In this talk, I will discuss Quantum Monte Carlo (QMC) calculations of the fundamental (charge) and optical (neutral) gaps in solid-state systems. Understanding finite size extrapolation is essential for accurately determining energy gaps in extended systems. We show that finite size effects of charge excitations decay slowly with 1/L, where L is the linear extension of the supercell. I will demonstrate how to correct both the leading and subleading finite-size errors. In the case of delocalized neutral excitations, the decay rate is also 1/L [1]. However, the particle-hole attraction will eventually localize neutral excitations for large enough supercells and the apparent 1/L decay for small supercells will be modified to a faster 1/L^3. Estimations of the scale where such excitonic effects set in can be used to provide simple finite size extrapolations for neutral gaps including electron-hole attraction [2].

I will illustrate the QMC method with examples of solid silicon and carbon. The method for determining fundamental gaps was already verified for solid hydrogen near metallization [3]. However, here a particular focus will be given to solid molecular hydrogen in phase I at ambient temperature and in the pressure range of 5GP a ≤ P ≤ 90GP a. This pressure range makes the system a wide gap insulator, but the nature of the excitation transforms from localized to delocalized upon increasing pressure, making it an ideal system to test our new method. I will discuss the results for neutral excitations obtained from QMC and compare them to recent experimental measurements. Additionally, I will compare the excitation spectra in hydrogen obtained with the Bethe-Salpeter equation to the experimental data and to QMC neutral gaps [4].

[1] Y. Yang, V.Gorelov et al., PRB 101, 085115 (2020)

[2] V. Gorelov et al., arXiv:2303.17944 (2023)

[3] V. Gorelov et al., PRL 124, 116401 (2020)

[4] V. Gorelov et al., in preparation (2023)