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Nonequilibrium dynamics of correlated disordered systems under interaction quenches

Eric Dohner

Published Jan 1, 2023• 0 citations
Publisher

Abstract

Correlated, disordered lattice fermion systems out of equilibrium have been understudied, at least in part for lack of tools to study them. The development of a new method, the NE-DMFT+CPA, is explicated, for investigating the properties of these systems. This method combines DMFT and the CPA using the Kadanoff-Baym-Keldysh contour-time formalism to study the dynamics of interacting disordered systems out of equilibrium. The method is detailed, then applied to the case of the Anderson-Hubbard model: first to replicate well-known equilibrium results, then to study the case of the model under the effect of an interaction quench. Here the effect of disorder on the relaxation of the system is studied by the time-dependence of the energies and momentum distribution function, which display the screening of the Coulomb interaction by disorder. It is then used to examine the thermalization of the same system by using the fluctuation-dissipation theorem to extract the distribution function and thereby the effective temperature after relaxation. This allows the characterization of the effect of the interplay of the disorder strength and the Coulomb interaction on the thermalization behavior. It is found that the heating of the system due to the quench is mitigated by stronger disorder. Finally, current progress of the work to apply the method to binary alloy disorder and an interaction ramp is summarized. Binary disorder appears to have the effect of damping the initial transient of the potential energy more strongly than the comparable uniformly-disordered system. The relaxation of the energies under the effect of an interaction ramp suggests that the crossing of the kinetic energies may be affected by the form of the time-dependent interaction.

DOI
10.54014/59zh-kvq0

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Correlated, disordered lattice fermion systems out of equilibrium have been understudied, at least in part for lack of tools to study them. The development of a new method, the NE-DMFT+CPA, is explicated, for investigating the properties of these systems. This method combines DMFT and the CPA using the Kadanoff-Baym-Keldysh contour-time formalism to study the dynamics of interacting disordered systems out of equilibrium. The method is detailed, then applied to the case of the Anderson-Hubbard model: first to replicate well-known equilibrium results, then to study the case of the model under the effect of an interaction quench. Here the effect of disorder on the relaxation of the system is studied by the time-dependence of the energies and momentum distribution function, which display the screening of the Coulomb interaction by disorder. It is then used to examine the thermalization of the same system by using the fluctuation-dissipation theorem to extract the distribution function and thereby the effective temperature after relaxation. This allows the characterization of the effect of the interplay of the disorder strength and the Coulomb interaction on the thermalization behavior. It is found that the heating of the system due to the quench is mitigated by stronger disorder. Finally, current progress of the work to apply the method to binary alloy disorder and an interaction ramp is summarized. Binary disorder appears to have the effect of damping the initial transient of the potential energy more strongly than the comparable uniformly-disordered system. The relaxation of the energies under the effect of an interaction ramp suggests that the crossing of the kinetic energies may be affected by the form of the time-dependent interaction.
We present a solution for the nonequilibrium dynamics of an interacting disordered system. The approach adapts the combination of the equilibrium dynamical mean-field theory and the equilibrium coherent potential approximation methods to the nonequilibrium many-body formalism, using the Kadanoff-Baym-Keldysh complex time contour, for the dynamics of interacting disordered systems away from equilibrium. We use our time domain solution to obtain the equilibrium density of states of the disordered interacting system described by the Anderson-Hubbard model, bypassing the necessity for the cumbersome analytical continuation process. We further apply the nonequilibrium solution to the interaction quench problem for an isolated disordered system. Here, the interaction is abruptly changed from zero (noninteracting system) to another constant (finite) value at which it is subsequently kept. We observe via the time dependence of the potential, kinetic, and total energies the effect of disorder on the relaxation of the system as a function of final interaction strength. The real-time approach has the potential to shed light on the fundamental role of disorder in the nonequilibrium dynamics of interacting quantum systems.
Although most studies of strongly correlated systems away from equilibrium have focused on clean systems, it is well known that disorder may significantly modify observed properties in various nontrivial ways. The nonequilibrium interplay of interaction and disorder in these systems thus requires further investigation. In the present paper, we use the recently developed nonequilibrium DMFT+CPA\text{DMFT}+\text{CPA}DMFT+CPA embedding scheme, that combines both the dynamical mean field theory (DMFT) and the coherent potential approximation (CPA) nonequilibrium extensions, to characterize the relaxation and the thermalization of a disordered interacting system described by the Anderson-Hubbard model under an interaction quench. The system, initially in equilibrium at a given temperature, has the interaction abruptly switched from zero to a finite value at a given time. To investigate the role of disorder, we use our effective medium approach to calculate, for different values of the final interaction and of the disorder strength, the distribution functions as the system evolves in time. This allows us to determine the effective temperature after the quench and to analyze the effects of disorder on the thermalization for various interaction strengths. We find that, for moderate interactions after the interaction quench, disorder can tune the final temperature of the system across a broad range of values with increased disorder strength leading to lower effective temperature.
We study the nonequilibrium dynamics of a binary disordered alloy when it is subjected to an interaction quench. Our study uses a nonequilibrium embedding scheme (DMFT+CPA) that combines the capacity of DMFT (dynamical mean field theory) to treat strongly correlated systems and the capacity of CPA (coherent potential approximation) to treat disordered systems, to effectively address the interplay of disorder and interaction for the nonequilibrium system. Our solution is applied on the equilibrium binary disordered and interacting alloy for the calculation of the density of states. This equilibrium density of states shows the modifications of the disorder-driven gap by the interaction and that of the interaction-driven Mott gap by the disorder. Next, we assess the relaxation of the system that is initially in equilibrium at a given temperature, and then abruptly has its interaction strength changed from zero to a finite value at which it is kept constant. The system undergoes a short time transient and then settles into a long-time state in a process that depends in a nontrivial manner on the disorder and interaction strength. In addition, we calculate the effective temperature of the system after it has gone through its initial transient and settled into its long-time state. The long-time effective temperature is increased with increasing final interaction strength. However, for a given final interaction, the final temperature can be tuned by the disorder strength over a broad range, with stronger disorder leading to lower temperature. Altogether, the results illustrate the nontrivial interplay between disorder and interaction for a binary alloy away from equilibrium.
We study the effect of spatially nonlocal correlations on the nonequilibrium dynamics of interacting fermions by constructing the nonequilibrium dynamical cluster theory, a cluster generalization of the nonequilibrium dynamical mean-field theory (DMFT). The formalism is applied to interaction quenches in the Hubbard model in one and two dimensions, and the results are compared with data from single-site DMFT, the time-dependent density matrix renormalization group, and lattice perturbation theory. Both in one and two dimensions the double occupancy quickly thermalizes, while the momentum distribution relaxes only on much longer time scales. For the two-dimensional square lattice we find a strongly momentum-dependent evolution of the momentum distribution around the Fermi energy, with a much faster relaxation near the momenta $(0,\ensuremath{\pi})$ and $(\ensuremath{\pi},0)$ than near $(\ensuremath{\pi}/2,\ensuremath{\pi}/2)$. This result is interpreted as reflecting the momentum-anisotropic quasiparticle lifetime of the marginal Fermi liquid. The method is further applied to the two-dimensional Hubbard model driven by a dc electric field, where the damping of the Bloch oscillation of the current is found to be less effective than predicted by DMFT and lattice perturbation theory.
We discuss a nonequilibrium dynamical mean-field framework for simulating inhomogeneous Hubbard models with local disorders. Our approach treats electron interactions and disorders on equal footing, by considering only local dynamical fluctuations. The theory reduces to nonequilibrium dynamical mean-field theory in the presence of only electron-electron interactions and to the coherent potential approximation in noninteracting systems with disorders. Both time-dependent and steady-state problems are treated by implementing the theory on the three-branch Kadanoff-Baym contour and two-branch Schwinger-Keldysh contour, respectively. Benchmarks on a eight-site cube show that the method yields rather accurate spectral functions in both the weakly and strongly interacting regimes. In a cubic lattice, we demonstrate energy conservation after an interaction quench and thermalization after just a few hopping times in both clean and disordered systems. As an application, we study transport through a serial double quantum-dot sandwiched between two leads, focusing on the current and dot occupations after a voltage quench.
The study of the interplay between interaction-induced correlations and nonequilibrium initial conditions in many-body systems has recently attracted a lot of attention. New experimental techniques provide high control over many-body systems and systematic access to their nonequilibrium regime. In many systems the nonequilibrium properties of a Fermi liquid can be relevant. A first approach to their understanding is the main content of this thesis. At the beginning, nonequilibrium phenomena, Landau's theory of a Fermi liquid, the Hubbard model, time evolution in quantum mechanics, related experiments, and the flow equation method are reviewed. The key observation of this thesis, namely a characteristic mismatch of expectation values in equilibrium and nonequilibrium, is first illustrated for the squeezed oscillator. For this one-particle model Hamiltonian the perturbative approach can be compared with the exact solution and it is shown that the mismatch holds even beyond perturbation theory. Afterwards these observations are generalized to a larger class of one-particle models. Then the nonequilibrium behavior of a Fermi liquid is examined by analyzing the Fermi liquid phase of the Hubbard model in more than one dimension. After a sudden switch-on of a weak two-particle interaction to the noninteracting Fermi gas the relaxation of the many-body system is observed. For this purpose, the flow equation transformation is implemented for the Hubbard Hamiltonian. Then the discussion of the momentum distribution function and of the kinetic energy displays a three-step relaxation behavior of the Fermi liquid from the initial perturbation until thermalization is reached. Firstly, the sudden switch inserts excitation energy into the system which drives the following dynamics. Then a rapid initial build-up of correlations is caused by initial dephasing and leads to the establishment of a quasiparticle picture. By that the system enters into a quasi-stationary state which can be long-lasting for weak interaction. This state shows prethermalization of the kinetic energy which already has relaxed to its final values. However, the momentum distribution still resembles a zero temperature Fermi liquid. Its later relaxation on a second time scale is caused by a residual two-particle interaction which allows for scattering processes and can be described by a quantum Boltzmann equation. The physical origin of this delayed relaxation can be traced back to an interplay of translational invariance of the Hubbard model and the Pauli principle for fermions; together they restrict the phase space for scattering events. Comparing with similar work I conjecture on the generic nature of the findings made for the quenched Fermi liquid in other many-body systems. Finally I point out to the potential relevance of the delayed relaxation for the observation of further nonequilibrium phenomena, for instance in BCS systems. In order to extend the study of sudden switching to arbitrary switching processes the calculation is repeated using the Keldysh perturbation theory. First evaluations for a linear increase of the interaction strength are given. Extensions of this work are suggested and may motivate further research.