Breitenlohner-Freedman Bound on Hyperbolic Tilings

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 $ϵ$, we solve the Klein-Gordon equation numerically on regular hyperbolic tilings. When $ϵ\to 0$, 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 us to further scan values of $m^2$ above the BF bound.

Figure 1

Hyperbolic {7,3} tiling in the Poincare disk representation. For the central node, the stencil of the discretized Laplace-Beltrami operator is highlighted. Red sites carry constant weights $w^{(7,3)}$, whereas the central node (blue) is weighted by $-7w^{(7,3)}$.

Figure 2

Field amplitude at the origin for different cutoff values $ϵ$. We observe Umklapp points appearing at any finite cutoff, the rightmost of which can be used as an indicator of unstable solutions. For smaller cutoffs, the Umklapp points become denser and converge toward the continuum BF bound (dotted line).

Figure 3

First Umklapp point for various hyperbolic tilings and radial cutoffs ${\theta }_c$. For small $ϵ$, the curves tend toward the continuum bound $m^2{\ell }^2=-1/4$, indicated by the dotted horizontal line. Inset: corresponding results from our hyperbolic electric circuit simulations.

Figure 4

Section of the hyperbolic electric circuit. The structure is repeated at every vertex in the lattice.