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→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.

Universality of continuous phase transitions on random Voronoi graphs.

The Voronoi construction is ubiquitous across the natural sciences and engineering. In statistical mechanics, however, only its dual, the Delaunay triangulation, has been considered in the investigation of critical phenomena. In this paper we set to fill this gap by studying three prominent systems of classical statistical mechanics, the equilibrium spin-1/2 Ising model, the nonequilibrium contact process, and the conserved stochastic sandpile model on two-dimensional random Voronoi graphs. Particular motivation comes from the fact that these graphs have vertices of constant coordination number, making it possible to isolate topological effects of quenched disorder from node-intrinsic coordination number disorder. Using large-scale numerical simulations and finite-size scaling techniques, we are able to demonstrate that all three systems belong to their respective clean universality classes. Therefore, quenched disorder introduced by the randomness of the lattice is irrelevant and does not influence the character of the phase transitions. We report the critical points to considerable precision and, for the Ising model, also the first correction-to-scaling exponent.

Violation of the Harris-Barghathi-Vojta Criterion.

In 1974, Harris proposed his celebrated criterion: Continuous phase transitions in d-dimensional systems are stable against quenched spatial randomness whenever dν>2, where ν is the clean critical exponent of the correlation length. Forty years later, motivated by violations of the Harris criterion for certain lattices such as Voronoi-Delaunay triangulations of random point clouds, Barghathi and Vojta put forth a modified criterion for topologically disordered systems: aν>1, where a is the disorder decay exponent, which measures how fast coordination number fluctuations decay with increasing length scale. Here we present a topologically disordered lattice with coordination number fluctuations that decay as slowly as those of conventional uncorrelated randomness, but for which the clean universal behavior is preserved, hence violating even the modified criterion.

Two-dimensional Ising model on random lattices with constant coordination number.

We study the two-dimensional Ising model on networks with quenched topological (connectivity) disorder. In particular, we construct random lattices of constant coordination number and perform large-scale Monte Carlo simulations in order to obtain critical exponents using finite-size scaling relations. We find disorder-dependent effective critical exponents, similar to diluted models, showing thus no clear universal behavior. Considering the very recent results for the two-dimensional Ising model on proximity graphs and the coordination number correlation analysis suggested by Barghathi and Vojta [Phys. Rev. Lett. 113, 120602 (2014)PRLTAO0031-900710.1103/PhysRevLett.113.120602], our results indicate that the planarity and connectedness of the lattice play an important role on deciding whether the phase transition is stable against quenched topological disorder.