Homogenization of the Dirac operator with position-dependent mass

TL;DR

This paper studies how a two-dimensional Dirac operator with a position-dependent, piecewise constant mass converges to an effective operator with a constant mass as the scale of the inclusions shrinks, providing convergence rates.

Contribution

It proves norm resolvent convergence of the Dirac operator with periodic mass inclusions to an effective operator with a constant mass, under general geometric assumptions.

Findings

Convergence in the norm resolvent sense as the period tends to zero.

Explicit estimates of the convergence speed based on geometric scaling.

Validation of the homogenization process for Dirac operators with periodic mass distributions.

Abstract

We address the homogenization of the two-dimensional Dirac operator with position-dependent mass. The mass is piecewise constant and supported on small pairwise disjoint inclusions evenly distributed along an $\varepsilon$-periodic square lattice. Under rather general assumptions on geometry of these inclusions we prove that the corresponding family of Dirac operators converges as $\varepsilon\to 0$ in the norm resolvent sense to the Dirac operator with a constant effective mass provided the masses in the inclusions are adjusted to the scaling of the geometry. We also estimate the speed of this convergence in terms of the scaling rates.