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Physical review. B./Physical review. B

Canonical derivation of the fermionic influence superoperator

Mauro Cirio, Po-Chen Kuo, Yueh-Nan Chen, Franco Nori, Neill Lambert

Published Jan 13, 2022• 24 citations
Publisher

Abstract

The Feynman-Vernon influence functional is a powerful result, based on path integrals, for describing the reduced state of a system interacting with a Gaussian environment. Here, the authors derive in detail a canonical (i.e., operatorial) form of the influence functional for fermionic environments, and show how it can be used to compute the dynamics under arbitrary parity-symmetry initial conditions as well as to derive generalized versions of the Lindblad master equation and the hierarchical equations of motion.

DOI
10.1103/physrevb.105.035121

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The Feynman-Vernon influence functional is a powerful result, based on path integrals, for describing the reduced state of a system interacting with a Gaussian environment. Here, the authors derive in detail a canonical (i.e., operatorial) form of the influence functional for fermionic environments, and show how it can be used to compute the dynamics under arbitrary parity-symmetry initial conditions as well as to derive generalized versions of the Lindblad master equation and the hierarchical equations of motion.
In this paper the so-called Feynman-Vernon influence functional theory has been generalized in order to include the possibility of dealing with initial conditions of the system plus environment other than the factorizable one. It has been shown that the new influence functional can now be written in terms of paths which represent the coupling of quantum-statistical to quantum-dynamical effects. Once the thermal paths are integrated out of this new influence functional, a generalization of the original Feynman-Vernon expression is obtained.
We present a nonperturbative and formally exact approach for the charge transport in interacting nanojunctions based on a real-time path-integral formulation of the reduced system dynamics. For reservoirs of noninteracting fermions, the exact trace over the leads' degrees of freedom results in the time-nonlocal Feynman-Vernon influence functional, a functional of the Grassmann-valued paths of the nanojunction, which induces correlations among the tunneling transitions in and out of the nanojunction. An expansion of the influence functional in terms of the number of tunneling transitions, and integration of the Grassmann variables between the tunneling times, allows us to obtain a still exact generalized master equation for the populations of the reduced density matrix in the occupation-number representation, as well as a formally exact expression for the current. By borrowing the nomenclature of the famous spin-boson model, we parametrize the two-state dynamics of each single-particle fermionic degree of freedom, in the occupation-number representation, in terms of blips and sojourns. We apply our formalism to the exactly solvable resonant level model (RLM) and to the single-impurity Anderson model (SIAM), the latter being a prototype system for studying strong correlations. For both systems, we demonstrate a hierarchical diagrammatic structure. While the hierarchy closes at the second tier for the RLM, this is not the case for the interacting SIAM. Upon inspection of the current kernel, known results from various perturbative and nonperturbative approximation schemes to quantum transport in the SIAM are recovered. Finally, a noncrossing approximation for the hierarchical kernel is developed, which enables us to systematically decrease temperature at each next level of the approximation. Analytical results for a simplified fourth-tier scheme are presented both in equilibrium and nonequilibrium and with an applied magnetic field.
The equation of motion for the reduced Wigner function of a system coupled to an external quantum system is presented for the specific case when the external quantum system can be modeled as a set of harmonic oscillators. The result is derived from the Wigner function formulation of the Feynman-Vernon influence functional theory. It is shown how the true self-energy for the equation of motion is connected with the influence functional for the path integral. Explicit expressions are derived in terms of the bare Wigner propagator. Finally, we show under which approximations the resulting equation of motion reduces to the Wigner-Boltzmann equation.
We establish a direct connection between the Feynman-Vernon path integral formalism for open quantum systems and the Wiener path integral used in classical stochastic dynamics. By considering a generalized influence functional in the strong decoherence limit, we demonstrate that integrating over the quantum coherence length leads to a derivation of stochastic Langevin dynamics. Specifically, we show that the quantum Feynman measure transforms into the stochastic Wiener measure. Applying this framework to the Wigner function representation, we show that the system follows a stochastic path interpretable via classical probability theory. Finally, we address the inverse problem: constructing an equivalent quantum influence functional from a given classical Langevin equation.
In the path integral approach we describe evolution of interacting electromagnetic and fermionic fields by the use of density matrix formalism. The equation for density matrix and transitions probability for fermionic field is obtained as average of electromagnetic field influence functional. We obtain a formula for electromagnetic field influence functional calculating for its various initial and final state. We derive electromagnetic field influence functional when its initial and final states are vacuum. We present Lagrangian for relativistic fermionic field under influence of electromagnetic field vacuum.