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arXiv:cond-mat/0606717 · Europhysics Letters (EPL)

Extended Gutzwiller wave function for the Hubbard-Holstein model

Paolo Barone, Roberto Raimondi, Massimo Capone, Claudio Castellani, Michele Fabrizio

Published Jul 20, 2007• 11 citations
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Abstract

We introduce a new type of Gutzwiller variational wave function for correlated electrons coupled to phonons, able to treat on equal footing electronic and lattice degrees of freedom. We benchmark the wave function in the infinite- U Hubbard-Holstein model away from half-filling on a Bethe lattice, where we can directly compare with exact results by the Dynamical Mean-Field Theory. For this model, we find that variational results agree perfectly well with the exact ones. In particular the wave function correctly describes the crossover to a heavy polaron gas upon increasing the electron-phonon coupling.

DOI
10.1209/0295-5075/79/47003

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Communications in Theoretical Physics•Feb 1, 2005•3 citations
We introduce a new type of Gutzwiller variational wave function for correlated electrons coupled to phonons, able to treat on equal footing electronic and lattice degrees of freedom. We benchmark the wave function in the infinite- U Hubbard-Holstein model away from half-filling on a Bethe lattice, where we can directly compare with exact results by the Dynamical Mean-Field Theory. For this model, we find that variational results agree perfectly well with the exact ones. In particular the wave function correctly describes the crossover to a heavy polaron gas upon increasing the electron-phonon coupling.
We analyze the ground-state properties of strongly correlated electrons coupled with phonons by means of a generalized Gutzwiller wave function, which includes phononic degrees of freedom. We study in detail the paramagnetic half-filled Hubbard-Holstein model, where the electron-electron interaction can lead to a Mott transition and the electron-phonon coupling to a bipolaronic transition. We critically discuss the quality of the proposed wave function in describing the various transitions and crossovers that occur as a function of the parameters. Previous variational attempts are recovered as particular choices of the wave function, while keeping all the variational freedom allows to access regions of the phase diagram, otherwise inaccessible within previous variational approaches.
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 describe a variational method to solve the Holstein model for an electron coupled to dynamical, quantum phonons on an infinite lattice. The variational space can be systematically expanded to achieve high accuracy with modest computational resources (12-digit accuracy for the one-dimensional polaron energy at intermediate coupling). We compute ground-state and low-lying excited-state properties of the model at continuous values of the wave vector k in essentially all parameter regimes. Our results for the polaron energy band, effective mass, and correlation functions compare favorably with those of other numerical techniques, including the density-matrix renormalization-group technique, the global-local method, and the exact diagonalization technique. We find a phase transition for the first excited state between a bound and unbound system of a polaron and an additional phonon excitation. The phase transition is also treated in strong-coupling perturbation theory.
We explore the phase diagram of the Hubbard model by generalizing the Gutzwiller variational wave function to take into account nearest-neighbor spin correlations. The energies of various variational states are calculated and compared. At half filling the lowest energy state has long-range antiferromagnetic correlations, in agreement with previous results. Away from half filling, when the antiferromagnet no longer has the lowest energy, the Gutzwiller approximation yields a lower energy than that found for small spin polarons.
A variational approach is developed to study the groundstate (GS) of the two-site Holstein model. By the extended coherent state, where the more phonon correlations are easily incorporated, we can get the very accurate ground state energy for all electron-phonon coupling range in typical values of hopping integral t = 0.5, 1.1, and 2.1 (in units of phonon frequency ωo ), which covers the crossover region from antiadiabatic limit to the adiabatic limit. Within a very wide t range [0, 2.7], the exact results for the GS energy are obtained with the twelfth (fourteenth) order corrections to the zeroth order wave function. Moreover, the present approach is more concise than any other analytical ones in this field, and hopefully can be easily generalized to many other Holstein models.
A variational approach is developed to study the groundstate (GS) of the two-site Holstein model. By the extended coherent state, where the more phonon correlations are easily incorporated, we can get the very accurate ground state energy for all electron-phonon coupling range in typical values of hopping integral t = 0.5,1.1, and 2.1 (in units of phonon frequency ω 0 ), which covers the crossover region from antiadiabatic limit to the adiabatic limit. Within a very wide t range [0,2.7], the exact results for the GS energy are obtained with the twelfth (fourteenth) order corrections to the zeroth order wave function. Moreover, the present approach is more concise than any other analytical ones in this field, and hopefully can be easily generalized to many other Holstein models.