Virtual walks in spin space: a study in a family of two-parameter models

TL;DR

This paper explores the dynamics of classical spins modeled as virtual walks in a two-parameter family of spin models, revealing different diffusive behaviors, crossover phenomena, and potential for efficient phase transition detection.

Contribution

It introduces a generalized two-parameter spin model and analyzes the virtual walk dynamics, uncovering crossover behaviors and phase transition detection capabilities.

Findings

Different diffusive regimes with b1  1 or 0.5 at large times

Crossover in the voter model point with system size scaling as L^2 log L

Virtual walk detects phase transitions more efficiently than other methods

Abstract

We investigate the dynamics of classical spins mapped as walkers in a virtual "spin" space using a generalised two-parameter family of spin models characterized by parameters $y$ and $z$ [M. J. de Oliveira, J. F. F. Mendes and M. A. Santos, J. Phys. A Math. Gen. \textbf{26}, 2317 (1993)]. The behavior of $S(x,t)$, the probability that the walker is at position $x$ at time $t$ is studied in detail. In general $S(x,t) \sim t^{-\alpha}f(x/t^{\alpha})$ with $\alpha \simeq 1$ or $0.5$ at large times depending on the parameters. In particular, $S(x,t)$ for the point $y=1, z=0.5$ corresponding to the voter model shows a crossover in time; associated with this crossover, two timescales can be defined which vary with the system size $L$ as $L^2\log L$. We also show that as the voter model point is approached from the disordered regions along different directions, the width of the Gaussian distribution $S(x,t)$ diverges in a power law manner with different exponents. For the majority voter case, the results indicate that the the virtual walk can detect the phase transition perhaps more efficiently compared to other non-equilibrium methods.