Mixed Quantum-Classical Methods for Polaron Spectral Functions

Abstract

In this work, using two distinct semiclassical approaches, namely the mean-field Ehrenfest (MFE) method and the mapping approach to surface hopping (MASH), we investigate the spectral function of a single charge interacting with phonons on a lattice. This quantity is relevant for the description of angle-resolved photoemission experiments. Focusing on the one-dimensional Holstein model, we compare the performance of these approaches across a range of coupling strengths and lattice sizes, exposing the relative strengths and weaknesses of each. We demonstrate that these approaches can be efficiently applied with reasonable accuracy to ab initio polaron models. Our work provides a route to the calculation of spectral properties in realistic electron-phonon-coupled systems in a computationally inexpensive manner with encouraging accuracy.

Figures (5)

• Figure 1: Spectral functions computed by MFE, MASH, and exact diagonalization, plotted across a range of hopping parameters $J$ and $k$-points at zero temperature. The upper two rows ((a)--(h)) depict results for the 2-site Holstein model with a bath frequency $\omega_0=0.91$ and coupling strength $g=1.23$, while the the lower two rows ((i)--(p)) show results for the 6-site Holstein model with $\omega_0=g=1.0$. For each regime, the top row presents results for $k=0$, and the bottom row present results for $k=\pi$.
• Figure 2: Spectral functions computed using MFE, MASH, and exact diagonalization. The upper two rows illustrate results for the Holstein model with the parameters $J=g=\omega_0=1.0$ across different temperatures. (a)--(d) show results for a 6-site lattice, while (e)--(h) show results for a 16-site lattice. For each lattice size, the left column represents low temperature while the right column represents high temperature. In the lower two rows, the model is evaluated at zero temperature with $J=\omega_0=1.0$ across different coupling strengths $g$, where (i)--(l) correspond to 6 sites and (m)--(p) to 16 sites. For each lattice size, the left column corresponds to $g=0.5$ while the right column corresponds to $g=1.0$.
• Figure 3: (a) Real and (b) imaginary parts of the correlation functions $C_k(t)$ computed using MFE and MASH for the 6-site Holstein model in the regime of $J=\omega_0=g=1.0$ at zero temperature and $k=\pi$.
• Figure 4: Spectral functions of a 6-site Holstein model with parameters $J = g = \omega_0 = 1.0$, evaluated at various temperatures and $k$-points. The left two columns correspond to $k=0$, while the right two columns correspond to $k=\pi$. For each $k$-point, the left column represents a temperature of 0.2, while the right column represents a temperature of 1.0. The first row ((a)--(d)) compares variational diagonalization (VD) results from Ref. Bona2019, full MFE, and cumulant calculations from Ref. Robinson2022. The second row ((e)--(h)) compares full MFE, frozen MFE, and VD. The third row ((i)--(l)) compares full MFE, MFE without the back-reaction force, and VD.
• Figure 5: (Color online) Spectral function at the $\Gamma$-point for an ab initio model of LiF computed using MFE, the CE, and variational Monte Carlo with only longitudinal optical phonons (LO-VMC) for varying $k$-grid sizes at zero temperature. CE data were digitally extracted from Nery et al.Nery2018 The broadening parameter is $0.01$ eV. Further details of the model can be found in Ref. robinson2025ab.