum is calculated numerically for the two-dimensional Hubbard model on square lattices with four electrons varying the interaction strength U in a wide range. Taking all symmetries explicitly into account a level statistical analysis is performed on this strongly correlated 2D system. The level spacing distribution and the number variance exhibit a random matrix (GOE) behavior for all positive values of U , the latter however only up to a medium energy scale λ* ≈ 3 independent of U .
The entire energy spectrum is calculated numerically for the two-dimensional Hubbard model on square lattices with four electrons varying the interaction strength U in a wide range. Taking all symmetries explicitly into account a level statistical analysis is performed on this strongly correlated 2D system. The level spacing distribution and the number variance exhibit a random matrix (GOE) behavior for all positive values of U , the latter however only up to a medium energy scale λ* ≈ 3 independent of U .
The energy level statistics of the Hubbard model for L\ifmmode\times\else\texttimes\fi{}L square lattices (L=3,4,5,6) at low filling (four electrons) is studied numerically for a wide range of coupling strength. All known symmetries of the model (space, spin, and pseudospin symmetry) have been taken into account explicitly from the beginning of the calculation by projecting into symmetry-invariant subspaces. The details of this group theoretical treatment are presented with special attention to the nongeneric case of L=4, where a particular complicated space group appears. For all the lattices studied, a significant amount of levels within each symmetry invariant subspaces remains degenerated, but except for L=4 the ground state is nondegenerate. We explain the remaining degeneracies, which occur only for very specific interaction-independent states, and we disregard these states in the statistical spectral analysis. The intricate structure of the Hubbard spectra necessitates a careful unfolding procedure, which is thoroughly discussed. Finally, we present our results for the level spacing distribution, the number variance ${\mathrm{\ensuremath{\Sigma}}}^{2}$, and the spectral rigidity ${\mathrm{\ensuremath{\Delta}}}_{3}$, which essentially all are close to the corresponding statistics for random matrices of the Gaussian ensemble independent of the lattice size and the coupling strength. Even very small coupling strengths approaching the integrable zero coupling limit lead to Gaussian ensemble statistics, stressing the nonperturbative nature of the Hubbard model.
A density-functional theory for many-body lattice models is considered in which the single-particle density matrix ${\ensuremath{\gamma}}_{\mathrm{ij}}$ is the basic variable. Eigenvalue equations are derived for solving Levy's constrained search of the interaction energy functional $W[{\ensuremath{\gamma}}_{\mathrm{ij}}].$ $W[{\ensuremath{\gamma}}_{\mathrm{ij}}]$ is expressed as the sum of Hartree-Fock energy ${E}_{\mathrm{HF}}[{\ensuremath{\gamma}}_{\mathrm{ij}}]$ and the correlation energy ${E}_{C}[{\ensuremath{\gamma}}_{\mathrm{ij}}].$ Exact results are obtained for ${E}_{C}({\ensuremath{\gamma}}_{12})$ of the Hubbard model on various periodic lattices, where ${\ensuremath{\gamma}}_{\mathrm{ij}}={\ensuremath{\gamma}}_{12}$ for all nearest neighbors i and j. The functional dependence of ${E}_{C}({\ensuremath{\gamma}}_{12})$ is analyzed by varying the number of sites Na, band filling Ne, and lattice structure. The infinite one-dimensional chain and one-, two-, or three-dimensional finite clusters with periodic boundary conditions are considered. The properties of ${E}_{C}({\ensuremath{\gamma}}_{12})$ are discussed in the limits of weak $({\ensuremath{\gamma}}_{12}\ensuremath{\simeq}{\ensuremath{\gamma}}_{12}^{0})$ and strong $({\ensuremath{\gamma}}_{12}\ensuremath{\simeq}{\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}})$ electronic correlations, and in the crossover region $({\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}}<~{\ensuremath{\gamma}}_{12}<~{\ensuremath{\gamma}}_{12}^{0}).$ Using an appropriate scaling we observe that ${\ensuremath{\varepsilon}}_{C}{(g}_{12}{)=E}_{C}{/E}_{\mathrm{HF}}$ has a pseudo-universal behavior as a function of ${g}_{12}=({\ensuremath{\gamma}}_{12}\ensuremath{-}{\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}})/({\ensuremath{\gamma}}_{12}^{0}\ensuremath{-}{\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}}).$ The fact that ${\ensuremath{\varepsilon}}_{C}{(g}_{12})$ depends weakly on Na,Ne, and lattice structure suggests that the correlation energy of extended systems could be obtained quite accurately from finite-cluster calculations. Finally, the behaviors of ${E}_{C}({\ensuremath{\gamma}}_{12})$ for repulsive (U>0) and attractive (U<0) interactions are contrasted.
In this work, we developed the unit step model as an approximate solution to the single-band Hubbard Hamiltonian to solve variationally strongly correlated interacting elections on a two-dimensional (2D) square lattice. We also showed primarily how to derive possible electronic states available for several 2D N x N square lattices, although, with special emphasis on a 2D 5 x 5 square lattice. The results emerging from our present study was compared with the results of Gutzwiller variational approach (GVA) and correlated variational approach (CVA), at the large limit of the Coulomb interaction strength (U/4t).The approximation to the Hubbard Hamiltonian study is actually necessary because of the strong limitation and difficulty pose by the Hubbard Hamiltonian as we move away from finite - size lattices to larger N - dimensional lattices. Thus this work has provided a means of overcoming the finite - size lattice defects as we pass on to a higher dimension. We have shown in this study, that the repulsive Coulomb interaction which in part leads to the strong electronic correlations, would indicate that the two electron system prefer not to condense into s-wave superconducting singlet state (s = 0), at high positive values of the interaction strength. This study reveals that when the Coulomb interaction is zero, that is, for free electron system (non-interacting), the variational parameters which describe the probability distribution of lattice electron system is the same. The spectra intensity increases with increase in the interaction strength and it decreases to zero when the interaction strength is made negatively large. Keywords : unit step Hamiltonian, Hubbard Hamiltonian, 3D cubic lattice, interaction strength, total energy, lattice separation.
Abstract We investigate the ground state properties and the energy level statistics of a two-dimensional 3 × 3 square lattice cluster of simple and extended Hubbard model in parameters space − 10 ≤ U, V ≤ 10 for various electron configurations (N e = 10, 8, 6 and 4) within the spin subspace S = 0 using exact diagonalization. In simple Hubbard model, we observe transition from a behavior where electrons remain correlated to lattice sites to electron-paired state as U is varied from positive to negative. In case of extended Hubbard model, we observe tendency for SDW ordering (enhanced by negative V) and CDW ordering (favored by positive V), modulated by electron density. The spectral properties of energy level statistics, analyzed using kernel density estimation, reveal a complex interplay between level attraction and level repulsion, highlighting the tunability of these properties through interaction parameters.
We investigate the ground state (T = 0 K) of the one-dimensional symmetrical (n = 1) Hubbard model formalized in terms of the system of integral equations, which we previously obtained using the method of the generating functional of Green’s functions with the subsequent Legendre transformation. In a wide range of variations in the parameter of Coulomb interaction U, the following characteristics of the system have been calculated: the electron density of states, the electron band spectrum, the number of doubly occupied lattice sites, the localized magnetic moment, the correlator of the square of the longitudinal component of spin at a site,<S Z 2>, and the internal energy of the system. It has been shown that, for all U > 0, the model yields two solutions, i.e., an antiferromagnetic insulator and a paramagnetic insulator, in which there are no single-electron quasi-particles at the Fermi level. The energy of the paramagnetic solution in the region of U < 1.1 is considerably less than that of the antiferromagnetic solution for the case of U > 1.1, we have the opposite situation.