Grassmann Time-Evolving Matrix Product Operators for Quantum Impurity Models

TL;DR

This paper introduces Grassmann time-evolving matrix product operators, a novel tensor network method for fermionic impurity models, enabling efficient non-equilibrium simulations by directly handling Grassmann path integrals.

Contribution

It develops the first full fermionic analog of TEMPO that directly manipulates Grassmann path integrals with comparable efficiency to bosonic methods.

Findings

Demonstrates superior performance on single impurity Anderson models

Achieves favorable complexity scaling over existing methods

Enables efficient non-equilibrium dynamics simulations for fermionic systems

Abstract

The time-evolving matrix product operators (TEMPO) method, which makes full use of the Feynman-Vernon influence functional, is the state-of-the-art tensor network method for bosonic impurity problems. However, for fermionic impurity problems the Grassmann path integral prohibits application of this method. We develop Grassmann time-evolving matrix product operators, a full fermionic analog of TEMPO, that can directly manipulates Grassmann path integrals with similar numerical cost as the bosonic counterpart. We further propose a zipup algorithm to compute expectation values on the fly without explicitly building a single large augmented density tensor, which boosts our efficiency on top of the vanilla TEMPO. Our method has a favorable complexity scaling over existing tensor network methods, and we demonstrate its performance on the non-equilibrium dynamics of the single impurity Anderson models. Our method solves the long standing problem of turning Grassmann path integrals into efficient numerical algorithms, which could significantly change the application landscape of tensor network based impurity solvers, and could also be applied for broader problems in open quantum physics and condensed matter physics.