Dynamics of the two-dimensional gonihedric spin model

TL;DR

This study investigates the dynamical behavior of the two-dimensional gonihedric spin model, revealing slow but non-glassy dynamics, and analyzes how parameters influence critical behavior and long-term evolution.

Contribution

It provides the first detailed numerical and analytical analysis of the 2D gonihedric spin model's dynamics, including effects of self-avoidance and finite size, and identifies non-glassy slow relaxation laws.

Findings

The model exhibits very slow dynamics without glassy behavior.

Finite size effects are significant and analyzed through a finite volume approach.

Long-time evolution follows a power-law law rather than a logarithmic law.

Abstract

In this paper we study dynamical aspects of the two-dimensional gonihedric spin model using both numerical and analytical methods. This spin model has vanishing microscopic surface tension and it actually describes an ensemble of loops living on a 2D surface. The self-avoidance of loops is parametrized by a parameter $\kappa$. The $\kappa=0$ model can be mapped to one of the six-vertex models discussed by Baxter and it does not have critical behavior. We have found that $\kappa\neq 0$ does not lead to critical behavior either. Finite size effects are rather severe, and in order to understand these effects a finite volume calculation for non self-avoiding loops is presented. This model, like his 3D counterpart, exhibits very slow dynamics, but a careful analysis of dynamical observables reveals non-glassy evolution (unlike its 3D counterpart). We find, also in this $\kappa=0$ case, the law that governs the long-time, low-temperature evolution of the system, through a dual description in terms of defects. A power, rather than logarithmic, law for the approximation to equilibrium has been found.