Breitenlohner-Freedman bound on hyperbolic tilings

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 $\epsilon$, we solve the Klein-Gordon equation numerically on regular hyperbolic tilings. When $\epsilon \to 0$, 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.

Comments: 11 pages, 9 figures, improved discussion and analysis, corrected typos