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arXiv:0711.0999 · Physical Review B

Quenching Bloch oscillations in a strongly correlated material: Nonequilibrium dynamical mean-field theory

James K. Freericks

Published Feb 12, 2008• 115 citations
arXivHTMLPDF

Abstract

Dynamical mean-field theory is generalized to solve the nonequilibrium Keldysh boundary problem: a system is started in equilibrium at a temperature $T=0.1$, a uniform electric field is turned on at $t=0$, and the system is monitored as it approaches the steady state. The focus here is on the Bloch oscillations of the current and how they decay after their initial appearance near $t=0$. The system is evolved out to the longest time allowed by our computational resources---in most cases we are unable to reach the steady state. The strongly correlated material is modeled by the spinless Falicov-Kimball model at half-filling on a hypercubic lattice in $d=\ensuremath{\infty}$ dimensions, which has a Mott-like metal-insulator transition at $U=\sqrt{2}$. The computational algorithm employed is highly efficient, parallelizes well, and scales to thousands of processors. For strong fields, we find beats develop with a period of $2\ensuremath{\pi}∕U$, while for strong interactions, the Bloch oscillations are sharply damped and become quite irregular in time.

DOI
10.1103/physrevb.77.075109

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Dynamical mean-field theory is generalized to solve the nonequilibrium Keldysh boundary problem: a system is started in equilibrium at a temperature T=0.1T=0.1T=0.1, a uniform electric field is turned on at t=0t=0t=0, and the system is monitored as it approaches the steady state. The focus here is on the Bloch oscillations of the current and how they decay after their initial appearance near t=0t=0t=0. The system is evolved out to the longest time allowed by our computational resources---in most cases we are unable to reach the steady state. The strongly correlated material is modeled by the spinless Falicov-Kimball model at half-filling on a hypercubic lattice in d=\ensuremath∞d=\ensuremath{\infty}d=\ensuremath∞ dimensions, which has a Mott-like metal-insulator transition at U=2U=\sqrt{2}U=2​. The computational algorithm employed is highly efficient, parallelizes well, and scales to thousands of processors. For strong fields, we find beats develop with a period of 2\ensuremathπ∕U2\ensuremath{\pi}∕U2\ensuremathπ∕U, while for strong interactions, the Bloch oscillations are sharply damped and become quite irregular in time.
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 present a review of our recent work in extending the successful dynamical mean-field theory from the equilibrium case to nonequilibrium cases. In particular, we focus on the problem of turning on a spatially uniform, but possibly time varying, electric field (neglecting all magnetic field effects). We show how to work with a manifestly gauge-invariant formalism, and compare numerical calculations from a transient-response formalism to different types of approximate treatments, including the semiclassical Boltzmann equation and perturbation theory in the interaction. In this review, we solve the nonequilibrium problem for the Falicov-Kimball model, which is the simplest many-body model and the easiest problem to illustrate the nonequilibrium behavior in both diffusive metals and Mott insulators. Due to space restrictions, we assume the reader already has some familiarity both with the Kadanoff-Baym-Keldysh nonequilibrium formalism and with equilibrium dynamical mean-field theory; we provide a guide to the literature where additional details can be found.
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.
Dynamical mean-field theory (DMFT) is a widely used, nonperturbative computational framework for strongly correlated electron systems. This thesis extends DMFT in two directions: treating disorder and higher-order correlators away from equilibrium, and using near-term quantum computers as impurity solvers. In the first part, we develop two methodological advances for the nonequilibrium DMFT treatment of disordered, interacting systems. The first overcomes the central computational bottleneck of the recently developed nonequilibrium DMFT plus coherent potential approximation (DMFT+CPA) by replacing its explicit numerical disorder averaging with an exact analytical integration, exploiting a normalization property of the Schwinger-Keldysh contour; applied to a disorder-and-interaction quench of the Anderson-Hubbard model, this reveals that weak random disorder promotes thermalization. The second demonstrates that the out-of-time-order correlator (OTOC) can be successfully computed within the existing DMFT+CPA framework, by generalizing the Schwinger-Keldysh contour to a double-folded structure; as a proof-of-principle demonstration, weak disorder is found to accelerate the OTOC's decay in the half-filled Anderson-Hubbard model in the intermediate-coupling regime. In the second part, we explore the use of near-term quantum computers as DMFT impurity solvers. A stable real-time DMFT iteration scheme, designed for compatibility with quantum hardware, is first developed and validated classically against the metal-to-insulator transition. A symmetry-adapted variational quantum eigensolver (VQE) is then benchmarked as the Anderson impurity solver for the converged DMFT Hamiltonian, and is found to reproduce ground-state energies accurately and to extract the single-particle Green's function reliably at intermediate to strong coupling, while accurate ground-state energies are shown not to guarantee accurate dynamics at weak coupling, a key challenge for extending such approaches to correlated metals. Together, we demonstrate that DMFT can be methodologically extended to study disordered and interacting systems away from equilibrium, and to incorporate near-term quantum hardware as a genuinely new computational resource to enhance its capabilities.
An exact formalism for calculating the retarded and advanced Green's functions of strongly correlated lattice models in a uniform electric field is derived within dynamical mean-field theory. To illustrate the method, we solve for the nonequilibrium density of states of the Hubbard model in both the metallic and Mott-insulating phases at half-filling (with an arbitrary strength electric field) by employing the approximate numerical renormalization group as the impurity solver. This general approach can be applied to any strongly correlated lattice model in the limit of large dimensions.