Stochastic Equation of Motion Approach to Fermionic Dissipative Dynamics. I. Formalism

TL;DR

This paper introduces a formalism for simulating fermionic open systems using stochastic equations of motion, overcoming numerical challenges with a new mapping scheme that enables efficient and accurate low-temperature dynamics modeling.

Contribution

The paper develops a minimal auxiliary space mapping scheme for stochastic Grassmann fields, making the stochastic equation of motion approach computationally feasible for fermionic systems.

Findings

MAS-SEOM method accurately captures fermionic dissipative dynamics.

The approach connects to perturbation and HEOM theories.

Potential for efficient simulation at ultra-low temperatures.

Abstract

In this work, we establish formally exact stochastic equations of motion (SEOM) theory to describe the dissipative dynamics of fermionic open systems. The construction of the SEOM is based on a stochastic decoupling of the dissipative interaction between the system and fermionic environment, and the influence of environmental fluctuations on the reduced system dynamics is characterized by stochastic Grassmann fields. Meanwhile, numerical realization of the time-dependent Grassmann fields has remained a long-standing challenge. To solve this problem, we propose a minimal auxiliary space (MAS) mapping scheme, with which the stochastic Grassmann fields are represented by conventional c-number fields along with a set of pseudo-levels. This eventually leads to a numerically feasible MAS-SEOM method. The important properties of the MAS-SEOM are analyzed by making connection to the well-established time-dependent perturbation theory and the hierarchical equations of motion (HEOM) theory. The MAS-SEOM method provides a potentially promising approach for accurate and efficient simulation of fermionic open systems at ultra-low temperatures.