Electric-circuit realization of a hyperbolic drum

The recent development of hyperbolic band theory, which describes energy spectra of particles on hyperbolic lattices, revived interest in crystalline models embedded in negatively curved spaces and sparked the search for suitable experimental realizations. We argue that electric circuits offer a highly versatile and easily fabricable platform to test this theory and to investigate a wide range of hyperbolic tight-binding models, while providing direct access to local degrees of freedom. 

 

To demonstrate this flexibility, we emulate a disk-shaped sample of the hyperbolic {3,7} lattice (i.e., the regular tessellation where seven equilateral triangles meet at each vertex), for which the low-energy modes are effectively described by the Laplace-Beltrami operator. Using this system, which we call “hyperbolic drum”, we reveal evidence of the negative curvature in both static and dynamical experiments. First, we measure the spectral ordering of the Laplacian eigenstates which is universally different in hyperbolic vs. flat two-dimensional spaces. Second, we verify signal propagation along curved geodesics. Our experiments showcase that electric circuits can be utilized to explore the propagation dynamics in curved spaces and to realize models of topological hyperbolic matter.