Simulating hyperbolic space on a circuit board

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 spec...

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Bibliographic Details
Published in:Nature communications 2022-07, Vol.13 (1), p.4373-4373, Article 4373
Main Authors: 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
Format: Article
Language:English
Subjects:
639/766/1130/2798
639/766/119
Circuits
Curvature
Eigenvectors
Euclidean space
Fluid flow
Geodesy
Gravity
Heat flow
Heat transmission
Humanities and Social Sciences
Hyperbolic coordinates
Laplace equation
Lattices
multidisciplinary
Physics
Regularization
Science
Science (multidisciplinary)
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Summary: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 the eigenstates of a ‘hyperbolic drum’, and in a time-resolved experiment we verify signal propagation along the curved geodesics. Our experiments showcase both a versatile platform to emulate hyperbolic lattices in tabletop experiments, and a set of methods to verify the effective hyperbolic metric in this and other platforms. The presented techniques can be utilized to explore novel aspects of both classical and quantum dynamics in negatively curved spaces, and to realise the emerging models of topological hyperbolic matter. Spaces with negative curvature are difficult to realise and investigate experimentally, but they can be emulated with synthetic matter. Here, the authors show how to do this using an electric circuit network, and present a method to characterize and verify the hyperbolic nature of the implemented model.
ISSN:2041-1723
2041-1723
DOI:10.1038/s41467-022-32042-4