[Paper Review] Aging and conformal invariance

Malte Henkel, Michel Pleimling+2 more|arXiv (Cornell University)|Jul 13, 2001

Theoretical and Computational Physics3 citations

TL;DR

This paper proposes that dynamical scaling in aging systems, governed by the two-time autoresponse function $ R(t,s) \approx s^{-1-a} F_2(t/s) $, can be extended to conformal invariance for any dynamical exponent $ z $, thereby fully determining the scaling function $ F_2 $. The prediction is confirmed numerically in 2D and 3D kinetic Ising models and exactly in spherical models with both short- and long-ranged interactions.

ABSTRACT

In situations such as phase ordering or non-equilibrium critical dynamics, the two-time autoresponse function scales as $R(t,s)\\approx s^{-1-a} F_2(t/s)$, where $a$ is a known exponent. For any value of the dynamical exponent $z$, we argue that this dynamical scaling can be extended towards conformal invariance. Thus the scaling function $F_2$ is determined. This prediction is confirmed by several examples, both for $T<T_c$ and at $T=T_c$: the two- and three-dimensional kinetic Ising model with Glauber dynamics is studied numerically, while exact results are available for the kinetic spherical model with a non-conserved order parameter, both for short-ranged and long-ranged interactions, as well as for the mean-field spherical spin glass.

Motivation & Objective

• To extend dynamical scaling in aging systems to include conformal invariance.
• To determine the scaling function $ F_2 $ in the two-time autoresponse function $ R(t,s) \approx s^{-1-a} F_2(t/s) $.
• To verify the conformal invariance prediction across diverse models, including non-equilibrium critical dynamics and phase ordering.
• To unify scaling behavior across different systems via conformal symmetry, regardless of the dynamical exponent $ z $.

Proposed method

• Apply conformal invariance to the two-time autoresponse function $ R(t,s) \approx s^{-1-a} F_2(t/s) $, assuming it holds for all $ z $.
• Use conformal mapping techniques to derive the functional form of $ F_2 $ from symmetry constraints.
• Test predictions numerically in two- and three-dimensional kinetic Ising models with Glauber dynamics.
• Compare results with exact solutions in the kinetic spherical model with non-conserved order parameter.
• Include both short-ranged and long-ranged interactions in the spherical model to test universality.
• Analyze mean-field spherical spin glass as an additional exact benchmark.

Experimental results

Research questions

• RQ1 Can conformal invariance be applied to aging systems with arbitrary dynamical exponent $ z $?
• RQ2 Does conformal symmetry fully determine the scaling function $ F_2 $ in the autoresponse function?
• RQ3 Is the predicted $ F_2 $ function consistent across different models, including non-equilibrium critical dynamics?
• RQ4 How does conformal invariance manifest in both numerical simulations and exact solutions?
• RQ5 Can the same conformal framework describe both $ T < T_c $ and $ T = T_c $ regimes?

Key findings

• The scaling function $ F_2 $ is fully determined by conformal invariance for any dynamical exponent $ z $.
• Numerical simulations of the 2D and 3D kinetic Ising model with Glauber dynamics confirm the predicted $ F_2 $ function.
• Exact solutions in the kinetic spherical model with non-conserved order parameter validate the conformal prediction for both short- and long-ranged interactions.
• The mean-field spherical spin glass model also exhibits the predicted $ F_2 $, confirming universality across different interaction types.
• The conformal framework successfully describes both $ T < T_c $ and $ T = T_c $ dynamics within a single theoretical structure.
• The results demonstrate that conformal invariance provides a universal mechanism for determining response functions in aging systems.