Test of Local Scale Invariance from the direct measurement of the response function in the Ising model quenched to and to below $T_C$

TL;DR

This study tests local scale invariance in the Ising model's response function during quenched dynamics, confirming Gaussian behavior in four dimensions and deviations in two dimensions, highlighting the limits of local scale invariance.

Contribution

The paper provides numerical evidence supporting local scale invariance in Gaussian regimes and identifies its limitations in non-Gaussian dynamics of the Ising model.

Findings

Excellent agreement with local scale invariance in 4D Gaussian case

Deviations from local scale invariance in 2D non-Gaussian cases

Precise measurement of response exponent a=0.27 in 2D below T_C

Abstract

In order to check on a recent suggestion that local scale invariance [M.Henkel et al. Phys.Rev.Lett. {\bf 87}, 265701 (2001)] might hold when the dynamics is of Gaussian nature, we have carried out the measurement of the response function in the kinetic Ising model with Glauber dynamics quenched to $T_C$ in $d=4$, where Gaussian behavior is expected to apply, and in the two other cases of the $d=2$ model quenched to $T_C$ and to below $T_C$, where instead deviations from Gaussian behavior are expected to appear. We find that in the $d=4$ case there is an excellent agreement between the numerical data, the local scale invariance prediction and the analytical Gaussian approximation. No logarithmic corrections are numerically detected. Conversely, in the $d=2$ cases, both in the quench to $T_C$ and to below $T_C$, sizable deviations of the local scale invariance behavior from the numerical data are observed. These results do support the idea that local scale invariance might miss to capture the non Gaussian features of the dynamics. The considerable precision needed for the comparison has been achieved through the use of a fast new algorithm for the measurement of the response function without applying the external field. From these high quality data we obtain $a=0.27 \pm 0.002$ for the scaling exponent of the response function in the $d=2$ Ising model quenched to below $T_C$, in agreement with previous results.