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arXiv:1102.1658 · Physical Review B

Quantum quenches in the Hubbard model: Time-dependent mean-field theory and the role of quantum fluctuations

Marco Schirò, Michele Fabrizio

Published Apr 12, 2011• 118 citations
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Abstract

We study the nonequilibrium dynamics in the fermionic Hubbard model after a sudden change of the interaction strength. To this scope, we introduce a time-dependent variational approach in the spirit of the Gutzwiller ansatz. At the saddle-point approximation, we find at half filling a sharp transition between two different regimes of small and large coherent oscillations, separated by a critical line of quenches where the system is found to relax. Any finite doping washes out the transition, leaving aside just a sharp crossover. In order to investigate the role of quantum fluctuations, we map the model onto an auxiliary quantum Ising model in a transverse field coupled to free fermionic quasiparticles. Remarkably, the Gutzwiller approximation turns out to correspond to the mean-field decoupling of this model in the limit of infinite coordination lattices. The advantage is that we can go beyond mean field and include Gaussian fluctuations around the non-equilibrium mean-field dynamics. Unlike at equilibrium, we find that quantum fluctuations become massless and eventually unstable before the mean-field dynamical critical line, which suggests they could even alter qualitatively the mean-field scenario.

DOI
10.1103/physrevb.83.165105

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We study the nonequilibrium dynamics in the fermionic Hubbard model after a sudden change of the interaction strength. To this scope, we introduce a time-dependent variational approach in the spirit of the Gutzwiller ansatz. At the saddle-point approximation, we find at half filling a sharp transition between two different regimes of small and large coherent oscillations, separated by a critical line of quenches where the system is found to relax. Any finite doping washes out the transition, leaving aside just a sharp crossover. In order to investigate the role of quantum fluctuations, we map the model onto an auxiliary quantum Ising model in a transverse field coupled to free fermionic quasiparticles. Remarkably, the Gutzwiller approximation turns out to correspond to the mean-field decoupling of this model in the limit of infinite coordination lattices. The advantage is that we can go beyond mean field and include Gaussian fluctuations around the non-equilibrium mean-field dynamics. Unlike at equilibrium, we find that quantum fluctuations become massless and eventually unstable before the mean-field dynamical critical line, which suggests they could even alter qualitatively the mean-field scenario.
A simple and very flexible variational approach to the out-of-equilibrium quantum dynamics in strongly correlated electron systems is introduced through a time-dependent Gutzwiller wave function. As an application, we study the simple case of a sudden change of the interaction in the fermionic Hubbard model and find at the mean-field level an extremely rich behavior. In particular, a dynamical transition between small and large quantum quench regimes is found to occur at half-filling, in accordance with the analysis of Eckstein, Phys. Rev. Lett. 103, 056403 (2009)10.1103/PhysRevLett.103.056403, obtained by dynamical mean-field theory, that turns into a crossover at any finite doping.
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.
The real-time dynamics of the Fermi-Hubbard model, driven out of equilibrium by quenching or ramping the interaction parameter, is studied within the framework of the nonequilibrium self-energy functional theory. A dynamical impurity approximation with a single auxiliary bath site is considered as a reference system, and the time-dependent hybridization is optimized as prescribed by the variational principle. The dynamical two-site approximation turns out to be useful to study the real-time dynamics on short and intermediate time scales. Depending on the strength of the interaction in the final state, two qualitatively different response regimes are observed. For both weak and strong couplings, qualitative agreement with previous results of nonequilibrium dynamical mean-field theory is found. The two regimes are sharply separated by a critical point at which the low-energy bath degree of freedom decouples in the course of time. We trace the dependence of the critical interaction of the dynamical Mott transition on the duration of the interaction ramp from sudden quenches to adiabatic dynamics and therewith link the dynamical to the equilibrium Mott transition.
We reformulate the time-dependent Gutzwiller approximation by M. Schir\'o and M. Fabrizio, [Phys. Rev. Lett. 105, 076401 (2010)] in the framework of slave-boson mean-field theory, which is used to investigate the dynamical Mott transition of the generic two-band Hubbard model at half filling upon an interaction quench. Interorbital fluctuations lead to notable changes with respect to the single-band case. The singular dynamical transition is replaced by a broad regime of long-lived fluctuations between metallic and insulating states, accompanied by intriguing precursor behavior. A mapping to a spin model proves helpful to analyze the different regions in terms of the evolution of an Ising-like order parameter. Contrary to the static case, singlet occupations remain vital in the Mott-insulating regime with finite Hund's exchange.
We investigate by means of the time-dependent Gutzwiller variational approach the out-of-equilibrium dynamics of an antiferromagnetic state evolved with the Hubbard model Hamiltonian after a sudden change of the repulsion strength UUU. We find that magnetic order survives more than what is expected on the basis of thermalization arguments, in agreement with recent dynamical mean field theory calculations. In addition, we find evidence of a dynamical transition for quenches to large values of UUU between a coherent antiferromagnet characterized by a finite quasiparticle residue to an incoherent one with vanishing residue, which finally turns into a paramagnet for even larger UUU.
We study the out-of-equilibrium dynamics of the fermionic Hubbard model induced by a linear ramp of the repulsive interaction UUU from the metallic state through the Mott transition. To this extent, we use a time-dependent Gutzwiller variational method and complement this analysis with the inclusion of quantum fluctuations at the leading order, in the framework of a Z2{Z}_{2}Z2​ slave-spin theory. We discuss the dynamics during the ramp and the issue of adiabaticity through the scaling of the excitation energy with the ramp duration $\ensuremath{\tau}$. In addition, we study the dynamics for time scales longer than the ramp time, when the system is again isolated and the total energy conserved. We establish the existence of a dynamical phase transition analogous to the one present in the sudden quench case and discuss its properties as a function of final interaction and ramp duration. Finally, we discuss the role of quantum fluctuations on the mean-field dynamics for both long ramps, where spin-wave theory is sufficient, and for very short ramps, where a self-consistent treatment of quantum fluctuations is required in order to obtain relaxation.