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Physical review. B, Condensed matter

Gutzwiller approximation in the Fermi hypernetted-chain theory

Xiao-Qian Wang, C. E. Campbell

Published Jun 15, 1993• 1 citation
Publisher

Abstract

We present a general formalism for the variational calculation of the one-body density-matrix and correlation functions for strongly correlated lattice electrons in terms of the Fermi hypernetted-chain (FHNC) diagrammatic expansion. It is shown that, in the limit of infinite spatial dimensions, the local-field-approximate FHNC scheme yields the exact solution for the Gutzwiller wave function, which is equivalent to the result of the Gutzwiller approximation.

DOI
10.1103/physrevb.47.15984

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Physical review. B, Condensed matter•May 1, 1990•157 citations
We present a general formalism for the variational calculation of the one-body density-matrix and correlation functions for strongly correlated lattice electrons in terms of the Fermi hypernetted-chain (FHNC) diagrammatic expansion. It is shown that, in the limit of infinite spatial dimensions, the local-field-approximate FHNC scheme yields the exact solution for the Gutzwiller wave function, which is equivalent to the result of the Gutzwiller approximation.
The Fermi hypernetted-chain (FHNC) theory is applied to spin-density-wave states combined with a Jastrow-type correlation operator. The commonly used approximation scheme FHNC/n is found to be unsuitable for dealing with this class of correlated wave functions. A new scheme, named FHNC-n, is proposed, and numerically tested for the one-dimensional (1D) Hubbard model and the Gutzwiller correlation operator. The results obtained are in very good agreement with the available variational Monte Carlo calculation. This scheme can be used to study 2D and 3D Hubbard models and can easily be extended to treat more general correlation operators, which include triplet and quartet correlations.
We introduce Gutzwiller-correlated wave functions for the variational investigation of general multi-band Hubbard models. We set up a diagrammatic formalism which allows us to evaluate analytically ground-state properties in the limit of infinite spatial dimensions. In this limit recent results obtained within the Gutzwiller approximation are seen to become exact for these wave functions. We further show that the slave-boson mean-field theory for degenerate bands becomes variationally controlled at zero temperature in infinite dimensions. Lastly, we briefly comment on the variational approach to the Anderson transition in strongly correlated electron systems.
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.
A method for the accurate evaluation of the diagonal and off-diagonal matrix elements of the Hamiltonian operator within the correlated basis function scheme is presented. Considering a basis consisting of Slater determinants, correlated with a Jastrow correlation factor, the expectation value of the kinetic energy operator has been expressed in the Jackson-Feenberg form. A diagrammatic analysis is presented for the construction and evaluation of the cluster expansions of the required quantities incorporating the Fermi hypernetted chain theory.
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'').