Simulating hyperbolic space on a circuit board

Lenggenhager, Patrick M; Stegmaier, Alexander; Upreti, Lavi K; Hofmann, Tobias; Helbig, Tobias; Vollhardt, Achim; Greiter, Martin; Lee, Ching Hua; Imhof, Stefan; Brand, Hauke; Kießling, Tobias; Boettcher, Igor; Neupert, Titus; Thomale, Ronny; Bzdušek, Tomáš

doi:10.1038/s41467-022-32042-4

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Lenggenhager, P. M., Stegmaier, A., Upreti, L. K., Hofmann, T., Helbig, T., Vollhardt, A., Greiter, M., Lee, C. H., Imhof, S., Brand, H., Kießling, T., Boettcher, I., Neupert, T., Thomale, R., & Bzdušek, T. (2022). Simulating hyperbolic space on a circuit board. Nature Communications, 13, 4373. https://doi.org/10.1038/s41467-022-32042-4

Abstract

The Laplace operator encodes the behavior of physical systems at vastly different scales, describing heat flow, fluids, as well as electric, gravitational, and quantum fields. A key input for the Laplace equation is the curvature of space. Here we discuss and experimentally demonstrate that the spectral ordering of Laplacian eigenstates for hyperbolic (negatively curved) and flat two-dimensional spaces has a universally different structure. We use a lattice regularization of hyperbolic space in an electric-circuit network to measure t