A topological transistor of waves via the Euler characteristic

Although topological materials have recently seen tremendous development, their applications have remained elusive. Simultaneously, there exists considerable interest in pushing the limits of topological materials, including the exploration of new forms of topological protection and the establishment of topologically protected order in non-electronic systems. Here we develop some novel forms of topological order (i.e. topological charges), primarily the Euler characteristic as well as manifold class. We further demonstrate that these topological orders can protect bulk current transmission, even when the topologically trivial phase possesses an arbitrarily large band gap. Such a transition between topologically trivial, periodic dispersion and topologically non-trivial, aperiodic dispersion can be obtained by the anomalous Doppler shift of waves in a gapped periodic medium. Our work suggests that additional topological charges may become relevant in moving beyond topological electronics.