Abstract
We review the recently proposed extension of the Gutzwiller approximation (SchirĂ² and 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 Z 2 symmetry breaking that can be identified as the Mott insulator-to-metal transition.
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Notes
- 1.
In reality, for the method to work it is enough that Wick’s theorem applies, hence ∣Ψ 0⟩ could even be a BCS wavefunction. Here, for sake of simplicity, we shall only consider Slater determinants.
- 2.
In fact, we can parametrize
$$\hat{{\Phi }}_{i}(t) =\hat{ {U}}_{i}(t)\,\sqrt{\hat{{P}}_{i } (t)},$$where \(\hat{{U}}_{i}(t)\) is a unitary matrix with elements U i Γ{n}, while \(\hat{{P}}_{i}(t)\) a positive definite matrix with elements P i {n}{m}(t), which can be represented as the density matrix of a local normalized state
$$\mid {\psi }_{i}(t)\rangle =\sum\limits_{\{n\}}\,{c}_{i\{n\}}(t)\mid i;\{ n\}\rangle ,$$with ⟨ψ i (t)∣ψ i (t)⟩ = 1, which automatically fulfills Eq. (16.26). In order to impose the constraint (16.27) it is then sufficient that, for α ≠ β
$$\langle {\psi }_{i}(t)\mid {d}_{i\alpha }^{\dag }{d}_{ i\beta }^{ }\mid {\psi }_{i}(t)\rangle = 0.$$This can be done by regarding ∣ψ i (t)⟩ as the eigenstate of a local many-body Hamiltonian that does not contain any term of the form c iα  † ∣i; { n}⟩⟨i; { n}∣c iβ for any ∣{n}⟩ including the vacuum.
- 3.
Once again, we must make sure that the effective Hamiltonian ℋ  ∗ (t), Eq. (16.57), is such that the local density matrix remains indeed diagonal in the operators c ia  † .
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Acknowledgements
These proceedings are based on the work that I have done in collaboration with Marco SchirĂ², whom I thank warmly. I am also grateful to Nicola LanatĂ for useful discussions. I also acknowledge support by the EU under the project GOFAST.
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Fabrizio, M. (2013). The Out-of-Equilibrium Time-Dependent Gutzwiller Approximation. In: Zlatic, V., Hewson, A. (eds) New Materials for Thermoelectric Applications: Theory and Experiment. NATO Science for Peace and Security Series B: Physics and Biophysics. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-4984-9_16
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