Field patterns: a new type of wave with infinitely degenerate band structure

Field pattern materials (FP-materials) are space-time composites with PT-symmetry in which the one-dimensional- spatial distribution of the constituents changes in time in such a special manner to give rise to a new type of waves, which we call field pattern waves (FP-waves) [G. W. Milton and O. Mattei, Proc. R. Soc. A 473, 20160819 (2017), O. Mattei and G. W. Milton, arXiv:1705.00539 (2017)]. Specifically, due to the special periodic space-time geometry of these materials, when an instantaneous disturbance propagates through the system, the branching of the characteristic lines at the space-time interfaces between phases does not lead to a chaotic cascade of disturbances but concentrates on an orderly pattern of disturbances: this is the field pattern. By applying Bloch-Floquet theory we find that the dispersion diagrams associated with these FP-materials are infinitely degenerate: associated with each point on the dispersion diagram is an infinite space of Bloch functions, a basis for which are generalized functions each concentrated on a field pattern, paramaterized by a variable that we call the launch parameter. The dynamics separates into independent dynamics on the different field patterns, each with the same dispersion relation.