Optimized Attenuated Interaction: Enabling Stochastic Bethe-Salpeter Spectra for Large Systems

TL;DR

This paper introduces an improved stochastic method for solving the Bethe-Salpeter equation that efficiently handles large systems by reducing the stochastic sampling complexity through an optimized interaction separation.

Contribution

It presents a novel stochastic formalism that separates the interaction to minimize sampling variance, enabling scalable Bethe-Salpeter spectra calculations for large systems.

Findings

Number of samples needed is system-size independent.

Method maintains accuracy with small stochastic orbital count.

Scaling prefactor remains small despite cubic theoretical complexity.

Abstract

We develop an improved stochastic formalism for the Bethe-Salpeter equation, based on an exact separation of the effective-interaction $W$ to two parts, $W=(W-v_W)+v_W$ where the latter is formally any translationally-invariant interaction $v_W(r-r')$. When optimizing the fit of $v_W$ exchange kernel to $W$, by using a stochastic sampling of $W$, the difference $W-v_W$ becomes quite small. Then, in the main BSE routine, this small difference is stochastically sampled. The number of stochastic samples needed for an accurate spectrum is then largely independent of system size. While the method is formally cubic in scaling, the scaling prefactor is small due to the constant number of stochastic orbitals needed for sampling $W$.