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arXiv:2209.00428 · Physical review. B./Physical review. B

Stochastic semiclassical theory for nonequilibrium electron-phonon coupled systems

Antonio Picano, Francesco Grandi, Philipp Werner, Martin Eckstein

Published Jul 10, 2023• 11 citations
arXivHTMLPDF

Abstract

We discuss a semiclassical approach to solve the quantum impurity model within nonequilibrium dynamical mean-field theory for electron-lattice models. The effect of electronic fluctuations on the phonon is kept beyond Ehrenfest dynamics, leading to a stochastic phonon evolution with damping and noise terms that are self-consistently determined by the electronic correlation functions in the fluctuating phonon field. Together with a solution of the electronic model based on a nonperturbative quantum Boltzmann equation, the approach can be used to address the coupled dynamics of the electrons and the lattice during photoinduced phase transitions. Results for the Anderson-Holstein model are benchmarked against numerically exact quantum Monte Carlo data. We find good agreement for the phonon distribution function at temperatures comparable to the charge ordering temperature. The general formulation can be extended to models with electron-electron interactions or multiorbital systems.

DOI
10.1103/physrevb.108.035115

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We discuss a semiclassical approach to solve the quantum impurity model within nonequilibrium dynamical mean-field theory for electron-lattice models. The effect of electronic fluctuations on the phonon is kept beyond Ehrenfest dynamics, leading to a stochastic phonon evolution with damping and noise terms that are self-consistently determined by the electronic correlation functions in the fluctuating phonon field. Together with a solution of the electronic model based on a nonperturbative quantum Boltzmann equation, the approach can be used to address the coupled dynamics of the electrons and the lattice during photoinduced phase transitions. Results for the Anderson-Holstein model are benchmarked against numerically exact quantum Monte Carlo data. We find good agreement for the phonon distribution function at temperatures comparable to the charge ordering temperature. The general formulation can be extended to models with electron-electron interactions or multiorbital systems.
A semiclassical method is presented for tunneling of self-trapped states in many-body systems with electrons and phonons. An overcomplete set of Slater determinants, lattice coordinates, and lattice momenta is used to represent a functional integral. Stationary phase equations are solved numerically without any constraint on the dynamics of electrons and phonons, i.e., without the use of the adiabatic approximation. To evaluate transition amplitudes, we integrate over small fluctuations in both electronic and phonon degrees of freedom, keeping their time order correctly. This method can be applied to general electron-phonon systems is and useful when self-trapped states have complex structures in charge or spin densities and lattice displacements. The effective hopping strength is calculated for a self-trapped kink in the commensurate charge-density-wave state in one dimension. At strong coupling in the Holstein and attractive Hubbard models, where tunneling involves effectively a single tightly bound bipolaron, this method reproduces previous analytic results. New results are obtained at intermediate coupling where the kink is extended over a couple of lattice sites and for models with both electron-electron and electron-phonon interactions.
Collective orders and photoinduced phase transitions in quantum matter can evolve on timescales which are orders of magnitude slower than the femtosecond processes related to electronic motion in the solid. Quantum Boltzmann equations can potentially resolve this separation of timescales, but are often constructed by assuming the existence of quasiparticles. Here we derive a quantum Boltzmann equation which only assumes a separation of timescales (taken into account through the gradient approximation for convolutions in time), but is based on a nonperturbative scattering integral, and makes no assumption on the spectral function such as the quasiparticle approximation. In particular, a scattering integral corresponding to nonequilibrium dynamical mean-field theory is evaluated in terms of an Anderson impurity model in a nonequilibrium steady state with prescribed distribution functions. This opens the possibility to investigate dynamical processes in correlated solids with quantum impurity solvers designed for the study of nonequilibrium steady states.
We study the relaxation of the Holstein model after a sudden switch-on of the interaction by means of the nonequilibrium dynamical mean field theory, with the self-consistent Migdal approximation as an impurity solver. We show that there exists a qualitative change in the thermalization dynamics as the interaction is varied in the weak-coupling regime. On the weaker interaction side of this crossover, the phonon oscillations are damped more rapidly than the electron thermalization time scale, as determined from the relaxation of the electron momentum distribution function. On the stronger interaction side, the relaxation of the electrons becomes faster than the phonon damping. In this regime, despite long-lived phonon oscillations, a thermalized momentum distribution is realized temporarily. The origin of the ``thermalization crossover'' found here is traced back to different behaviors of the electron and phonon self-energies as a function of the electron-phonon coupling. In addition, the importance of the phonon dynamics is demonstrated by comparing the self-consistent Migdal results with those obtained with a simpler Hartree-Fock impurity solver that neglects the phonon self-energy. The latter scheme does not properly describe the evolution and thermalization of isolated electron-phonon systems.
We introduce a simple ansatz for the wave function of a many-body system based on coupled forward and backward propagating semiclassical trajectories. This method is primarily aimed at, but not limited to, treating nonequilibrium dynamics in electron-phonon systems. The time evolution of the system is obtained from the Euler-Lagrange variational principle, and we show that this ansatz yields Ehrenfest mean-field theory in the limit that the forward and backward trajectories are orthogonal, and in the limit that they coalesce. We investigate accuracy and performance of this method by simulating electronic relaxation in the spin-boson model and the Holstein model. Although this method involves only pairs of semiclassical trajectories, it shows a substantial improvement over mean-field theory, capturing quantum coherence of nuclear dynamics as well as electron-nuclear correlations. This improvement is particularly evident in nonadiabatic systems, where the accuracy of this coupled trajectory method extends well beyond the perturbative electron-phonon coupling regime. This approach thus provides an attractive route forward to the ab initio description of relaxation processes, such as thermalization, in condensed phase systems.
An investigation is presented of the utility of semiclassical approximations for solving the quantum-impurity problems arising in the dynamical-mean-field approach to correlated-electron models. The method is based on performing an exact numerical integral over the zero-Matsubara-frequency component of the spin part of a continuous Hubbard-Stratonovich field, along with a spin-field-dependent steepest descent treatment of the charge part. We test this method by applying it to one- or two-site approximations to the single-band Hubbard model with different band structures, and comparing the results to quantum Monte Carlo and simplified exact diagonalization calculations. The resulting electron self-energies, densities of states, and magnetic transition temperatures show reasonable agreement with the quantum Monte Carlo simulation over wide parameter ranges, suggesting that the semiclassical method is useful for obtaining a reasonable picture of the physics in situations where other techniques are too expensive.
We study equilibrium and nonequilibrium properties of electron-phonon systems described by the Hubbard-Holstein model using dynamical mean-field theory. In equilibrium, we benchmark the results for impurity solvers based on the one-crossing approximation and slave-rotor approximation against non-perturbative numerical renormalization group reference data. We also examine how well the low-energy properties of the electron-boson coupled systems can be reproduced by an effective static electron-electron interaction. The one-crossing and slave-rotor approximations are then used to simulate insulator-to-metal transitions induced by a sudden switch-on of the electron-phonon interaction. The slave-rotor results suggest the existence of a critical electron-phonon coupling above which the system is transiently trapped in a nonthermal metallic state with coherent quasiparticles. The same quench protocol in the one-crossing approximation results in a bad metallic state.