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

Abstract

We develop a hierarchical equations of motion (HEOM) approach to compute real-time singleparticle 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.

Cumulant Methods for Electron-Phonon Problems I: Perturbative Expansions

“…Two recent versions of HEOM have provided practical routes to circumventing instabilities. 39,68 To compute G (k, t) with HEOM we rewrite…”

“…[25][26][27] DQMC has provided the temperature dependent mobilities for the Holstein and Fröhlich models; however, as with spectral information this approach is restricted by an ill-conditioned analytic continuation procedure. 4,28 More recently, numerically exact dynamical methods based on DMRG+VD, 29 a generalized cluster expansion, 30 and the Hierarchical Equations of Motion (HEOM) [31][32][33][34][35][36][37][38][39] approach have been introduced for real-time dynamics in lattice models with EPIs.…”

Real-time non-adiabatic dynamics in the one-dimensional Holstein model: Trajectory-based vs exact methods

“…These methods allow to study both single polarons 87,101 and charge-density wave states 88 in systems with a finite electronic density. For the polaron problem, alternative quantum-chemistry methods are the hierarchical equation-of-motion method 114,115,157 or the Davydov D 1 or D 2 ansatz. 116,117,158,159 Recently, the multiconfigurational Ehrenfest method has been applied to the polaron problem with promising results.…”