Matrix-valued Quantum Lattice Boltzmann Method

TL;DR

This paper introduces a novel lattice Boltzmann method tailored for matrix-valued quantum Boltzmann equations, enabling efficient simulation of spin systems with potential applications in spintronics.

Contribution

It develops a matrix-valued LBM incorporating Fermi-Dirac distributions and validates the BGK approximation for quantum spin systems.

Findings

Validates the BGK collision operator in the quantum matrix setting

Provides a framework for simulating complex spin systems

Potential applications in spintronics simulations

Abstract

We devise a lattice Boltzmann method (LBM) for a matrix-valued quantum Boltzmann equation, with the classical Maxwell distribution replaced by Fermi-Dirac functions. To accommodate the spin density matrix, the distribution functions become 2 x 2 matrix-valued. From an analytic perspective, the efficient, commonly used BGK approximation of the collision operator is valid in the present setting. The numerical scheme could leverage the principles of LBM for simulating complex spin systems, with applications to spintronics.