Schrödinger-invariance in non-equilibrium critical dynamics

TL;DR

The paper develops a time-dependent non-equilibrium Schrödinger invariance framework for ageing at criticality with dynamical exponent $z=2$, deriving explicit covariance constraints that fix the forms of two-time and single-time observables in terms of a small set of exponents and the composite field $\widetilde{\phi^2}$. By applying the framework to six exactly solvable models (voter, spherical, Edwards–Wilkinson, Arcetri, bosonic contact, and bosonic pair-contact processes), it demonstrates that their ageing correlators and responses are precisely described by the predicted scaling functions, often expressible in terms of hypergeometric functions, and identifies universal dimensions $(\delta,\xi,\tilde{\xi},\tilde{\delta}_2)$ with $\nu=0$ across models. The results show that EW acts as a mean-field description for these $z=2$ systems in suitable dimensions, while the Arcetri model connects to KPZ-like growth through the KPZ ansatz; overall, the non-equilibrium Schrödinger framework provides a unifying, exact account of non-equilibrium critical dynamics in diverse interacting systems. The work highlights the central role of composite environmental couplings and points toward extensions to quantum/hydro dynamics as a promising direction.

Abstract

The scaling functions of single-time and two-time correlators in systems undergoing non-equilibrium critical dynamics with dynamical exponent ${z}=2$ are predicted from a new time-dependent non-equilibrium representation of the Schrödinger algebra. These explicit predictions are tested and confirmed in the ageing of several exactly solvable models.

Figures (3)

• Figure 1: Ageing of the single-time correlator $C(s;r)$ in the voter model. (a) Dependence of $C(s;r)$ on $r$ for several values of the time $s$ for $d=1$ and (b) the data collapse when replotted over against $r/\sqrt{s\,}$. (c) Form of the scaling function $F_C(1,r/\sqrt{s\,})=s^b C(s;r)$ for $d=[\frac{1}{3},1,\frac{5}{3}]$ (full lines) from top to bottom and for $d=[\frac{7}{3},3,\frac{11}{3}]$ (dashed lines) from bottom to top.
• Figure 2: Ageing of the two-time auto-correlator $C(t,s)$ in the voter model. (a) Dependence of $C(s+\tau,s)$ on $\tau$ for several waiting times $s$ for $d=1$ and (b) the data collapse when replotted over against $y=t/s$. (c) Form of the scaling function $f_C(y)=s^b C(ys,s)$ for $d=[\frac{1}{3},1,\frac{5}{3}]$ (full lines) from top to bottom and for $d=[\frac{7}{3},3,\frac{11}{3}]$ (dashed lines) from bottom to top.
• Figure 3: Evolution step of a growing interface with the rsos constraint.