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arXiv:2510.25429

Schrödinger-invariance in non-equilibrium critical dynamics

Malte Henkel, Stoimen Stoimenov

Published Oct 29, 2025• 0 citations
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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.

DOI
10.48550/arxiv.2510.25429

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The scaling functions of single-time and two-time correlators in systems undergoing non-equilibrium critical dynamics with dynamical exponent z=2{z}=2z=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.
The explicit calculation of the scaling form of the two-time autocorrelation function in phase-ordering kinetics and in those cases of non-equilibrium critical dynamics where the dynamical exponent z = 2 through the extension of dynamical scaling to local scale invariance is reviewed. Conceptually, this mainly requires an extension from the usually considered d -dimensional ageing or Schrödinger algebras to a new kind of representation of the conformal algebra in d + 2 dimensions. Explicit tests in several exactly solved models of simple magnets and through simulations in the 2D Ising and q -states Potts models ( q = 2, 3, 8) quenched to T < T c are presented and the extension to systems with non-equilibrium steady states is discussed through two exactly solvable models as well. In the context of surface growth models, possible generalizations for a dynamical exponent z = 4 and beyond are discussed.
Abstract For ageing systems, quenched onto a critical temperature T = T c such that the dominant noise comes from the thermal bath, with a non-conserved order-parameter and in addition with dynamical exponent z = 2 , the form of the two-time autocorrelator as well as the time-space form of the single-time correlator are derived from Schrödinger-invariance, generalised to non-equilibrium ageing. These findings reproduce the exact results in the 1 D Glauber–Ising model at T = 0 and the critical spherical model in d > 2 dimensions.
For ageing systems, quenched onto a critical temperature T=TcT=T_cT=Tc​ such that the dominant noise comes from the thermal bath, with a non-conserved order-parameter and in addition with dynamical exponent z=2{z}=2z=2, the form of the two-time auto-correlator as well as the time-space form of the single-time correlator are derived from Schr\"odinger-invariance, generalised to non-equilibrium ageing. These findings reproduce the exact results in the 1D1D1D Glauber-Ising model at T=0T=0T=0 and the critical spherical model in d>2d>2d>2 dimensions.
The generalization of dynamical scaling to local scale invariance is reviewed. Starting from a recapitulation of the phenomenology of ageing phenomena, the generalization of dynamical scaling to local scale transformation for any given dynamical exponent zzz is described and the two distinct types of local scale invariance are presented. The special case z=2z=2z=2 and the associated Ward identity of Schrödinger invariance is treated. Local scale invariance predicts the form of the two-point functions. Existing confirmations of these predictions for (I) the Lifshitz points in spin systems with competing interactions such as the ANNNI model and (II) non-equilibrium ageing phenomena as occur in the kinetic Ising model with Glauber dynamics are described.
The generic shape of the single-time and two-time correlators in non-equilibrium phase-ordering kinetics with z=2{z}=2z=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.