Simulating hyperbolic space on a circuit board

TL;DR

This paper demonstrates a method to simulate hyperbolic space using electric circuits, allowing experimental exploration of properties like eigenstates and signal propagation in negatively curved geometries.

Contribution

The authors develop and experimentally validate a circuit-based platform to emulate hyperbolic lattices and verify hyperbolic metrics in tabletop experiments.

Findings

Spectral ordering of Laplacian eigenstates differs between hyperbolic and flat spaces.

Electric circuit networks can measure eigenstates of hyperbolic geometries.

Signal propagation along hyperbolic geodesics is experimentally verified.

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 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.