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A NEW EXPANSION FOR GENERALIZED GUTZWILLER WAVE-FUNCTIONS - ANTIFERROMAGNETIC CASE

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Published Sep 10, 2026• 0 citations
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Abstract

We propose an alternative approach to the 1/D-type expansions for evaluating properties of generalized Gutzwiller wave functions (GWF's) in low dimensions. Our expansion uses the sum rule and symmetries of the trial wave function explicitly. We apply the scheme to an antiferromagnetic generalization of the GWF for the ID Hubbard model. We find good agreement with known results from variational Monte Carlo calculations and a significant improvement over previous approximations.

DOI
10.82444/warw.33550918

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Physical review. B, Condensed matter•May 1, 1990•24 citations
We propose an alternative approach to the 1/D-type expansions for evaluating properties of generalized Gutzwiller wave functions (GWF's) in low dimensions. Our expansion uses the sum rule and symmetries of the trial wave function explicitly. We apply the scheme to an antiferromagnetic generalization of the GWF for the ID Hubbard model. We find good agreement with known results from variational Monte Carlo calculations and a significant improvement over previous approximations.
A systematic expansion scheme that goes beyond the Gutzwiller-type approximation is developed for generalized Gutzwiller wave functions. The expansion satisfies sum rules and symmetries of the trial wave function in a self-consistent way. We apply the scheme to an antiferromagnetic generalization of the Gutzwiller wave function for the one-dimensional Hubbard model. We find good agreement with known results from variational Monte Carlo calculations. We also demonstrate that the expansion scheme converges rapidly.
A systematic expansion scheme that goes beyond the Gutzwiller-type approximation is developed for generalized Gutzwiller wave functions. The expansion satisfies sum rules and symmetries of the trial wave function in a self-consistent way. We apply the scheme to an antiferromagnetic generalization of the Gutzwiller wave function for the one-dimensional Hubbard model. We find good agreement with known results from variational Monte Carlo calculations. We also demonstrate that the expansion scheme converges rapidly.
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'').
A new analytic approach to the evaluation of ground-state properties of Hubbard-type models in terms of the Gutzwiller variational wave function is presented. It is based on the observation that expectation values in terms of this wave function may be expressed by sums over different lattice sites. This makes the application of Wick's theorem and the resulting contractions extremely simple, since the latter involve only anticommuting numbers as in a Grassmann algebra. Expressions for the momentum distribution ${n}_{\mathrm{k}\ensuremath{\sigma}}$ and the Hubbard interaction in terms of a power series in a particular correlation parameter are derived which are valid for all dimensions. An explicit diagrammatic evaluation of the coefficients is described. In one dimension these coefficients may be determined to all orders which yields an approximation-free calculation of nk{n}_{\mathrm{k}}nk​ and ground-state energy E for arbitrary density n and interaction strength U. In the case of a half-filled band and large U, hitherto unexpected nonanalyticities are found. The results allow for the first approximation-free assessment of the properties of the Gutzwiller wave function. It is shown that the well-known Gutzwiller approximation may be derived diagrammatically, too. The approximation is seen to yield the exact results for expectation values in terms of this wave function in the limit of infinite dimensionality.
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.