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 Z2 symmetry breaking
that can be identified as the Mott insulator-to-metal transition.Comment: 25 pages. Proceedings of the Hvar 2011 Workshop on 'New materials for
thermoelectric applications: theory and experiment