Decay of Correlations in a Topological Glass

TL;DR

This paper investigates the low-temperature dynamics of a topological glass model, revealing defect mobility, diffusive behavior, and a power-law decay of energy correlations over time.

Contribution

It introduces a detailed analysis of defect dynamics and correlation decay in a topological glass, highlighting new mechanisms of defect mobility and energy relaxation.

Findings

Defects move along 1D paths with high mobility.

Single defects diffuse and annihilate upon encounter.

Energy correlations decay as a power law, approximately t^{-0.4}.

Abstract

In this paper we continue the study of a topological glassy system. 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 other numbers of neighbors are supposed to be positive. The dynamics is that of flipping the diagonal between two adjacent triangles, with a temperature dependent probability. We consider the system at very low temperatures. We concentrate on several new aspects of this model: Starting from a detailed description of the stationary state, we conclude that pairs of defects (nodes with the "wrong" degree) move with very high mobility along 1-dimensional paths. As they wander around, they encounter single defects, which they then move "sideways" with a geometrically defined probability. This induces a diffusive motion of the single defects. If they meet, they annihilate, lowering the energy of the system. We both estimate the decay of energy to equilibrium, as well as the correlations. In particular, we find a decay like $t^{-0.4}$.