When can phonon modes be trapped by arbitrarily weak local defects? Rigorous sufficient conditions in a monatomic lattice

Abstract

Although many studies have shown that localized defects in vibrational lattices can trap localized phonon modes for particular geometries, there has been no general proof of localization for arbitrary localized defects of arbitrarily weak strength. This is distinct from the problem of Anderson localization, where random perturbations are spread across the entire lattice and one aims to prove whether all states become localized. Here, we study the former problem for monatomic 1d and 2d crystals with nearest-neighbor harmonic interactions, corresponding in 2d to square lattices of masses and springs, and prove that arbitrary localized perturbations (such as any finite set of defect masses) that decrease the net mass by any amount must lead to localized states. In contrast, the analogous statement is generally false in 3d. Our proof employs a straightforward variational method, inspired by analogous results for the Schrödinger and Maxwell equations, that should be extensible to other discrete systems.