  This work addresses the problem of elastic scattering through a localized
impurity in a one-dimensional crystal with sublattice freedom degrees. The
impurity yields long-range interferences in the local density of states known
as Friedel oscillations. Here, we show that the internal degrees of freedom of
Bloch waves are responsible for a geometrical phase shift in Friedel
oscillations. The Fourier transform of the energy-resolved interference pattern
reveals a topological property of this phase shift, which is intrinsically
related to the Bloch band structure topology in the absence of impurity.
Therefore, Friedel oscillations in the local density of states can be regarded
as a probe of wave topological properties in a broad class of classical and
quantum systems, such as acoustic and photonic crystals, ultracold atomic gases
in optical lattices, and electronic compounds.
