Phase and Topology

The primary objective of this chapter is to present and review the introductory concepts related to phase and topology that will serve as foundation for the development of the more complex concepts to be presented in subsequent chapters. In particular, we illustrate the concept of geometric phase in the case of two prototypical elastic systems, namely the one-dimensional harmonic oscillator and a one-dimensional binary superlattice. We demonstrate formally the relationship between the variation of the geometric phase in the spectral and wave number domains and the parallel transport of a vector field along paths on curved manifolds possessing helicoidal twists which exhibit non-conventional topology. The formal mapping of the evolution of the geometric phase on the spectral or wave number domains onto the parallel transport of a vector field on curved manifold spanning the frequency and wave number spaces will be used to help interpret topological features of elastic waves in more complex media such as two-dimensional or three-dimensional phononic crystals and acoustic metamaterials.We relate the notion of geometric phase and Green’s functions. In particular, we show for Hermitian operators that the Berry connection is proportional to the imaginary part of its Green’s function. We illustrate this notion in the case of simple elastic systems composed of mass and springs, namely the one-dimensional harmonic crystal and the one-dimensional harmonic crystal with a side branch. In the latter case, the notions of Friedel phase and transmission (reflection) phase are introduced and related to the notion of geometric phase. Finally, we sketch how topological modes arise at interfaces between media with different topologies.