The fermion sign problem, when severe, prevents the computation of physical quantities of a system of interacting fermions via stochastic evaluation of its path integral. This is due to the oscillatory nature of the integrand $\exp (-S),$ where $S$ is the imaginary-time action. This issue is a major obstacle to first-principles lattice quantum Monte Carlo studies of excited states of electrons in matter. However, in the Kohn-Sham orbital basis, which is the output of a density-functional theory simulation, the path integral for electrons in a semiconductor nanoparticle has only a mild fermion sign problem and is amenable to evaluation by standard stochastic methods. This is evidenced by our simulations of silicon hydrogen-passivated nanocrystals such as ${\mathrm{Si}}_{35}{\mathrm{H}}_{36},{\mathrm{Si}}_{87}{\mathrm{H}}_{76},{\mathrm{Si}}_{147}{\mathrm{H}}_{100}$, and ${\mathrm{Si}}_{293}{\mathrm{H}}_{172},$ which range in size $1.0-2.4$ nm and contain 176 to 1344 valence electrons. We find that approximating the fermion action by its leading order polarization term results in a positive-definite integrand in the functional integral, and is a very good approximation of the full action. We compute imaginary-time electron propagators and extract the energies of low-lying electron and hole levels. Our quasiparticle gap predictions agree with the results of previous high-precision $G_0{W}_0$ calculations. This formalism allows calculations of more complex excited states such as excitons and trions.

• Received 29 May 2020
• Revised 19 August 2020
• Accepted 3 May 2021

DOI:https://doi.org/10.1103/PhysRevResearch.3.023173

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Published by the American Physical Society