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arXiv:1302.2108 · Physical Review B

Gutzwiller variational method for intersite Coulomb interactions: The spinless fermion model in one dimension

Mohammad Sherafati, S. Satpathy, Dix H. Pettey

Published Jul 11, 2013• 3 citations
arXivHTMLPDF

Abstract

We study the Gutzwiller method for the spinless fermion model in one dimension, which is one of the simplest models that incorporates the intersite Coulomb interaction. The Gutzwiller solution of this model has been studied in the literature but with differing results. We obtain the Gutzwiller solution of the problem by a careful enumeration of the many-particle configurations and explain the origin of the discrepancy in the literature to be due to the neglect of the correlation between the neighboring bond occupancies. The correct implementation of the Gutzwiller approach yields results different from the slave-boson mean-field theory, unlike for the Hubbard model with on-site interaction, where both methods are known to be equivalent. The slave-boson and the Gutzwiller results are compared to the exact solution, available for the half-filled case, and to the numerical exact diagonalization results for the case of general filling.

DOI
10.1103/physrevb.88.035114

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We study the Gutzwiller method for the spinless fermion model in one dimension, which is one of the simplest models that incorporates the intersite Coulomb interaction. The Gutzwiller solution of this model has been studied in the literature but with differing results. We obtain the Gutzwiller solution of the problem by a careful enumeration of the many-particle configurations and explain the origin of the discrepancy in the literature to be due to the neglect of the correlation between the neighboring bond occupancies. The correct implementation of the Gutzwiller approach yields results different from the slave-boson mean-field theory, unlike for the Hubbard model with on-site interaction, where both methods are known to be equivalent. The slave-boson and the Gutzwiller results are compared to the exact solution, available for the half-filled case, and to the numerical exact diagonalization results for the case of general filling.
We investigate the ground state properties of the one- and the two-dimensional Hubbard models using a new variational Monte Carlo method. By generalizing the Gutzwiller variational wave function (GF) to take into account an off-diagonal intersite correlation, we examine the metal-insulator transition mechanism at the half filling. As Kaplan et al. first pointed out, the binding mechanism between doubly occupied sites and empty sites plays a crucial role for this transition; we confirm this by examining the real space correlation function between these two kinds of sites. Results for the spin and the charge structure factors are presented as well as the momentum distribution function and are discussed in connection with the pairing effect. We also find that in less-than half filling, the improvements of the structure factors at 2 k F are realized as compared with GF results.
[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT AUTHOR'S REQUEST.] In this dissertation, we have presented our research on new developments in the Gutzwiller variational method as well as the applications of the method to two specific correlated electron systems of current interest: (i) the spinless fermion model with non-local Coulomb interaction in one dimension and (ii) the pressure-induced insulator-metal transition in the colossal magneto-resistive compound LaMnO3. Our results for the ground-state energy of the spinless fermion model are obtained by careful enumeration of the many-body configurations of the system and calculating the Gutzwiller reduction factor to the kinetic energy for general filling. We resolve the inconsistency in the existing literature for this problem as we find that our result for the half-filled case agrees with one of the three papers and attribute the discrepancy between them to the incorrect counting method and the assumption that the Gutzwiller approximation is equivalent to the mean-field slave-boson approach, which is although correct for the on-site Hubbard model, turns out not to be so in the present case. Compared with the exact diagonalization results, we show that the Gutzwiller method indeed offers a better solution than the slave-boson approach. As the second problem, we have studied the ground-state properties of paramagnetic LaMnO3 under hydrostatic pressure at room temperature. We apply the Gutzwiller method to an extended Hubbard model for spinless eg electrons on Mn3+ ion including all important ingredients such as cohesive energy as well as two main competing Coulomb and Jahn-Teller (JT) interactions. The plot of the energy of the paramagnetic LaMnO3 as a function of volume clearly confirms that 1) contrary to the existing accepted theoretical picture for vanishing lattice distortion as the metallic state emerges, the JT distortion survives even beyond the transition pressure into the metallic region and 2) the application of pressure leads to a structural phase separation and breaks the material into two blocks composed of domains of undistorted (metallic character) and distorted (insulating character) MnO6 octahedra; both of these findings have been already observed in experiments with no theoretical explanations. As the key unifying connection between both of these findings we suggest that the pressure-induced transition and the phase coexistence is indeed percolative. Finally, applying the idea of universality in percolation quantities near the threshold, we support our hypothesis by proposing a scaling law for the pressure-dependence of the resistance and use it as a best fit to recently-acquired experimental data. We find good agreement between our proposed model and the experimental results, confirming that the pressure-induced insulator-metal transition in LaMnO3 is indeed percolative.
We review the recently proposed extension of the Gutzwiller approximation, M. Schiro' and M. Fabrizio, Phys. Rev. Lett. 105, 076401 (2010), designed to describe the out-of-equilibrium time-evolution of a Gutzwiller-type variational wave function for correlated electrons. The method, which is strictly variational in the limit of infinite lattice-coordination, is quite general and flexible, and it is applicable to generic non-equilibrium conditions, even far beyond the linear response regime. As an application, we discuss the quench dynamics of a single-band Hubbard model at half-filling, where the method predicts a dynamical phase transition above a critical quench that resembles the sharp crossover observed by time-dependent dynamical mean field theory. We next show that one can actually define in some cases a multi-configurational wave function combination of a whole set of mutually orthogonal Gutzwiller wave functions. The Hamiltonian projected in that subspace can be exactly evaluated and is equivalent to a model of auxiliary spins coupled to non-interacting electrons, closely related to the slave-spin theories for correlated electron models. The Gutzwiller approximation turns out to be nothing but the mean-field approximation applied to that spin-fermion model, which displays, for any number of bands and integer fillings, a spontaneous Z2Z_2Z2​ symmetry breaking that can be identified as the Mott insulator-to-metal transition.
We investigate an extended version of the periodic Anderson model (the so-called periodic Anderson-Hubbard model) with the aim to understand the role of interaction between conduction electrons in the formation of the heavy-fermion and mixed-valence states. Two methods are used: (i) variational calculation with the Gutzwiller wave function optimizing numerically the ground-state energy and (ii) exact diagonalization of the Hamiltonian for short chains. The fff-level occupancy and the renormalization factor of the quasiparticles are calculated as a function of the energy of the fff orbital for a wide range of the interaction parameters. The results obtained by the two methods are in reasonably good agreement for the periodic Anderson model. The agreement is maintained even when the interaction between band electrons, Ud{U}_{d}Ud​, is taken into account, except for the half-filled case. This discrepancy can be explained by the difference between the physics of the one- and higher-dimensional models. We find that this interaction shifts and widens the energy range of the bare fff level, where heavy-fermion behavior can be observed. For large-enough Ud{U}_{d}Ud​ this range may lie even above the bare conduction band. The Gutzwiller method indicates a robust transition from Kondo insulator to Mott insulator in the half-filled model, while Ud{U}_{d}Ud​ enhances the quasiparticle mass when the filling is close to half filling.
We introduce a new lattice version of the correlated-basis-function approach for the study of strongly interacting electrons. As a first application we have chosen the one-dimensional Hubbard model. For this system, we have implemented different types of correlations: namely, (a) Gutzwiller; (b) Jastrow; (c) nearest-neighbor spin-spin; (d) excitonic (e-d); and (f) pairing (BCS). The resulting energies, momentum distribution, and other quantities are compared with exact results (when available) and are discussed, both from the methodological and physical point of view. The most important correlations that are needed to obtain good ground-state energies are found to be Gutzwiller (reduced double occupancy) plus e-d correlations (between empty and doubly occupied sites), in agreement with other studies.
Two very different methods – exact diagonalization on finite chains and a variational method – are used to study the possibility of a metal-insulator transition in the symmetric half-filled periodic Anderson-Hubbard model. With this aim we calculate the density of doubly occupied d sites (νd) as a function of various parameters. In the absence of on-site Coulomb interaction (Uf ) between f electrons, the two methods yield similar results. The double occupancy of d levels remains always finite just as in the one-dimensional Hubbard model. Exact diagonalization on finite chains gives the same result for finite Uf , while the Gutzwiller method leads to a Brinkman-Rice transition at a critical value (U c d ), which depends on Uf and V .