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arXiv:1008.0021

Statistically-consistent Gutzwiller approach and its equivalence with the mean-field slave-boson method for correlated systems

Jakub Jȩdrak, Jan Kaczmarczyk, Józef Spałek

Published Jul 30, 2010• 5 citations
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Abstract

We propose a new method of solving a class of mean-field (MF) models, which is based on the Maximum Entropy (MaxEnt) principle with additional constraints included. Next, we show equivalence of our method when applied to the Gutzwiller approximation (GA), with the mean-field slave-boson (SB) formalism (on the example of the single-band Hubbard model). This equivalence provides thus an alternative justification of the results obtained within the SB approach which, however, contains ad hoc assumptions to position it in agreement with GA. Our approach implies that all predictions of the MF SB method can be obtained in a simpler, transparent, and controllable manner within GA when supplemented with the statistical-consistency conditions. We call the method as the Statistically-consistent Gutzwiller Approximation (SGA). Explicitly, the present formulation does not require introducing the condensed amplitudes of auxiliary Bose fields, which do not have a direct physical meaning and do not appear in the present formulation. Although the results of SGA are in the present case equivalent to SB, one can improve them further by utilizing more advanced schemes of calculating averages beyond the standard GA. To illustrate our approach, as well as to outline its advantages over alternative treatments of GA, we select the case of almost localized Fermi liquid (ALFL) in two dimensions and analyze it in detail within the tight-binding approximation. We also comment on significance of our method for describing correlated fermions. Namely, the reasoning used here can be applied to the corresponding MF treatment of the multiband Hubbard, the periodic Anderson, the $t-J$, and the $t-J-U$ models. In this manner our method may be applied for strongly correlated electron systems, optical lattices, and other related situations.

DOI
10.48550/arxiv.1008.0021

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We propose a new method of solving a class of mean-field (MF) models, which is based on the Maximum Entropy (MaxEnt) principle with additional constraints included. Next, we show equivalence of our method when applied to the Gutzwiller approximation (GA), with the mean-field slave-boson (SB) formalism (on the example of the single-band Hubbard model). This equivalence provides thus an alternative justification of the results obtained within the SB approach which, however, contains ad hoc assumptions to position it in agreement with GA. Our approach implies that all predictions of the MF SB method can be obtained in a simpler, transparent, and controllable manner within GA when supplemented with the statistical-consistency conditions. We call the method as the Statistically-consistent Gutzwiller Approximation (SGA). Explicitly, the present formulation does not require introducing the condensed amplitudes of auxiliary Bose fields, which do not have a direct physical meaning and do not appear in the present formulation. Although the results of SGA are in the present case equivalent to SB, one can improve them further by utilizing more advanced schemes of calculating averages beyond the standard GA. To illustrate our approach, as well as to outline its advantages over alternative treatments of GA, we select the case of almost localized Fermi liquid (ALFL) in two dimensions and analyze it in detail within the tight-binding approximation. We also comment on significance of our method for describing correlated fermions. Namely, the reasoning used here can be applied to the corresponding MF treatment of the multiband Hubbard, the periodic Anderson, the t−Jt-Jt−J, and the t−J−Ut-J-Ut−J−U models. In this manner our method may be applied for strongly correlated electron systems, optical lattices, and other related situations.
We investigate Gutzwiller-correlated wave functions as variational ground states for the periodic Anderson model. We extend a recently developed technique for the Hubbard model to the case of two hybridized bands with degeneracy N. We calculate expectation values in the following limits: dilute system (single-impurity case), large degeneracy N, and large dimension d. The mean-field results using the slave-boson approach of Coleman and of Kotliar and Ruckenstein are reproduced in the large-N and large-d limit, respectively. These techniques, therefore, obey variational principles at T=0 in the respective limits and their applicability for T>0 is naturally limited to the Fermi-liquid regime. Evaluations of Gutzwiller-correlated wave functions using the ``Gutzwiller approximation'' are seen to become exact in the limit of large dimensions. A systematic expansion in 1/d can be performed for N=2. In general, corrections are small, of order 1/(dN).
A systematic extension of the model of almost localized fermions to finite temperatures is presented. It consists in accounting for the quantum fluctuations beyond the Gutzwiller approximation (GA). We use the slave-boson formulation of the Hubbard model which has been proved to be equivalent at the saddle-point level to the GA. We determine the values of the Lagrange multipliers introduced to enforce the constraints, and show how they involve a Mott gap at nearly half filling for U>Uc{U}_{c}Uc​ (localization edge). The results of the GA for the correlation functions are reproduced at the saddle point when taking into account the implicit dependence of the boson fields with the external excitation. The Gaussian fluctuations are then considered in a renormalized basis of the boson fields which brings out two distinct channels (symmetric s and antisymmetric a). We show how the free energy and the correlation functions can be simply expressed as a function of the same Landau parameters F0s{F}_{0}^{s}F0s​ and F0a{F}_{0}^{a}F0a​ defined in the GA.
We consider slave boson formulations of the Hubbard model, the t–J model and the Anderson model, in which spin and charge fermion degrees of freedom are represented by Bose fields. The approach has been applied to the Hubbard model generalizing the formulation of Kotliar and Ruckenstein. There the Gutzwiller solution is recovered as a saddle point. The method allows to find other types of mean field solutions in a systematic way. Dynamical correlation functions in mean field approximation as well as fluctuation contributions are also calculated.
For weakly interacting many-body quantum systems one can apply the extensive framework of perturbative quantum field theory. On the other hand, strong interactions can not be treated practically in perturbation theory because of the absence of a small control parameter. In this context variational methods are a powerful tool to obtain reliable informations regarding the properties of the ground state of the strongly correlated electron system. In this thesis we present novel formulations of two widely used methods: the Gutzwiller wave function and the slave-spin theory. These techniques share a similar philosophy and the extensions are based on the common attitude towards a more detailed description of the high-energy incoherent excitations characterizing interacting electron systems together with low-energy quasiparticles excitations, captured by standard approaches. Our task is not only crucial for a faithful description of the insulating phase but also improves the variational characterization of the low-energy quasiparticle excitations. We apply the novel ghost-Gutzwiller wave function technique to tackle an intriguing phenomenon, namely the exciton Mott transition in photoexcited semiconductors. Despite being a quite old topic that goes back to the 1970's, the nature of the exciton Mott transition still defies a complete understanding. By considering an idealised model of photoexcited semiconductors we unveil the important role of the exciton binding energy in determining the nature of the transition. Moreover, our results uncover rather anomalous electron-hole liquid phase next to the transition, which still sustains excitonic excitations although being a degenerate Fermi liquid of heavy mass quasiparticles. By means of the Dirac-Frenkel variational principle, we generalize the ghost-Gutzwiller wave function to study the out of equilibrium dynamics of strongly correlated electron systems. Numerical results on the single-band Hubbard model show a remarkable agreement with those obtained with time-dependent dynamical mean field theory. We believe that the method opens the way to several promising developments and future applications. Concerning the slave-particle approach, we consider Anderson impurity models and we show that, within our formulation, the constraint on the slave-spin variable can be removed, leaving thermal average free by any projection on the "physical" subspace of the enlarged Hilbert space. To the best of our knowledge, this represents an exception and shows that our formulation is more convenient than other slave-particle approaches, where the constraint have to be implemented explicitly. The method, suited to deal with impurity models, finds direct application in studying time-dependent transport across an interacting quantum dot. To this aim we formulate, by means of the Keldysh Green's function, the slave-spin mean-field theory in the out of equilibrium framework and we apply the method to the dynamics of a driven quantum dot. Finally we study the current-voltage characteristic of an interacting quantum dot tunnel-coupled to the edge of a superconductive nanowire. The appearance of the topological Majorana edge mode is signalized by distinctive features in the charge transport properties of the junction. Apart from the well known half-integer zero bias conductance we find that the topological region is characterized by a vanishing Fano factor, that can be measured in experiments to detect the presence of a Majorana zero mode hybridized with the quantum dot.
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.