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Critical ageing correlators from Schrödinger-invariance

Malte Henkel, Stoimen Stoimenov

TL;DR

The paper tackles universal ageing correlators in non-equilibrium critical dynamics for systems with dynamical exponent $z=2$. It employs Schrödinger-invariance, extended to ageing via a representation change, to derive the two-time auto-correlator $C(t,s)$ and the time-space single-time correlator without model-specific details, linking them to three-point responses and a composite operator $\tilde{\phi}^2$. The authors validate the approach by reproducing exact results for the $1D$ Glauber-Ising model at $T=0$ and the spherical model at Tc in $d>2$, and by establishing key exponent relations such as $\lambda_C=\lambda_R$. They also elucidate how the composite scaling dimensions govern the structure of the correlators and discuss the scope and limitations of Schrödinger-invariance in non-equilibrium ageing, outlining paths to extend the framework beyond $z=2$ and Tc. Overall, the work provides a model-independent, symmetry-based route to universal ageing functions with explicit cross-checks against exact solvable models.

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 auto-correlator 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 $1D$ Glauber-Ising model at $T=0$ and the critical spherical model in $d>2$ dimensions.

Critical ageing correlators from Schrödinger-invariance

TL;DR

The paper tackles universal ageing correlators in non-equilibrium critical dynamics for systems with dynamical exponent . It employs Schrödinger-invariance, extended to ageing via a representation change, to derive the two-time auto-correlator and the time-space single-time correlator without model-specific details, linking them to three-point responses and a composite operator . The authors validate the approach by reproducing exact results for the Glauber-Ising model at and the spherical model at Tc in , and by establishing key exponent relations such as . They also elucidate how the composite scaling dimensions govern the structure of the correlators and discuss the scope and limitations of Schrödinger-invariance in non-equilibrium ageing, outlining paths to extend the framework beyond and Tc. Overall, the work provides a model-independent, symmetry-based route to universal ageing functions with explicit cross-checks against exact solvable models.

Abstract

For ageing systems, quenched onto a critical temperature such that the dominant noise comes from the thermal bath, with a non-conserved order-parameter and in addition with dynamical exponent , the form of the two-time auto-correlator 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 Glauber-Ising model at and the critical spherical model in dimensions.

Paper Structure

This paper contains 12 sections, 41 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Non-equilibrium field-theory
  4. Schrödinger invariance
  5. Far from equilibrium observables
  6. Calculation of correlators
  7. Two-time auto-correlator
  8. Single-time correlator
  9. Comparison with models
  10. Example 1: $1D$ Glauber-Ising model
  11. Example 2: critical spherical model
  12. Conclusions

Figures (2)

  • Figure 1: Illustration of physical ageing in single-time and two-time correlators. A typical single-time correlator $C(s;r)$ is in panel (a) for different times $s_1<s_2<s_3$ but collapses in panel (b), when replotted over against rescaled lengths $r/\ell(s)$. The dynamical length scale is $\ell(s)\sim s^{1/\mathpzc{z}}$. In panel (c) a typical two-time auto-correlator $C(s+\tau,s)$ is displayed over against $\tau=t-s$, for different waiting times $s_1<s_2<s_3$ which collapse when replotted in panel (d) over against $y=t/s$. The inset shows the asymptotic power-law form $f_C(y)\sim y^{-\lambda_C/z}$.
  • Figure 2: Auto-correlator scaling function $C(ys,s)=f_C(y)$ according to (\ref{['gl:autoC-GI1D']}) with $C_{\infty}=\frac{\sqrt{8\,}}{\pi}$ and for several values of $\nu$. The exact solution (\ref{['1DGIC']}) corresponds to $\nu=0$.