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Topological modes in radiofrequency resonator arrays

Topological properties of solid states have sparked considerable recent interest due to their importance in the physics of lattices with a non-trivial basis and their potential in the design of novel materials. Here we describe an experimental and an accompanying numerical toolbox to create and analyze topological states in coupled radiofrequency resonator arrays. The arrays are coupled harmonic oscillator systems that are very easily constructed, offer a variety of geometric configurations, and whose eigenfunctions and eigenvalues are amenable to detailed analysis. These systems offer well defined analogs to coupled oscillator systems in general in that they are characterized by resonances whose frequency spectra depend on the individual resonators, the interactions between them, and the geometric and topological symmetries and boundary conditions. In particular, we describe an experimental analog of a small one-dimensional system, with excellent agreement with theory. The numerical part of the toolbox allows for simulations of larger mesoscopic systems with a semi-continuous band structure, in which all resonators still exhibit individual signatures. Systematic parameter variation yields an astonishing richness of band structures in this simple linear system, allowing for further explorations into novel phenomena of topological modes.