A Topological Glass

Abstract

We propose and study a model with glassy behavior. The state space of the model is given by all triangulations of a sphere with $n$ nodes, half of which are red and half are blue. Red nodes want to have 5 neighbors while blue ones want 7. Energies of nodes with different numbers of neighbors are supposed to be positive. The dynamics is that of flipping the diagonal of two adjacent triangles, with a temperature dependent probability. We show that this system has an approach to a steady state which is exponentially slow, and show that the stationary state is unordered. We also study the local energy landscape and show that it has the hierarchical structure known from spin glasses. Finally, we show that the evolution can be described as that of a rarefied gas with spontaneous generation of particles and annihilating collisions

Complex quantum network geometries: Evolution and phase transitions

“…This dynamics can explore networks embedded in surfaces of different genus, and displays a dynamical slowing down as a function of the inverse temperature of the Monte Carlo algorithm. Therefore in these simulations the ground state, ordered network is not observed, similarly to what has been observed in models of glass and foam dynamics [88,89]. This approach is very different from the one proposed in the present article, although the focus is always the characterization of the network geometry.…”

“…This approach is very different from the one proposed in the present article, although the focus is always the characterization of the network geometry. For example the phase transitions present in the Quantum Complex Network Geometries are very different from the one observed in [29,88,89]. In fact in the Complex Quantum Network Geometries the observed phase transitions are non-equilibrium phase transitions, they are determined by the quenched disorder in the network.…”

“…Other models for complex networks embedded into surfaces have been recently proposed [29] extending approaches used already in the study of glasses and foams [88,89]. In [29] maximal embedded graphs have been characterized using a Monte Carlo algorithm determined by an Hamiltonian which is a function of the degree of the nodes.…”

Random and frozen states in complex triangulations

“…At very low temperatures (β → ∞), the probability allows only moves that lower the energy or leave it unchanged, driving therefore the system towards the ground state where all vertices have degree equal to k (if it is an integer number). However, it was observed that, for low genus embeddings, a 'freezing' transition takes place and the system cannot reach its ground state in any finite time [9,22,23]. The general cases with arbitrary genus g have not been studied yet, and this is indeed the topic of the present paper.…”

“…The appealing aspect of such a topological glass is that the typical features of structural glasses are observed in a simple triangulation of a surface [9]. It has been shown that this model has some of the typical properties of a strong glass with aging behaviour, but without breakdown of the fluctuation-dissipation relation [9,[20][21][22][23] . In this paper, we extend the topological glass model, originally developed in [9], to the general case of triangulations embedded on complex surfaces with different genus.…”