Tight-Binding Reduction and Topological Equivalence in Strong Magnetic Fields

Topological insulators (TIs) are a class of materials which are insulating in their bulk form yet, upon introduction of an a boundary or edge, e.g. by abruptly terminating the material, may exhibit spontaneous current along their boundary. This property is quantified by topological indices associated with either the bulk or the edge system. In the field of condensed matter physics, tight binding (discrete) approximate models, parametrized by hopping coefficients, have been used successfully to capture the topological behavior of TIs in many settings. However, whether such tight binding models capture the same topological features as the underlying continuum models of quantum physics has been an open question. We resolve this question in the context of the archetypal example of topological behavior in materials, the integer quantum Hall effect. We study a class of continuum Hamiltonians, $H^\lambda$, which govern electron motion in a two-dimensional crystal under the influence of a perpendicular magnetic field. No assumption is made on translation invariance of the crystal. We prove, in the regime where both the magnetic field strength and depth of the crystal potential are sufficiently large, $\lambda\gg1$, that the low-lying energy spectrum and eigenstates (and corresponding large time dynamics) of $H^\lambda$ are well-described by a scale-free discrete Hamiltonian, $H^{\rm TB}$; we show norm resolvent convergence. The relevant topological index is the Hall conductivity, which is expressible as a Fredholm index. We prove that for large $\lambda$ the topological indices of $H^\lambda$ and $H^{\rm TB}$ agree. This is proved separately for bulk and edge geometries. Our results justify the principle of using discrete models in the study of topological matter.