Schrödinger-invariance in phase-ordering kinetics

TL;DR

This work establishes that ageing in phase-ordering kinetics with $z=2$ can be understood from Schrödinger-invariance by employing a non-equilibrium representation of the Schrödinger algebra. It introduces non-equilibrium scaling operators with pairs of dimensions $(\delta,\xi)$ for the order parameter and $(\widetilde{\delta},\widetilde{\xi})$ for the response, linked by a representation change of the symmetry generators. From covariance of four-point response functions, the authors derive universal scaling forms for the single-time and two-time correlators, including a key exponent relation $\lambda/2 = 2\delta-\xi$ which, together with $\delta=\xi$ for phase-ordering, yields $\lambda=2\delta$ and the non-equilibrium constraint $2\delta = d/(1+\zeta_p) = \lambda$. They also obtain finite-size scaling predictions, Porod-law behaviour in the structure factor, and a global ageing exponent $\Theta = (d-\lambda)/2$, demonstrating a coherent symmetry-based framework for phase-ordering dynamics.

Abstract

The generic shape of the single-time and two-time correlators in non-equilibrium phase-ordering kinetics with ${z}=2$ is obtained from the co-variance of the four-point response functions. Their non-equilibrium scaling forms follow from a new non-equilibrium representation of the Schrödinger algebra.

Figures (3)

• Figure 1: Ageing of the phase-ordering in the single-time correlator $C(s;r)$ in the (mean) spherical model in $d>2$ dimensions. The inset in panel (b) shows the form of the scaling function for a scalar order-parameter and the agreement with Porod's law.
• Figure 2: Ageing of the phase-ordering in the two-time auto-correlator $C(t,s)$ in the $3D$ (mean) spherical model, see Henk25V. The inset in panel (b) shows the form of the scaling function for dimensions $d=[3,5,7]$ from top to bottom.
• Figure 3: Autocorrelator $C(ys,s)$ in a fully finite system.