Asymptotic gauge-invariant Hybrid High-Order method for magnetic Schr\"odinger equations

TL;DR

This paper presents a gauge-invariant Hybrid High-Order method for the magnetic Schrödinger equation, ensuring physical invariance at the discrete level with optimal convergence and stability.

Contribution

It introduces a discrete covariant gradient operator on polyhedral meshes that guarantees gauge covariance asymptotically, a novel approach in this context.

Findings

Achieves optimal convergence rates.

Preserves a discrete Garding inequality.

Successfully reproduces physical phenomena like the Aharonov-Bohm effect.

Abstract

We introduce a Hybrid High-Order (HHO) method for the Schr\"odinger equation in the presence of a magnetic vector potential. In quantum mechanics, physical observables are invariant under continuous gauge transformations, which must be kept at the discrete level to avoid unphysical artifacts. To address this, we construct a discrete covariant gradient operator on arbitrary polyhedral meshes. We prove that the resulting discrete bilinear form guarantees gauge covariance asymptotically at the discrete level. The resulting scheme achieves optimal convergence rates and preserves a discrete Garding inequality, guaranteeing a stable ground state. The theoretical properties of the scheme are corroborated by numerical experiments, including the computation of the Fock-Darwin fundamental energy and replicating the Aharonov-Bohm effect.