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arXiv:1401.4089 · Physical Review B

Long-lived nonequilibrium states in the Hubbard model with an electric field

A. V. Joura, James K. Freericks, Alexander I. Lichtenstein

Published Jun 26, 2015• 17 citations
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Abstract

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 $U$. 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.

DOI
10.1103/physrevb.91.245153

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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.
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.
The Keldysh boundary problem in a nonequilibrium Falicov-Kimball model in infinite dimensions is studied within the truncated and self-consistent perturbation theories and the dynamical mean-field theory. Within the model the system is started in equilibrium, and later a uniform electric field is turned on. The Kadanoff-Baym-Wagner equations for the nonequilibrium Green's functions are derived and numerically solved. The contributions of initial correlations are studied by monitoring the system evolution. It is found that the initial correlations are essential for establishing the full electron correlations of the system and are independent of the starting time of preparing the system in equilibrium. By examining the contributions of the initial correlations to the electric current and the double occupation, we find that the contributions are small in relation to the total value of those physical quantities when the interaction is weak, and significantly increase when the interaction is strong. The neglect of initial correlations may cause artifacts in the nonequilibrium properties of the system, especially in the strong-interaction case.
We perform a perturbative analysis for the nonequilibrium Green's functions of the spinless Falicov-Kimball model in the presence of an arbitrary external time-dependent but spatially uniform electric field. The conduction-electron self-energy is found from a strictly truncated second-order perturbative expansion in the local electron-electron repulsion UUU. We examine the current at half-filling and compare to both the semiclassical Boltzmann equation and exact numerical solutions for the contour-ordered Green's functions from a transient-response formalism (in infinite dimensions) on the Kadanoff-Baym-Keldysh contour. We find that a strictly truncated perturbation theory in the two-time formalism cannot reach the long-time limit of the steady state; instead, it illustrates pathological behavior for times larger than approximately 2∕U2∕U2∕U.
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 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.
A model for nonequilibrium dynamical mean-field theory is constructed for the infinite-dimensional Hubbard lattice. We impose nonequilibrium by expressing the physical orbital as a superposition of a left (L)(L)(L)-moving and right (R)(R)(R)-moving electronic state with the respective chemical potentials ${\ensuremath{\mu}}_{L}$ and ${\ensuremath{\mu}}_{R}$. Using the second-order iterative perturbation theory we calculate the quasiparticle properties as a function of the chemical potential bias between the LLL and RRR movers, i.e., $\ensuremath{\Phi}={\ensuremath{\mu}}_{L}\ensuremath{-}{\ensuremath{\mu}}_{R}$. The evolution of the nonequilibrium quasiparticle spectrum is mapped out as a function of the bias and temperature. The quasiparticle states with the renormalized Fermi-energy scale ${\ensuremath{\epsilon}}_{QP}^{0}$ disappear at $\ensuremath{\Phi}\ensuremath{\sim}{\ensuremath{\epsilon}}_{QP}^{0}$ in the low-temperature limit. The second-order perturbation theory predicts that in the vicinity of the Mott-insulator transition at the Coulomb-parameter U=UcU={U}_{c}U=Uc​, there exists another critical Coulomb-parameter Ud{U}_{d}Ud​ (<Uc)(<{U}_{c})(<Uc​) such that, for Ud<U<Uc{U}_{d}<U<{U}_{c}Ud​<U<Uc​, quasiparticle states are destroyed abruptly when ${({\ensuremath{\epsilon}}_{QP}^{0})}^{2}\ensuremath{\sim}a{(\ensuremath{\pi}{k}_{B}{T}_{c})}^{2}+b{\ensuremath{\Phi}}_{c}^{2}$ with the critical temperature Tc{T}_{c}Tc​, the critical bias ${\ensuremath{\Phi}}_{c}$, and the numerical constants aaa and bbb on the order of unity.