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Journal of the Physical Society of Japan

Analytical Solution of Scaling Equations in the Two-Dimensional Half-Filled Hubbard Model

Satoru Inagaki

Published Feb 15, 1997• 0 citations
Publisher

Abstract

We solve analytically the scaling equations in the half-filled Hubbard model on the two-dimensional square lattice. The equations are coupled differential equations for the vertex functions, which were derived independently by Schulz and Dzyaloshinskii to take the electron-correlation effects into account beyond the random phase approximation. Our analytical solution confirms the results so far obtained on the scaling equations.

DOI
10.1143/jpsj.66.408

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We solve analytically the scaling equations in the half-filled Hubbard model on the two-dimensional square lattice. The equations are coupled differential equations for the vertex functions, which were derived independently by Schulz and Dzyaloshinskii to take the electron-correlation effects into account beyond the random phase approximation. Our analytical solution confirms the results so far obtained on the scaling equations.
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In this paper we investigate effects of a lattice dimension on strongly correlated electronic systems at T=0KT=0 KT=0K. The model for numerical calculations is formalized in terms of the integral equations which were obtained previously for the half -- filled paramagnetic Hubbard model. The assumption that all relevant functions of the system have a magnitude dependence of momentus vectors makes ultimately possible treating a lattice dimension DDD as an external parameter. We here submit for consideration the number of doubly occupied lattice sites, the electronic density of states, and the momentum distribution in electronic occupation numbers at DDD=1, DDD=2, DDD=3
The ground-state energy E and momentum distribution n k of the electrons are expanded from the atomic limit for the half-filled Hubbard model. The coefficients of expansion are represented by the spin correlation functions of the spin 1 / 2 Heisenberg model. Using the spin-wave theory, approximate values of coefficients are calculated for the square lattice and the simple cubic lattice. In one dimension, the theory shows a good agreement with the exact solution. An effective spin Hamiltonian for the half-filled Hubbard model with arbitrary hopping integrals is obtained up to the fifth order. It is shown that the fourth term contains four spin interactions.
An exact analytical diagonalization is used to solve the two-dimensional extended Hubbard model (EHM) for a system with finite size. We have considered an EHM including on-site and off-site interactions with interaction energies U and V , respectively, for a square lattice containing 4 × 4 sites at one-eighth filling with periodic boundary conditions, recently treated by Kovacs and Gulacsi (2006 Phil. Mag. 86 2073). Taking into account the symmetric properties of this square lattice and using a translation linear operator, we have constructed a r-space basis only with 85 state-vectors which describe all possible distributions for four electrons in the 4 × 4 square lattice. The diagonalization of the 85 × 85 matrix energy allows us to study the local properties of the above system as a function of the on-site and off-site interactions energies, where we have shown that the off-site interaction encourages the existence of the double occupancies at the first excited state and induces a supplementary conductivity of the system.
We investigate the grand potential of the one-dimensional Hubbard model in the high-temperature limit, calculating the coefficients of the high-temperature expansion $(\ensuremath{\beta}$-expansion) of this function up to order ${\ensuremath{\beta}}^{4}$ by an alternative method. The results derived are analytical and do not involve any perturbation expansion in the hopping constant, being valid for arbitrary density of electrons in the one-dimensional model. In the half-filled case, we compare our analytical results for the specific heat and the magnetic susceptibility, in the high-temperature limit, with the ones obtained by Beni et al. [Phys. Rev. B 8, 3329 (1973)] and Takahashi's integral equations, showing that the latter result does not take into account the complete energy spectrum of the one-dimensional Hubbard model. The exact integral solution by J\"uttner et al. [Nucl. Phys. B 522, 471 (1998)] is applied to the determination of the range of validity of our expansion in $\ensuremath{\beta}$ in the half-filled case, for several different values of U.