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arXiv:cond-mat/9610142 · Physical review. B, Condensed matter

Energy level statistics of the two-dimensional Hubbard model at low filling

Henrik Bruus, J C Anglès d'Auriac

Published Apr 1, 1997• 43 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.55.9142

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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 \ensuremathΣ2{\mathrm{\ensuremath{\Sigma}}}^{2}\ensuremathΣ2, and the spectral rigidity \ensuremathΔ3{\mathrm{\ensuremath{\Delta}}}_{3}\ensuremathΔ3​, 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.
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 .
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.
The Hubbard model has been widely used to describe magnetism due to itinerant electrons. However, except for the one-dimensional case, no exact solution of the model exists. We have studied the two-dimensional Hubbard model on a square lattice using a Monte Carlo simulation technique. Simulations were performed on lattices of up to 8×8 in spatial size. We will discuss results for thermodynamic properties and correlation functions as a function of temperature and interaction strength. For the half-filled band case, our results indicate that the system exhibits long-range antiferromagnetic order in the ground state for any nonzero interaction, although substantially reduced from the classical Néel state due to both charge and spin fluctuations. For the non-half-filled band case, our results suggest that no long-range magnetic order exists. We also don’t find any indication of short-ranged ferromagnetic correlations in the parameter range studied. We have also studied the predictions of various approximate treatments for this model, particularly the static approximation and the coherent potential approximation (CPA). By comparison with the Monte Carlo results, we discuss the validity of these schemes. Finally, we discuss the effect of changing the band structure on the magnetic properties of the system.
Abstract The statistical properties of level spacings provide valuable insights into the dynamical properties of a many-body quantum systems. We investigate the level statistics of the Fermi–Hubbard model with dimerized hopping amplitude and find that after taking into account translation, reflection, spin and η pairing symmetries to isolate irreducible blocks of the Hamiltonian, the level spacings in the limit of large system sizes follow the distribution expected for hermitian random matrices from the Gaussian orthogonal ensemble. We show this by analyzing the distribution of the ratios of consecutive level spacings in this system, its cumulative distribution and quantify the deviations of the distributions using their mean, standard deviation and skewness.
Using exact diagonalization, we calculate the density of states of the two-dimensional Hubbard model on a 4\ifmmode\times\else\texttimes\fi{}4 square lattice at U/t=0.5, 4, and 10, and even number of electrons with filling factors n ranging from a quarter filling up to half filling. We compare the ground-state energy and density of states at U/t=0.5 and 4 with second-order perturbation theory in U/t in the paramagnetic phase, and find that while the agreement is reasonable at U/t=0.5, it becomes worse as the perturbatively determined (i.e., using Stoner's criterion) boundary of the paramagnetic to spin-density-wave instability is approached. In the strong coupling regime (U/t=10), we find reasonable agreement between the density of states of the Hubbard and the t-J model especially for low doping fractions. In general, we find that at half filling the filled states are separated from the empty states by a gap. At U/t=10, the density of states shows two bands clearly separated by a Mott-Hubbard gap of order U.
We present exact explicit analytical results describing the exact ground state of four electrons in a two-dimensional square Hubbard cluster containing 16 sites taken with periodic boundary conditions. The presented procedure, which works for arbitrary even particle number and lattice sites, is based on explicitly given symmetry adapted base vectors constructed in r space. The Hamiltonian acting on these states generates a closed system of 85 linear equations, providing by its minimum eigenvalue, the exact ground state of the system. The presented results, described with the aim to generate further creative developments, not only show how the ground state can be exactly obtained and what kind of contributions enter in its construction, but also emphasize further characteristics of the spectrum. On this line (i) possible explications are found regarding why weak coupling expansions often provide a good approximation for the Hubbard model at intermediate couplings, or (ii) explicitly given low-lying energy states of the kinetic energy, avoiding double occupancy, suggest new roots for pairing mechanism attracting decrease in the kinetic energy, as emphasized by kinetic energy driven superconductivity theories.