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Physical review. B, Condensed matter

Systematic study on antiferromagnetic states of the Hubbard model with the dimensiond=1, 2, 3 and ∞

Hideo Hasegawa

Published May 1, 1990• 24 citations
Publisher

Abstract

Antiferromagnetic states of the half-filled Hubbard model on the d=1, 2, 3, and \ensuremath{\infty} dimensional lattices are systematically studied by using the Gutzwiller-type analytical approximation based on the slave-boson functional-integral approach. In all the cases investigated, antiferromagnetic states have lower energies than paramagnetic states even for infinitesimally small interactions, U. Calculated results of the ground-state energy E and the sublattice moment m for d=2 and 3 are in very good agreement with those of the Monte Carlo simulations. Even for d=1 the antiferromagnetic state in our Gutzwiller-type approximation has energy much closer to the exact one than the paramagnetic states in Gutzwiller's approximation and in Metzner-Vollhardt's approximation-free method. On the basis of the 1/d plot of E and m, a possibility of the 1/d expansion starting from the limit of d=\ensuremath{\infty} is discussed. It is also shown that the Brinkman-Rice metal-insulator transition does not take place at finite U for all d.

DOI
10.1103/physrevb.41.9168

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Antiferromagnetic states of the half-filled Hubbard model on the d=1, 2, 3, and \ensuremath{\infty} dimensional lattices are systematically studied by using the Gutzwiller-type analytical approximation based on the slave-boson functional-integral approach. In all the cases investigated, antiferromagnetic states have lower energies than paramagnetic states even for infinitesimally small interactions, U. Calculated results of the ground-state energy E and the sublattice moment m for d=2 and 3 are in very good agreement with those of the Monte Carlo simulations. Even for d=1 the antiferromagnetic state in our Gutzwiller-type approximation has energy much closer to the exact one than the paramagnetic states in Gutzwiller's approximation and in Metzner-Vollhardt's approximation-free method. On the basis of the 1/d plot of E and m, a possibility of the 1/d expansion starting from the limit of d=\ensuremath{\infty} is discussed. It is also shown that the Brinkman-Rice metal-insulator transition does not take place at finite U for all d.
Antiferromagnetic and metal to insulator transitions in the ground state of a two dimensional half-filled Hubbard model with the nearest neighbor and the second nearest neighbor transfers (- t and t ′ , respectively) are studied by using the Hartree-Fock approximation. The results indicate that the phase diagram depends on the band structure or t ′ / t ratio; a discontinuous transition between the paramagnetic metallic (PM) and the antiferromagnetic insulator (AI) phases takes place for t ′ / t =0.2, while continuous transitions from PM to AM (antiferromagnetic metal) and then to AI phases take place for t ′ / t =0.5. The results are compared with those of the Gutzwiller approximation and discussed in the light of the other existing theories. The general importance of considering antiferromagnetism in this problem is emphasized.
We present a systematic formalism for the variational evaluation of ground-state properties of Hubbard-type models in finite dimensions d. The formalism starts from generalized Gutzwiller correlated wave functions, which are then studied in a systematic (1/d) expansion around the limit of high dimensions (d=\ensuremath{\infty}). The limit of d\ensuremath{\rightarrow}\ensuremath{\infty} has recently been introduced by Metzner and Vollhardt (MV) for itinerant lattice fermion systems. The approach, presented in this paper, is particularly efficient since results in d=\ensuremath{\infty} are obtained without having to calculate a single graph. Our results confirm the finding of MV that counting approximations in the spirit of the Gutzwiller approximation become exact in d=\ensuremath{\infty} for translationally invariant wave functions. This type of approximation is no longer exact for more complicated (e.g., antiferromagnetic) wave functions. In addition, we completely reproduce the results of the Kotliar-Ruckenstein path-integral approach to the Hubbard model. Performing a (1/d) expansion for the Gutzwiller wave function, we show that the lowest orders in (1/d) are sufficient to reproduce all numerical findings in d=2,3 quantitatively. We therefore conclude that the limit of d=\ensuremath{\infty} is a very fruitful starting point for the study of finite-dimensional systems. On the basis of our study we propose new variational wave functions for the numerical investigation of antiferromagnetism in the Hubbard, the t-J, and the spin-1/2 Heisenberg model. For the first two models we calculate in d=\ensuremath{\infty} only, for the Heisenberg model we also derive corrections up to order (1/d). To this order we obtain complete agreement with linear spin-wave theory. Since our trial state is based on a fermionic description of the Heisenberg model, we interpret this analytically determined wave function as Fermi sea of spin-1/2 quasiparticles (``spinons'').
We have investigated the antiferromagnetic phase of two-dimensional (2D), three-dimensional (3D), and extended Hubbard models on a bipartite cubic lattice by means of the composite operator method within a two-pole approximation. This approach yields a fully self-consistent treatment of the antiferromagnetic state that respects the symmetry properties of both the model and the algebra. The complete phase diagram, as regards the antiferromagnetic and paramagnetic phases, has been drawn. We first reported, within a pole approximation, three kinds of transitions at half-filling: Mott-Hubbard, Mott-Heisenberg, and Heisenberg transitions. We have also found a metal-insulator transition, driven by doping, within the antiferromagnetic phase. This latter is restricted to a very small region near half-filling, and has, in contrast to what has been found by similar approaches, a finite critical Coulomb interaction as a lower bound at half-filling. Finally, it is worth noting that our antiferromagnetic gap has two independent components: one due to the antiferromagnetic correlations, and another coming from the Mott-Hubbard mechanism.
We explore the phase diagram of the Hubbard model by generalizing the Gutzwiller variational wave function to take into account nearest-neighbor spin correlations. The energies of various variational states are calculated and compared. At half filling the lowest energy state has long-range antiferromagnetic correlations, in agreement with previous results. Away from half filling, when the antiferromagnet no longer has the lowest energy, the Gutzwiller approximation yields a lower energy than that found for small spin polarons.
Antiferromagnetic and metal to insulator transitions in the ground state of two-dimensional half-filled and nearly half-filled Hubbard models with the nearest neighbor and the second nearest neighbor transfers (- t and t ′ , respectively) are studied by using the Green's function approach with the self-energy up to the second order in the coupling constant U . For a choice of the parameter value t ′ / t =0.2 the half-filled model undergoes the first order phase transition between paramagnetic metallic (PM) phase and antiferromagnetic insulator (AFI) phase at U / t =4.0 while the first order transition between PM and antiferromagnetic metallic (AFM) phases takes place for nearly half-filled band. The one-electron spectral density is calculated for the PM, AFM, and AFI phases near the phase boundaries. In the PM phase the quasi-particle dispersion indicates the band narrowing from the Hartree-Fock results and the flat band behavior as was indicated by previous stadies and was observed in high- T c cuprates. The quasi-particle dispersion in the AFM phase is seen to be close to the Hartree-Fock results.
We study the Hubbard model in infinite dimension d= infinity within an approximation reproducing simultaneously the atomic limit and the weak-coupling limit up to second order in the interaction strength U. Depending on U we find a paramagnetic metal, a paramagnetic insulator, or an antiferromagnetic insulator. The stability of the antiferromagnetic phase is investigated by calculating the grand canonical potential. We present the phase diagram in the T-U plane and results for the magnetization m as a function of temperature T. For small values of U the antiferromagnetic phase transition is continuous, but for larger values of U the approximation yields a first-order magnetic phase transition. Results for the density of states N( epsilon ) are presented; it either exhibits one single band or an upper and a lower Hubbard band separated by the Mott gap. The upper and lower Hubbard band may exhibit additional magnetic structure, depending on the value of T and U.