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When semi-infinite phononic crystals (PCs) are in contact, localized modesmay exist at their boundary. The central question is generally to predict theirexistence and to determine their stability. With the rapid expansion of thefield of topological insulators, powerful tools have been developed to addressthese questions. In particular, when applied to one-dimensional systems withmirror symmetry, the bulk-boundary correspondence claims that the existence ofinterface modes is given by a topological invariant computed from the bulkproperties of the phononic crystal, which ensures strong stability properties.This one-dimensional bulk-boundary correspondence has been proven in variousworks. Recent attempts have exploited the notion of surface impedance, relyingon analytical calculations of the transfer matrix. In the present work, themonotonic evolution of surface impedance with frequency is proven for allone-dimensional phononic crystals with mirror symmetry. This result allows usto establish a stronger version of the bulk-boundary correspondence thatguarantees not only the existence but also the uniqueness of a topologicallyprotected interface state. The method is numerically illustrated in thephysically relevant case of PCs with imperfect interfaces, where analyticalcalculations would be out of reach.

Surface impedance and topologically protected interface modes in one-dimensional phononic crystals