  Topological phononic crystals (PCs) are periodic artificial structures which
can support nontrivial acoustic topological bands, and their topological
properties are linked to the existence of topological edge modes. Most previous
studies focused on the topological edge modes in Bragg gaps which are induced
by lattice scatterings. While local resonant gaps would be of great use in
subwavelength control of acoustic waves, whether it is possible to achieve
topological interface states in local resonant gaps is a question. In this
article, we study the topological bands near local resonant gaps in a
time-reversal symmetric acoustic systems and elaborate the evolution of band
structure using a spring-mass model. Our acoustic structure can produce three
band gaps in subwavelength region: one originates from local resonance of unit
cell and the other two stem from band folding. It is found that the topological
interface states can only exist in the band folding induced band gaps but never
appear in the local resonant band gap. The numerical simulation perfectly
agrees with theoretical results. Our study provides an approach of localizing
the subwavelength acoustic wave.

Mesoscale and Nanoscale Physics;Applied Physics