Spectral and thermodynamic properties of the Holstein polaron: Hierarchical equations of motion approach

TL;DR

This paper introduces a symmetry-adapted hierarchical equations of motion (HEOM) approach to accurately compute spectral and thermodynamic properties of the Holstein polaron at finite temperature, overcoming numerical challenges of previous methods.

Contribution

The paper develops a momentum-space HEOM method that improves numerical stability and accuracy in studying the Holstein model, validated by comparison with quantum Monte Carlo results.

Findings

HEOM accurately computes spectral functions and thermodynamics for chains up to ten sites.

Finite-size effects are weak, making small-chain results representative of larger systems.

HEOM results compare favorably with existing literature and QMC data.

Abstract

We develop a hierarchical equations of motion (HEOM) approach to compute real-time single-particle correlation functions and thermodynamic properties of the Holstein model at finite temperature. We exploit the conservation of the total momentum of the system to formulate the momentum-space HEOM whose dynamical variables explicitly keep track of momentum exchanges between the electron and phonons. Our symmetry-adapted HEOM enable us to overcome the numerical instabilities inherent to the commonly used real-space HEOM. The HEOM method is then used to study the spectral function and thermodynamic quantities of chains containing up to ten sites. The HEOM results compare favorably to existing literature. To provide an independent assessment of the HEOM approach and to gain insight into the importance of finite-size effects, we devise a quantum Monte Carlo (QMC) procedure to evaluate finite-temperature single-particle correlation functions in imaginary time and apply it to chains containing up to twenty sites. QMC results reveal that finite-size effects are quite weak, so that the results on 5 to 10-site chains, depending on the parameter regime, are representative of larger systems. A detailed comparison between the HEOM and QMC data place our HEOM method among reliable methods to compute real-time finite-temperature correlation functions in parameter regimes ranging from low- to high-temperature, and weak- to strong-coupling regime.