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arXiv:2508.08963 · Nuclear Physics B

Correlators in phase-ordering from Schrodinger-invariance

Malte Henkel, Stoimen Stoimenov

Published Oct 5, 2025• 3 citations
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Abstract

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.

DOI
10.1016/j.nuclphysb.2025.117151

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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.
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.
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.
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 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.
The persistence probability P g ( t ) of the global order parameter of a simple ferromagnet undergoing phase ordering kinetics after a quench from a fully disordered state to below the critical temperature, T < T c , is analysed. It is argued that the persistence probability decays algebraically with time in the entire low-temperature phase. For Markov processes, the associated global persistence exponent θ g = (2λ C − d )/(2 z ) is related to the autocorrelation exponent λ C . This relationship is confirmed for phase ordering in the exactly solved 1D Ising model and the d -dimensional spherical model. For the 2D Glauber–Ising model, the temperature-independent estimate θ g = 0.063(2) indicates that the dynamics of the global order parameter is described by a non-Markovian process.
The phase-ordering kinetics of the ferromagnetic two-dimensional Ising model with uniform disorder is investigated by intensive Monte Carlo simulations. Taking into account finite-time corrections to scaling, simple ageing behaviour is observed in the two-time responses and correlators. The dynamical exponent z and the form of the scaling functions only depend on the ratio eps/T, where eps describes the width of the distribution of the disorder. The agreement of the predictions of local scale-invariance generalised to z<> 2 for the two-time scaling functions of response and correlations with the numerical data provides a direct test of generalised Galilei-invariance.