Minority spin dynamics in non-homogeneous Ising model: diverging timescales and exponents

TL;DR

This study explores the minority spin dynamics in the non-homogeneous Ising model at zero temperature, revealing diverging timescales and exponents, with distinct behaviors in one and two dimensions, and introduces new scaling relations.

Contribution

It provides a detailed analysis of persistence probabilities and timescales in the non-homogeneous Ising model, identifying new exponents and scaling behaviors, especially in two dimensions.

Findings

In 1D, persistence decays algebraically with different exponents for up and down spins.

In 2D, minority spins show stretched exponential decay, while majority spins approach a finite value.

The timescale diverges as (x_c - x)^{-λ} with λ ≈ 2.31, indicating critical behavior.

Abstract

We investigate the dynamical behaviour of the Ising model under a zero temperature quench with the initial fraction of up spins $0\leq x\leq 1$. In one dimension, the known results for persistence probability are verified; it shows algebraic decay for both up and down spins asymptotically with different exponents. It is found that the conventional finite size scaling is valid here. In two dimensions however, the persistence probabilities are no longer algebraic; in particular for $x\leq 0.5$, persistence for the up (minority) spins shows the behaviour $P_{min}(t) \sim t^{-\gamma}\exp(-(t/\tau)^{\delta})$ with time $t$, while for the down (majority) spins, $P_{maj}(t)$ approaches a finite value. We find that the timescale $\tau$ diverges as $(x_c-x)^{- \lambda}$, where $x_c=0.5$ and $\lambda\simeq2.31$. The exponent $\gamma$ varies as $\theta_{2d}+c_0(x_c-x)^{\beta}$ where $\theta_{2d}\simeq0.215$ is very close to the persistence exponent in two dimensions; $\beta\simeq1$. The results in two dimensions can be understood qualitatively by studying the exit probability, which for different system size is found to have the form $E(x) = f\big[(\frac{x-x_c}{x_c})L^{1/\nu}\big]$, with $\nu \approx 1.47$. This result suggests that $\tau \sim L^{\tilde{z}}$, where $\tilde{z} = \frac{\lambda}{\nu} = 1.57 \pm 0.11$ is an exponent not explored earlier.