Tight binding approximation of continuum 2D quantum materials

Abstract: We consider 2D quantum materials, modeled by a continuum Schroedinger operator whose potential is composed of an array of identical potential wells centered on the vertices of a discrete subset, \Omega, of the plane. We study the low-lying spectrum in the regime of very deep potential wells.

We present results on scaled resolvent norm convergence to a discrete (tight-binding) operator and, in the translation invariant case, corresponding results on the scaled convergence of low-lying dispersion surfaces. Examples include the single electron model for bulk graphene ($\Omega$=honeycomb lattice), and a sharply terminated graphene half-space, interfaced with the vacuum along an arbitrary line-cut. We also apply our methods to the case of strong constant perpendicular magnetic fields. This is joint work with CL Fefferman and J Shapiro.

A detailed analysis of the spectrum of the limiting tight binding model on a honeycomb lattice, which is terminated along an arbitrary rational line-cut (joint work with CL Fefferman and S Fliss), will be presented in the upcoming lecture of CL Fefferman.

mathematical physicsanalysis of PDEsquantum physics

Audience: researchers in the topic

Mathematical Challenges in Quantum Mechanics 2021 Workshop