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arXiv:0707.0956 · Physical Review Letters

Real-Time Path Integral Approach to Nonequilibrium Many-Body Quantum Systems

Lothar Mühlbacher, Eran Rabani

Published May 1, 2008• 348 citations
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Abstract

A real-time path-integral Monte Carlo approach is developed to study the dynamics in a many-body quantum system coupled to a phonon background until reaching a nonequilibrium stationary state. The approach is based on augmenting an exact reduced equation for the evolution of the system in the interaction picture which is amenable to an efficient path integral (worldline) Monte Carlo approach. Results obtained for a model of inelastic tunneling spectroscopy reveal the applicability of the approach to a wide range of physically important regimes, including high (classical) and low (quantum) temperatures, and weak (perturbative) and strong electron-phonon couplings.

DOI
10.1103/physrevlett.100.176403

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A real-time path-integral Monte Carlo approach is developed to study the dynamics in a many-body quantum system coupled to a phonon background until reaching a nonequilibrium stationary state. The approach is based on augmenting an exact reduced equation for the evolution of the system in the interaction picture which is amenable to an efficient path integral (worldline) Monte Carlo approach. Results obtained for a model of inelastic tunneling spectroscopy reveal the applicability of the approach to a wide range of physically important regimes, including high (classical) and low (quantum) temperatures, and weak (perturbative) and strong electron-phonon couplings.
A nonperturbative technique is developed to solve for the quantum dynamics of an electron interacting strongly with its environment. Real-time path-integral methods are used to solve numerically a quantum dissipation problem that simulates nonequilibrium electron transport in solids. A linearly coupled electron-phonon interaction is used to model the dissipation. This method is nonperturbative in both the electron-phonon coupling and the external electric field, making it useful for electron-transport calculations in cases of both high fields and large phonon-scattering rates. Monte Carlo sampling over the electron paths is done using the Feynman-Vernon influence functional as a weight. Importance-sampling methods are developed for the numerical solution of the quantum propagation. Results of the calculations are shown which include the motion of the electron, the wave-packet spread, and temperature effects.
We formulate the quasielastic response of a nonrelativistic many-body system at zero temperature in terms of ground-state density-matrix elements and real-time path integrals that embody the final-state interactions. While the former provide the weight for a conventional Monte Carlo calculation, the latter require a more sophisticated treatment. We argue that the stationary-phase Monte Carlo technique recently developed by Doll et al. can be used to study the approach to ``Y scaling.'' We perform calculations for a particle in a potential well in one and three dimensions and compare them with the exact results available for these models.
We developed a path-integral quantum Monte Carlo-based methodology for calculation of polaron mobility in systems with electron-phonon interaction. Within the method, the current-current correlation function in both the imaginary and real time is calculated in a numerically exact way. The choice of basis for representation of the path integral enabled us to reduce the sign problem and perform real-time calculations for longer times. The DC polaron mobility was extracted by performing analytic continuation that makes use of both the real and imaginary-time data. The method was applied to the Holstein polaron model in one dimension. We obtained reliable results for the temperature dependence of the Holstein polaron mobility for interactions ranging from weak to strong and temperatures that are not too low.
We formulate the quasielastic response of a non-relativistic many-body system at zero temperature in terms of ground state density matrix elements and real time path integrals that embody the final state interactions. While the former provide the weight for a conventional Monte Carlo calculation, the latter require a more sophisticated treatment. We argue that the recently developed Stationary Phase Monte Carlo technique can be used to study the approach to "Y-scaling." We perform calculations for a particle in a potential well in one and three dimensions and compare them to the exact results available for these models. We then derive an eikonal approximation to the Path Integrals. This method is suitably generalized to treat strongly repulsive interactions, and allows comparison to Silver's theory of final state interactions in a straightforward way. We also give an exact prescription to calculate the scaling limit for potentials comprising a hard core. Finally, we study the approach to scaling in a model 4He nucleus, and find good agreement with experimental data.
Abstract Monte Carlo techniques have played an important role in understanding strongly correlated systems across many areas of physics, covering a wide range of energy and length scales. Among the many Monte Carlo methods applicable to quantum mechanical systems, the path integral Monte Carlo approach with its variants has been employed widely. Since semi-classical or classical approaches will not be discussed in this review, path integral based approaches can for our purposes be divided into two categories: approaches applicable to quantum mechanical systems at zero temperature and approaches applicable to quantum mechanical systems at finite temperature. While these two approaches are related to each other, the underlying formulation and aspects of the algorithm differ. This paper reviews the path integral Monte Carlo ground state (PIGS) approach, which solves the time-independent Schrödinger equation. Specifically, the PIGS approach allows for the determination of expectation values with respect to eigen states of the few- or many-body Schrödinger equation provided the system Hamiltonian is known. The theoretical framework behind the PIGS algorithm, implementation details, and sample applications for fermionic systems are presented.