We consider 2D quantum materials, modeled by a continuum Schrödinger operator whose potential is an array of identical potential wells centered on the vertices of discrete subset, $\Omega$, of the plane (not necessarily translation invariant). We discuss the regime of deep wells (strong binding), where the spectrum is approximated by that of a discrete (tight-binding) Hamiltonian. In the translation invariant case, our results imply the convergence of low-lying Floquet-Bloch dispersion surfaces to those of the limiting tight-binding operator. 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 approach to the study the case of a strong constant perpendicular magnetic field. In this case, strong resolvent convergence is used to prove the equality of topological indices (for bulk and edge systems) of discrete (tight binding) and continuum (strongly bound) Hamiltonians.
Finally, we present recent results on the spectrum of the tight-binding Hamiltonian obtained from the sharp termination of a graphene half-space, interfaced with the vacuum along an arbitrary rational line-cut.
This lecture focuses on joint works with C.L. Fefferman, S. Fliss and J. Shapiro.
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