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arXiv:2505.22301 · Journal of Physics A Mathematical and Theoretical

Critical ageing correlators from Schrödinger-invariance

Malte Henkel, Stoimen Stoimenov

Published Nov 17, 2025• 2 citations
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Abstract

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.

DOI
10.1088/1751-8121/ae2075

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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.
Systems undergoing phase-ordering kinetics after a quench into the ordered phase with 0 < T < T c from a fully disordered initial state and with a non-conserved order-parameter have the dynamical exponent z = 2 . The long-time behaviour of their single-time and two-time correlators, determined by the noisy initial conditions, is derived from Schrödinger-invariance. It is shown that the generic ageing scaling forms of the single-time and two-time correlators follow from the Schrödinger-covariance of the four-point response functions. The autocorrelation exponent λ is related to the passage exponent ζ p which describes the time-scale for the cross-over into the ageing regime. Both Porod’s law and the bounds d / 2 ≤ λ ≤ d are reproduced in a simple way. The dynamical finite-size scaling in fully finite systems is established. The dynamical scaling of global correlators are found and the low-temperature generalisation λ = d − 2 Θ of the Janssen-Schaub-Schmittmann scaling relation is derived.
A generalised form of time-translation-invariance permits to re-derive the known generic phenomenology of ageing, which arises in classical many-body systems after a quench from an initially disordered system to a temperature T ≤ T c , at or below the critical temperature T c . Generalised time-translation-invariance is obtained, out of equilibrium, from a change of representation of the Lie algebra generators of the dynamical symmetries of scale-invariance and time-translation-invariance. Observable consequences include the algebraic form of the scaling functions for large arguments of the two-time auto-correlators and auto-responses, the equality of the auto-correlation and the auto-response exponents λ C = λ R , the cross-over scaling form for an initially magnetised critical system and the explanation of a novel finite-size scaling if the auto-correlator or auto-response converge for large arguments y = t / s ≫ 1 to a plateau. For global two-time correlators, the time-dependence involving the initial critical slip exponent Θ is confirmed and is generalised to all temperatures below criticality and to the global two-time response function, and their finite-size scaling is derived as well. This also includes the time-dependence of the squared global order-parameter. The celebrate Janssen-Schaub-Schmittmann scaling relation with the auto-correlation exponent is thereby extended to all temperatures below the critical temperature. A simple criterion on the relevance of non-linear terms in the stochastic equation of motion is derived, taking the dimensionality of couplings into account. Its applicability in a wide class of models is confirmed, for temperatures T ≤ T c . Relevance to experiments is also discussed.
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.