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arXiv:1106.3483 · Physical Review B

Approach to a stationary state in a driven Hubbard model coupled to a thermostat

A. Amaricci, Cédric Weber, Massimo Capone, Gabriel Kotliar

Published Aug 8, 2012• 67 citations
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Abstract

We investigate the dynamics of the Hubbard model in a static electric field in order to identify the conditions necessary to reach a nonequilibrium stationary state. We show that, for a generic electric field, the convergence to a stationary state requires coupling to a thermostatting bath that absorbs the work done by the external field. Following the real-time dynamics of the system, we show that a nonequilibrium stationary state is reached for essentially any value of the coupling to the bath. We characterize the properties of such nonequilibrium stationary states by studying suitable physical observables, pointing out the existence of an analog of the Pomeranchuk effect as a function of the electric field. We map out a phase diagram in terms of dissipation and electric field strengths and identify the dissipation values at which the steady current is largest for a given field.

DOI
10.1103/physrevb.86.085110

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We investigate the dynamics of the Hubbard model in a static electric field in order to identify the conditions necessary to reach a nonequilibrium stationary state. We show that, for a generic electric field, the convergence to a stationary state requires coupling to a thermostatting bath that absorbs the work done by the external field. Following the real-time dynamics of the system, we show that a nonequilibrium stationary state is reached for essentially any value of the coupling to the bath. We characterize the properties of such nonequilibrium stationary states by studying suitable physical observables, pointing out the existence of an analog of the Pomeranchuk effect as a function of the electric field. We map out a phase diagram in terms of dissipation and electric field strengths and identify the dissipation values at which the steady current is largest for a given field.
We study the steady-state dynamics of the Hubbard model driven out of equilibrium by a constant electric field and coupled to a dissipative heat bath. For a very strong field, we find a dimensional reduction: the system behaves as an equilibrium Hubbard model in lower dimensions. We derive steady-state equations for the dynamical mean-field theory in the presence of dissipation. We discuss how the electric field induced dimensional crossover affects the momentum resolved and integrated spectral functions, the energy distribution function, as well as the steady current in the nonlinear regime.
We study the nonequilibrium steady state of a Mott insulator coupled to a thermostat and driven by a constant electric field, starting from weak fields, until the dielectric breakdown, and beyond. We find that the conventional Zener picture does not describe the steady-state physics. In particular, the current at weak field is found to be controlled by the dissipation. Moreover, in connection with the electric-field-driven dimensional crossover, we find that the dielectric breakdown occurs when the field strength is on the order of the Mott gap of the corresponding lower-dimensional system. We also report a resonance and the meltdown of the quasiparticle peak when the field strength is half of this Mott gap.
We study the single-band Hubbard model in the presence of a large spatially uniform electric field out of equilibrium. Using the Keldysh nonequilibrium formalism, we solve the problem using perturbation theory in the Coulomb interaction UUU. We present numerical results for the charge current, the total energy of the system, and the double occupancy on an infinite-dimensional hypercubic lattice with nearest-neighbor hopping. The system is isolated from an external bath and is in the paramagnetic state. We show that an electric field pulse can drive the system to a steady nonequilibrium state, which does not evolve into a thermal state. We compare results obtained within second-order perturbation theory (SOPT), self-consistent SOPT, and iterated perturbation theory (IPT). We also discuss the importance of initial conditions for a system which is not coupled to an external bath.
Understanding strongly correlated systems driven out of equilibrium is a challenging task necessitating the simultaneous treatment of quantum mechanics, dynamical constraints, and strong interactions. A Mott insulator subjected to a uniform and static electric field is prototypical, raising key questions such as the fate of Bloch oscillations with increasing correlation strength, the approach to a steady-state DC transport regime and the role of dissipation in it, and electric-field-driven phase transitions. Despite tremendous efforts over the past decade employing various numerical and analytical approaches, the manner in which a nonequilibrium steady state gets established has remained an unresolved problem. We develop here an effective large-N\mathcal{N}N Keldysh field theory for studying nonequilibrium transport in a regular one-dimensional dissipative Mott insulator system subjected to a uniform electric field. Upon abruptly turning on the electric field (a quench), a transient oscillatory current response reminiscent of Bloch oscillations is found. In the regime of small tunneling conductance the amplitude of these oscillations, over a large time window, decreases as an inverse square power law in time, ultimately going over to an exponential decay beyond a large characteristic time ${\ensuremath{\tau}}_{d}$ that increases with N.\mathcal{N}.N. Such a relaxation to a steady-state DC response is absent in the dissipation-free Hubbard chain at half filling. The steady-state current at small fields is governed by large-distance cotunneling, a process absent in the equilibrium counterpart. The low-field DC current has a Landau-Zener-Schwinger form but qualitatively differs from the expression for pair-production probability for the dissipation-free counterpart. The breakdown of perturbation theory in the Mott phase possibly signals a nonequilibrium phase transition to a metallic phase. Our study sheds light on the approach of a driven, dissipative strongly correlated system to a nonequilibrium steady state and also provides a general analytic microscopic framework for understanding other nonequilibrium phenomena in these systems.
We study the spectral and transport properties of a Mott insulator driven by a static electric field into a nonequilibrium steady state. For the dissipation, we consider two mechanisms: wide band fermion reservoirs and phonons included within the Migdal approximation. The electron correlations are treated via nonequilibrium dynamical mean field theory with an impurity solver suitable for strong correlations. We find that dissipation via phonons is limited to restricted ranges of field values around Wannier-Stark resonances. To cover the full range of field strengths, we allow for a small coupling with fermionic baths, which stabilizes the solution. When considering both dissipation mechanisms, we find that phonons enhance the current for field strengths close to half of the gap while lowering it at the gap resonance as compared to the purely electronic dissipation used by Murakami and Werner [Murakami and Werner, Phys. Rev. B 98, 075102 (2018)]. Once phonons are the only dissipation mechanism, the current in the metallic phase is almost one order of magnitude smaller than the typical values obtained by coupling to a fermionic bath. In this case, the transport regime is characterized by an accumulation of charge in the upper Hubbard band described by two effective chemical potentials.
We use a self-consistent strong-coupling expansion for the self-energy (perturbation theory in the hopping) to describe the nonequilibrium dynamics of strongly correlated lattice fermions. We study the three-dimensional homogeneous Fermi-Hubbard model driven by an external electric field showing that the damping of the ensuing Bloch oscillations depends on the direction of the field and that for a broad range of field strengths a long-lived transient prethermalized state emerges. This long-lived transient regime implies that thermal equilibrium may be out of reach of the time scales accessible in present cold atom experiments but shows that an interesting new quasiuniversal transient state exists in nonequilibrium governed by a thermalized kinetic energy but not a thermalized potential energy. In addition, when the field strength is equal in magnitude to the interaction between atoms, the system undergoes a rapid thermalization, characterized by a different quasiuniversal behavior of the current and spectral function for different values of the hopping.