  When semi-infinite phononic crystals (PCs) are in contact, localized modes
may exist at their boundary. The central question is generally to predict their
existence and to determine their stability. With the rapid expansion of the
field of topological insulators, powerful tools have been developed to address
these questions. In particular, when applied to one-dimensional systems with
mirror symmetry, the bulk-boundary correspondence claims that the existence of
interface modes is given by a topological invariant computed from the bulk
properties of the phononic crystal, which ensures strong stability properties.
This one-dimensional bulk-boundary correspondence has been proven in various
works. Recent attempts have exploited the notion of surface impedance, relying
on analytical calculations of the transfer matrix. In the present work, the
monotonic evolution of surface impedance with frequency is proven for all
one-dimensional phononic crystals with mirror symmetry. This result allows us
to establish a stronger version of the bulk-boundary correspondence that
guarantees not only the existence but also the uniqueness of a topologically
protected interface state. The method is numerically illustrated in the
physically relevant case of PCs with imperfect interfaces, where analytical
calculations would be out of reach.
