Efficient low temperature simulations for fermionic reservoirs with the hierarchical equations of motion method: Application to the Anderson impurity model

TL;DR

This paper introduces a barycentric rational approximation method to improve the efficiency and accuracy of the hierarchical equations of motion for simulating low-temperature fermionic reservoirs, enabling better modeling of complex baths.

Contribution

The authors develop a novel rational decomposition technique using barycentric representation to enhance HEOM simulations at low temperatures and with complex bath structures.

Findings

Reduces the number of basis functions needed for reservoir correlation functions.

Enables accurate HEOM simulations at ultra-low temperatures.

Demonstrates stability and efficiency in the Anderson impurity model.

Abstract

The hierarchical equations of motion (HEOM) approach is an accurate method to simulate open system quantum dynamics, which allows for systematic convergence to numerically exact results. To represent the effects of the bath, the reservoir correlation functions are usually decomposed into the summation of multiple exponential terms in the HEOM method. Since the reservoir correlation functions become highly non-Markovian at low temperatures or when the bath has complex band structures, a present challenge is to obtain accurate exponential decompositions that allow efficient simulation with the HEOM. In this work, we employ the barycentric representation to approximate the Fermi function and hybridization functions in the frequency domain. The new method, by approximating these functions with optimized rational decomposition, greatly reduces the number of basis functions in decomposing the reservoir correlation functions, which further allows the HEOM method to be applied to ultra-low temperature and general bath structures. We demonstrate the efficiency, accuracy, and long-time stability of the new decomposition scheme by applying it to the Anderson impurity model (AIM) in the low-temperature regime with the Lorentzian and tight-binding hybridization functions.