Grassmann time-evolving matrix product operators: An efficient numerical approach for fermionic path integral simulations

TL;DR

This paper introduces a Grassmann time-evolving matrix product operator method for simulating fermionic open quantum systems, effectively handling Grassmann algebra and enabling accurate studies of non-Markovian dynamics and strong coupling physics.

Contribution

The paper presents a novel Grassmann tensor network approach for fermionic path integrals, extending tensor network methods to fermionic environments in open quantum systems.

Findings

Demonstrates robustness and accuracy in simulating fermionic dynamics

Provides benchmarks for structured fermionic environments

Offers an alternative impurity solver for strongly correlated systems

Abstract

Developing numerical exact solvers for open quantum systems is a challenging task due to the non-perturbative and non-Markovian nature when coupling to structured environments. The Feynman-Vernon influence functional approach is a powerful analytical tool to study the dynamics of open quantum systems. Numerical treatments of the influence functional including the quasi-adiabatic propagator technique and the tensor-network-based time-evolving matrix product operator method, have proven to be efficient in studying open quantum systems with bosonic environments. However, the numerical implementation of the fermionic path integral suffers from the Grassmann algebra involved. In this work, we present a detailed introduction of the Grassmann time-evolving matrix product operator method for fermionic open quantum systems. In particular, we introduce the concepts of Grassmann tensor, signed matrix product operator, and Grassmann matrix product state to handle the Grassmann path integral. Using the single-orbital Anderson impurity model as an example, we review the numerical benchmarks for structured fermionic environments for real-time nonequilibrium dynamics, real-time and imaginary-time equilibration dynamics, and its application as an impurity solver. These benchmarks show that our method is a robust and promising numerical approach to study strong coupling physics and non-Markovian dynamics. It can also serve as an alternative impurity solver to study strongly-correlated quantum matter with dynamical mean-field theory.