Breitenlohner-Freedman bound on hyperbolic tilings

TL;DR

This paper investigates the Breitenlohner-Freedman bound in hyperbolic tilings of Euclidean Anti-de Sitter space, combining numerical solutions and circuit simulations to understand stability conditions for fluctuation modes.

Contribution

It provides a numerical analysis of the BF bound on hyperbolic tilings and introduces a novel active circuit to explore stability beyond the traditional bound.

Findings

BF bound approached independently of tiling as cutoff goes to zero

Numerical solutions confirm stability conditions for fluctuation modes

Active circuit design enables scanning of $m^2$ values above the BF bound

Abstract

We establish how the Breitenlohner-Freedman (BF) bound is realized on tilings of two-dimensional Euclidean Anti-de Sitter space. For the continuum, the BF bound states that on Anti-de Sitter spaces, fluctuation modes remain stable for small negative mass-squared $m^2$. This follows from a real and positive total energy of the gravitational system. For finite cutoff $\varepsilon$, we solve the Klein-Gordon equation numerically on regular hyperbolic tilings. When $\varepsilon\to0$, we find that the continuum BF bound is approached in a manner independent of the tiling. We confirm these results via simulations of a hyperbolic electric circuit. Moreover, we propose a novel circuit including active elements that allows to further scan values of $m^2$ above the BF bound.