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arXiv:1605.00121 · Fortschritte der Physik

The role of average time dependence on the relaxation of excited electron populations in nonequilibrium many‐body physics

Alexander F. Kemper, H. R. Krishnamurthy, James K. Freericks

Published Aug 22, 2016• 4 citations
arXivHTMLPDF

Abstract

We examine the exact equation of motion for the relaxation of populations of strongly correlated electrons after a nonequilibrium excitation by a pulsed field, and prove that the populations do not change when the Green's functions have no average time dependence. We show how the average time dependence enters into the equation of motion to lowest order and describe what governs the relaxation process of the electron populations in the long‐time limit. While this result may appear, on the surface, to be required by any steady‐state solution, the proof is nontrivial, and provides new critical insight into how nonequilibrium populations relax, which goes beyond the assumption that they thermalize via a simple relaxation rate determined by the imaginary part of the self‐energy, or that they can be described by a quasi‐equilibrium condition with a Fermi‐Dirac distribution and a time‐dependent temperature. We also discuss the implications of this result to approximate theories, which may not satisfy the exact relation in the equation of motion.

DOI
10.1002/prop.201600042

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Bulletin of the American Physical Society•Mar 6, 2015•0 citations
We examine the exact equation of motion for the relaxation of populations of strongly correlated electrons after a nonequilibrium excitation by a pulsed field, and prove that the populations do not change when the Green's functions have no average time dependence. We show how the average time dependence enters into the equation of motion to lowest order and describe what governs the relaxation process of the electron populations in the long‐time limit. While this result may appear, on the surface, to be required by any steady‐state solution, the proof is nontrivial, and provides new critical insight into how nonequilibrium populations relax, which goes beyond the assumption that they thermalize via a simple relaxation rate determined by the imaginary part of the self‐energy, or that they can be described by a quasi‐equilibrium condition with a Fermi‐Dirac distribution and a time‐dependent temperature. We also discuss the implications of this result to approximate theories, which may not satisfy the exact relation in the equation of motion.
We compare the decay rates of excited populations directly calculated within a Keldysh formalism to the equation of motion of the population itself for a Hubbard-Holstein model in two dimensions. While it is true that these two approaches must give the same answer, it is common to make a number of simplifying assumptions, within the differential equation for the populations, that allows one to interpret the decay in terms of hot electrons interacting with a phonon bath. Here, we show how care must be taken to ensure an accurate treatment of the equation of motion for the populations due to the fact that there are identities that require cancellations of terms that naively look like they contribute to the decay rates. In particular, the average time dependence of the Green’s functions and self-energies plays a pivotal role in determining these decay rates.
From the early days of many-body physics, it was realized that the self-energy governs the relaxation or lifetime of the retarded Green’s function. So it seems reasonable to directly extend those results into the nonequilibrium domain. But experiments and calculations of the response of quantum materials to a pump show that the relationship between the relaxation and the self-energy only holds in special cases. Experimentally, the decay time for a population to relax back to equilibrium and the linewidth measured in a linear-response angle-resolved photoemission spectroscopy differ by large amounts. Theoretically, aside from the weak-coupling regime where the relationship holds, one also finds deviations and additionally one sees violations of Mathiessen’s rule. In this work, we examine whether looking at an effective transport relaxation time helps to analyze the decay times of excited populations as they relax back to equilibrium. We conclude that it may do a little better, but it has a fitting parameter for the overall scale which must be determined.
Many-body quantum systems in nonequilibrium remain one of the frontiers of many-body physics. While there has been significant advances in describing the short-time evolution of these systems using a variety of different numerical algorithms, it has been quite difficult to evolve a system from an equilibrium state prior to the application of a driving field, to the long-time steady (or periodically oscillating) state. These dynamics are complex: the retarded quantities tend to approach their long-time limit much faster than the lesser (or greater) quantities. Recent work on strongly correlated electrons in DC electric fields illustrated that the system may evolve through successive quasi-thermal states obeying an effective fluctuation-dissipation theorem in time. We demonstrate an extrapolation scheme that uses the short-time transient calculation to obtain the retarded quantities and to extract how the lesser/greater quantities vary with time and then extend the numerical solutions all the way to the steady state, with minimal additional computational cost. Our approach focuses on extrapolating the electronic self-energy and then employing that to determine the Green's function and various experimentally relevant expectation values.
It is important that any dynamics method approaches the correct population distribution at long times. In this paper, we derive a one-body reduced density matrix dynamics for electrons in energetic contact with a bath. We obtain a remarkable equation of motion which shows that in order to reach equilibrium properly, rates of electron transitions depend on the density matrix. Even though the bath drives the electrons towards a Boltzmann distribution, hole blocking factors in our equation of motion cause the electronic populations to relax to a Fermi-Dirac distribution. These factors are an old concept, but we show how they can be derived with a combination of time-dependent perturbation theory and the extended normal ordering of Mukherjee and Kutzelnigg for a general electronic state. The resulting non-equilibrium kinetic equations generalize the usual Redfield theory to many-electron systems, while ensuring that the orbital occupations remain between zero and one. In numerical applications of our equations, we show that relaxation rates of molecules are not constant because of the blocking effect. Other applications to model atomic chains are also presented which highlight the importance of treating both dephasing and relaxation. Finally, we show how the bath localizes the electron density matrix.
We examine electron-electron mediated relaxation following ultrafast electric field pump excitation of the fermionic degrees of freedom in the Falicov-Kimball model for correlated electrons. The results reveal a dichotomy in the temporal evolution of the system as one tunes through the Mott metal-to-insulator transition: in the metallic regime relaxation can be characterized by evolution toward a steady state well described by Fermi-Dirac statistics with an increased effective temperature; however, in the insulating regime this quasithermal paradigm breaks down with relaxation toward a nonthermal state with a complicated electronic distribution as a function of momentum. We characterize the behavior by studying changes in the energy, photoemission response, and electronic distribution as functions of time. This relaxation may be observable qualitatively on short enough time scales that the electrons behave like an isolated system not in contact with additional degrees of freedom which would act as a thermal bath, especially when using strong driving fields and studying materials whose physics may manifest the effects of correlations.
Nonequilibrium dynamical mean field theory and nonequilibrium self-consistent strong coupling expansion[1] are used to study the relaxation of correlated quantum systems driven out of equilibrium by DC electric fields[2]. Both the FalicovKimball and the Hubbard model are found to exhibit regimes of monotonic or oscillatory themalization as well as regimes where they evolve in a monotonic or oscillatory manner towards a non-thermal state. This suggests that driven quantum systems have a richer behavior than their quenched counterparts and that integrability does not play as critical a role. In the monotonic thermalization scenario, the system evolves through successive quasi-thermal states and it is possible to extrapolate its long time properties from its transient; bridging the gap between the transient and the steady state with very little computational cost. Furthermore, regardless of the relaxation scenario, it is interesting to ask how the particles are distributed as the system evolves in time. We will show that non-trivial parameter-dependent patterns are formed when the system is visualized in momentum space [3]. These features should be observable in current cold atom experiments.