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Physical review. B, Condensed matter

Analytic calculation of ground-state properties of correlated fermions with the Gutzwiller wave function

Walter Metzner, D. Vollhardt

Published May 1, 1988• 178 citations
Publisher

Abstract

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 ${n}_{\mathrm{k}}$ 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.

DOI
10.1103/physrevb.37.7382

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Physical review. B, Condensed matter•May 1, 1990•157 citations
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 nk\ensuremathσ{n}_{\mathrm{k}\ensuremath{\sigma}}nk\ensuremathσ​ 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.
The properties of the Gutzwiller variational wave function, which is frequently used to study ground-state properties of correlated fermions with a short-range interaction, are investigated by use of a new, analytically tractable approach. As a first application several ground-state quantities are evaluated exactly in dimensionality d=1 for arbitrary band-filling and interaction strengths. The results allow for the first approximation-free assessment of the wave function. The method itself is applicable to arbitrary space dimensions.
We make use of a recently developed diagrammatic theory to calculate correlation functions for interacting fermions of Hubbard-type models in terms of the Gutzwiller wave function. Of the eleven nontrivial correlation functions involving the spin, density, empty and doubly occupied sites, and local Cooper pairs, four are shown to be independent. They are expressed as a power series in a suitably chosen correlation parameter, whose terms are represented diagrammatically. In one dimension these terms may be evaluated to arbitrary order by employing symmetry relations. This allows for an analytic, approximation-free calculation of the correlation functions for arbitrary momentum, particle density, and interaction strength. In the atomic limit the momentum-dependent spin-correlation function shows an antiferromagnetic divergence at half filling in all dimensions. In one dimension the behavior is in very good agreement with all exact analytic and numerical results for the antiferromagnetic Heisenberg chain. Hole-hole correlations also compare very well with exact results. However, correlations between holes and doubly occupied sites appear insufficient. Superconducting correlations involving on-site, singlet Cooper pairs are suppressed. The results allow for an analytic evaluation of the ground-state energy of a large class of extended Hubbard models in terms of the Gutzwiller wave function. Thus they provide exact upper bounds for their ground-state energies.
This work covers three lectures on Gutzwiller wave functions for correlated electron systems. In the first lecture, some basic ideas on model Hamiltonians and their motivation are outlined, Gutzwiller's variational approach to Landau Fermi-liquid theory is sketched, and, as a first application, the theory is applied to the one-band Hubbard model where the Brinkman–Rice metal-to-insulator transition is found. The second lecture is devoted to the perturbative calculation of expectation values. In this rather technical part the variational analog of Wick's theorem, Feynman diagrams, and the linked-cluster theorem are introduced. The important differences between the variational approach and the standard Green function technique are the absence of a time-dependence and the flexibility to select the expansion parameter of the variational perturbation theory. When this flexibility is used appropriately, not a single self-energy diagram must be calculated in the limit of large dimensions, i.e. in the limit of large lattice coordination number the expectation values can be evaluated exactly for general Hubbard-type models for all interaction strengths and band fillings. The third lecture covers the application of the theory to ferromagnetism of nickel. After a discussion of the nickel problem, multi-band Hubbard models and Gutzwiller wave functions are introduced and parameterized. The favourable comparison between the results from Gutzwiller theory and a large number of experiments shows that correlations play an important role in itinerant ferromagnets.
We present analytic results for ground-state properties of Hubbard-type models in terms of the Gutzwiller variational wave function with nonzero values of the magnetization m. In dimension D=1D=1D=1 approximation-free evaluations are made possible by appropriate canonical transformations and an analysis of umklapp processes. We calculate the double occupation and the momentum distribution, as well as its discontinuity at the Fermi surface, for arbitrary values of the interaction parameter g, density n, and magnetization m. These quantities determine the expectation value of the one-dimensional Hubbard Hamiltonian for any symmetric, monotonically increasing dispersion ${\ensuremath{\epsilon}}_{k}.$ In particular for nearest-neighbor hopping and densities away from half filling the Gutzwiller wave function is found to predict ferromagnetic behavior for sufficiently large interaction U.
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'').