Phonon modes in a Möbius band

It is well known that phonon modes become sensitive to the geometry of an object when the phonon wavelengths are comparable to the objects physical length scale. In contrast, the sensitivity of phonon modes toward topology is much less explored and understood. In this paper we discuss the effects of topology on phonon modes using a finite thickness Möbius band of centerline radius a as the model system. The phonon modes are derived using the xyz algorithm based on Riemannian geometry. From the boundary conditions and paritywe identify two sets ofmodeswithwave numbers q = n 2a described by odd and even integers n. Modes characterized by odd integers have flexural vibrations whereas those characterized by even integers exhibit dilatational and shear/torsional motion. While the phonon dispersion at large wave numbers agrees with that of structures having simple topology (rings and wires), at low frequencies and wave numbers the Möbius topology introduces significant differences. Uniquely, we find three of the four phonon branches do not go to zero frequency with decreasing wave number, but converge on a finite frequency.Weidentify a new form of vibrational pattern resembling incomplete breathing modes and discuss the ramifications of the modified spectrum, including a local increase in the density of states and the existence of a phonon band gap.