Hannay Angle: Yet Another Symmetry-Protected Topological Order Parameter in Classical Mechanics

The topological way of thinking now goes beyond quantum solids, and topological characters of classical mechanical systems obeying Newton's law are attracting current interest. To provide a physical insight into the topological numbers in mechanics, we demonstrate the use of the Hannay angle, a classical Berry phase, as a symmetry-protected topological order parameter. The Hannay angle is derived using a canonical transformation that maps Newton's equation to a Schrodinger-type equation, and the condition for the quantization is discussed in connection with the symmetry in mechanics. Also, we demonstrate the use of the Hannay angle for a topological characterization of a spring-mass model focusing on the bulk-edge correspondence.