PDE and Discrete Models of 2D Quantum Materials

We consider 2D quantum materials (non-magnetic and constant magnetic field cases), modeled by a continuum Schroedinger operator, whose potential is a sum of translates of an atomic well, centered on the vertices of a discrete subset of the plane, Omega. In the regime of deep potential wells (strong binding —greater than tight binding limit), a centered and rescaled resolvent operator converges to the resolvent of a discrete (tight-binding) effective Hamiltonian, yielding a blown up picture of the low-lying energy spectrum. Examples include the tight binding limit of the single electron model for bulk graphene (Omega=honeycomb lattice), a sharply terminated graphene half-space, interfaced with the vacuum along an arbitrary line-cut, and the Harper model. For magnetic Hamiltonians, recent results on lower bounds for the tunneling probability between identical wells, in the presence of a strong constant magnetic field, play a central role. Finally, we present our current understanding of the spectrum of the tight binding Hamiltonian which arises for the case of a graphene half-space, terminated along an arbitrary rational line-cut.