Tight-binding reduction and topological equivalence in strong magnetic fields

Topological insulators (TIs) are a class of materials which insulating in their bulk form yet, upon introduction an boundary or edge, e.g. by abruptly terminating the material, may exhibit spontaneous current along boundary. This property is quantified topological indices associated with either edge system. In field condensed matter physics, tight binding (discrete) approximate models, parametrized hopping coefficients, have been used successfully to capture behavior TIs many settings. However, whether such models same features as underlying continuum quantum physics has open question. We resolve this question context archetypal example materials, integer Hall effect. study Hamiltonians, Hλ, govern electron motion two-dimensional crystal under influence perpendicular magnetic field. No assumption made on translation invariance crystal. prove, regime where both strength and depth potential sufficiently large, λ≫1, that low-lying energy spectrum eigenstates (and corresponding large time dynamics) Hλ well-described scale-free discrete Hamiltonian, HTB; we show norm resolvent convergence. The relevant index conductivity, expressible Fredholm index. prove for λ HTB agree. proved separately geometries. Our results justify principle using matter.