Schur complement solver for Quantum Monte-Carlo simulations of strongly interacting fermions

TL;DR

This paper introduces a non-iterative Schur complement solver for sparse linear systems in Quantum Monte-Carlo simulations of strongly interacting fermions, offering significant speed advantages over traditional iterative methods especially near quantum phase transitions.

Contribution

The paper presents a novel non-iterative Schur complement solver tailored for QMC simulations of fermions, outperforming iterative solvers in speed for large lattice sizes and multiple right-hand sides.

Findings

Solver scales as the cube of lattice sites but is faster in practice.

Significant speed-up over iterative methods near quantum phase transitions.

GPU acceleration further enhances computational efficiency.

Abstract

We present a non-iterative solver based on the Schur complement method for sparse linear systems of special form which appear in Quantum Monte-Carlo (QMC) simulations of strongly interacting fermions on the lattice. While the number of floating-point operations for this solver scales as the cube of the number of lattice sites, for practically relevant lattice sizes it is still significantly faster than iterative solvers such as the Conjugate Gradient method in the regime of strong inter-fermion interactions, for example, in the vicinity of quantum phase transitions. The speed-up is even more dramatic for the solution of multiple linear systems with different right-hand sides. We present benchmark results for QMC simulations of the tight-binding models on the hexagonal graphene lattice with on-site (Hubbard) and non-local (Coulomb) interactions, and demonstrate the potential for further speed-up using GPU.