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Physical review. B, Condensed matter

Generalized spin-wave theory for the Hubbard model at half filling

Xin-Zhong Yan

Published Mar 1, 1992• 9 citations
Publisher

Abstract

By taking into account the contribution of transverse spin fluctuations deviating from the mean-field approximation, we develop a generalized spin-wave theory for the square-lattice Hubbard model at half filling. The present theory reproduces very accurately the ground-state energy of Lieb and Wu for the one-dimensional case; the discrepancy is typically 1%. In the very-strong-coupling limit, it reduces exactly to Anderson's spin-wave theory for the antiferromagnetic Heisenberg model.

DOI
10.1103/physrevb.45.4741

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By taking into account the contribution of transverse spin fluctuations deviating from the mean-field approximation, we develop a generalized spin-wave theory for the square-lattice Hubbard model at half filling. The present theory reproduces very accurately the ground-state energy of Lieb and Wu for the one-dimensional case; the discrepancy is typically 1%. In the very-strong-coupling limit, it reduces exactly to Anderson's spin-wave theory for the antiferromagnetic Heisenberg model.
The square-lattice extended Hubbard model at half filling is studied by a perturbation scheme. By taking into account the ground state of the mean-field theory as a reference state, the scheme treats the collective fluctuations beyond the mean field as the predominant perturbation. The system at ground state can be a spin-density wave (SDW) or a charge-density wave (CDW) depending on the relative strengths of the on-site electron interaction and the nearest-neighbor interaction. The excitation modes above the ground state are analyzed through diagonalizing an effective Hamiltonian. The energy spectra and the wave functions are self-consistently given by the formalism. By comparing the ground-state energies, we investigate the SDW-CDW transition.
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.
By using the classic approximation method of Singwi, Tosi, Land, and Sj\"olander (STLS), we calculate the ground-state energy of the one- and two-dimensional Hubbard model in the half-filled antiferromagnetic state. In the one-dimensional case, by comparison with the exact result of Lieb and Wu, we find that the STLS correlation energy is generally better than that obtained from the random-phase approximation (RPA) and competes in quality with the best values obtained from variational calculations based on the antiferromagnetic Gutzwiller wave function (AFGWF). In particular, the STLS approximation yields a maximum in the absolute value of the correlation energy as a function of U, which is also found in the exact treatment, but not in the RPA. However, the numerical values of the STLS correlation energy are not as accurate as in the case of the uniform electron gas. In two dimensions, where the exact energy is not known, we find a similar relationship between the results of STLS, RPA, and AFGWF calculations, but the absolute value of the correlation energy is smaller. For large values of U, in both dimensions, the STLS method (like the RPA) runs into difficulties, due to the insufficient treatment of transverse spin fluctuations.
A recently proposed projection operator technique based on the introduction of cumulants is used to investigate the ground-state energy of the Hubbard model at half filling on a square lattice in two dimensions, in the antiferromagnetic phase. Our results in the intermediate and large regime of the ratio U/t are below those of variational Monte Carlo calculations, which set an upper bound for the energy. By including the most dominant spin-fluctuation processes, the present calculation also obeys the correct infinite U/t limit, where the Hubbard model at half filling reduces to the Heisenberg antiferromagnet. The relation of our approach to recent coupled-cluster calculations for the Hubbard model is also discussed. \textcopyright{} 1996 The American Physical Society.
We develop series expansions for the ground state properties of the Hubbard model by introducing an Ising anisotropy into the Hamiltonian. For the two-dimensional square lattice half-filled Hubbard model, the ground state energy, local moment, sublattice magnetization, uniform magnetic susceptibility, and spin stiffness are calculated as a function of U/t, where U is the Coulomb constant and t is the hopping parameter. Magnetic susceptibility data indicate a crossover around U\ensuremath{\approxeq}4 between spin density wave antiferromagnetism and Heisenberg antiferromagnetism. Comparisons with Monte Carlo simulations, random phase approximation result, and mean-field solutions are also made.
We introduce a local formalism to deal with the Hubbard model on an N × N square lattice (for even N ) in terms of eigenstates of number operators, having well defined point symmetry. For U →0, the low-lying shells of the kinetic energy are filled in the ground state. At half-filling, using the 2 N -2 one-body states of the partially occupied shell hf , we build a set of degenerate unperturbed ground states with S z = 0 which are then resolved by the Hubbard interaction Ŵ = U ∑ r r ↑ r ↓ . We study the many-body eigenstates in hf of the kinetic energy with vanishing eigenvalue of the Hubbard repulsion ( W = 0 states). In the S z = 0 sector, this is an N -times-degenerate multiplet. From the singlet component one obtains the ground state of the Hubbard model for U = 0 + , which is unique, in agreement with a theorem of Lieb. The wave function demonstrates an antiferromagnetic order, a lattice step translation being equivalent to a spin flip. We show that the total momentum vanishes, while the point symmetry is s or d for even or odd N /2, respectively.