Janssen–Schaub–Schmittmann Scaling Relation

• The Janssen–Schaub–Schmittmann scaling relation defines a link between the initial slip exponent, autocorrelation exponent, and dynamic exponent in ageing systems.
• It employs a generalized time-translation invariance with a modified generator to extend the scaling framework to quenches at and below the critical temperature.
• This relation offers a unified approach for analyzing experimental and numerical data in diverse systems like spin glasses, liquid crystals, and classical magnets.

The Janssen–Schaub–Schmittmann (JSS) scaling relation is a fundamental result in the theory of non-equilibrium critical dynamics, particularly for systems exhibiting ageing following a quench to or below the critical temperature. Originally derived in the context of short-time non-equilibrium relaxation after quenches from a disordered initial condition, the JSS relation connects the initial slip exponent of the order parameter $\Theta$, the dynamical exponent $z$, the dimension $d$, and the autocorrelation exponent $\lambda$. Recent theoretical developments have generalized the JSS relation, revealing its validity well beyond the critical point, with wide-ranging implications for the scaling theory of ageing in classical many-body systems.

1. Historical Origin and Context

The JSS scaling relation was introduced by H. K. Janssen, B. Schaub, and B. Schmittmann in the context of critical dynamics, specifically non-equilibrium relaxation and ageing phenomena following a sudden quench to the critical temperature $T_c$. In these scenarios, physical observables display dependence on both the waiting time $s$ and the observation time $t > s$, breaking time-translation invariance and giving rise to new universal scaling laws.

At the heart of this phenomenology are two-time quantities such as the auto-correlation function

$$C(t, s) = \langle \phi(t, \mathbf{r}) \phi(s, \mathbf{r}) \rangle$$

and the auto-response function

$$R(t, s) = \left. \frac{\delta \langle \phi(t, \mathbf{r}) \rangle}{\delta h(s, \mathbf{r})} \right|_{h=0}$$

where $\phi$ is the order parameter field and $h$ an applied field.

2. Formulation of the Scaling Relation

The celebrated JSS scaling relation connects the initial slip exponent $\Theta$ to the autocorrelation exponent $\lambda$ and dynamical exponent $z$ for spatial dimension $d$:

$\Theta = \frac{d - \lambda}{z}$

This relation was initially derived^{\text{(Henkel, 23 Apr 2025)}} for quenches to $T_c$, linking the short-time increase of the global order parameter

$$m(t) = m_0 t^{\Theta} \mathcal{F}_m(m_0 t^{-x_0/z}),$$

(where $m_0$ is the initial magnetization and $x_0$ its scaling dimension) to the long-time scaling of $m(t)$ and to the scaling behavior of two-time functions.

The exponent $\lambda$ characterizes the decay of equal-site autocorrelations at long times:

$$C(t, s) \sim s^{-b} F_C\left( \frac{t}{s} \right), \quad F_C(y) \sim y^{-\lambda/z} \text{ for } y \gg 1,$$

with $b$ a scaling dimension determined by underlying field theory.

3. Extension to Generalized Time-Translation Invariance and Temperatures $T \leq T_c$

Recent developments^{\text{(Henkel, 23 Apr 2025)}} have shown that the JSS relation is underpinned by a generalization of time-translation invariance. In equilibrium, the time-translation generator $X_{-1} = -\partial_t$ ensures invariance, but after a quench the symmetry breaks. By introducing a “dressed” generator

$\bar{X}_{-1} = -\partial_t + \frac{\xi}{t}$

accompanied by a field transformation $$\bar{\phi}(t, \mathbf{r}) = t^{\xi} \phi(t, \mathbf{r})$$, the scaling theory generalizes to all $T \leq T_c$. This approach modifies the effective scaling dimensions and constrains two-time quantities to universal algebraic forms:

$$C(t, s) = s^{-b} F_C(t/s), \qquad R(t, s) = s^{-1-a} F_R(t/s),$$

with the large-$y$ behavior $F_{C/R}(y) \sim y^{-\lambda/z}$. The equality $\lambda_C = \lambda_R = \lambda$ emerges as a consequence of these symmetries, not requiring $T = T_c$.

By a scaling analysis of global two-time observables, such as the integrated auto-correlator

$$Q(t, s) = \frac{1}{|\Lambda|} \sum_{n,m} \langle \sigma_n(t) \sigma_m(s) \rangle,$$

it is shown that the early-time growth

$Q(t, 0) \sim t^{\Theta}$

persists both at and below the critical temperature. Consequently, the JSS relation $\Theta = (d-\lambda)/z$ retains its validity throughout the entire ageing regime for $T \leq T_c$.

4. Analytical Mechanism: Lie Algebra Structure and Scaling Functions

The extension of time-translation invariance modifies not only the exponent structure but the entire form of the scaling functions. The generalized generator leads to unique solutions for the scaling functions:

$$C(t, s) = s^{-2\delta + 2\xi}(t/s)^{-2\delta + \xi} \mathcal{F}_C(0),$$

where $\delta$ is the canonical scaling dimension and $\xi$ encodes deviation from equilibrium time-translation invariance. The effective scaling dimension $\delta_{\text{eff}} = \delta - \xi$ determines the exponents $b = 2(\delta-\xi)$ and $\lambda = 2\delta - \xi$, ensuring that the scaling forms derived for the two-time observables are consistent with the generalized symmetry and reducing to established critical results in equilibrium.

5. Physical Implications: Ageing, Initial Slip, and Universality

The initial critical slip exponent $\Theta$ characterizes the short-time power-law growth of the order parameter following a quench from a disordered initial state. The JSS relation establishes a direct bridge between this slip exponent, the system dimensionality, the autocorrelation decay, and the dynamical critical scaling:

$\Theta = (d-\lambda)/z.$

Global observables such as the total magnetization or global two-time correlators provide experimentally accessible quantities from which $\Theta$ can be measured. Combined with estimates of $z$ and $\lambda$ (or the analysis of the long-time decay of correlators), this allows a complete characterization of the non-equilibrium scaling sector.

This universality extends to coarsening and ageing dynamics in a broad range of systems, including spin glasses, liquid crystals, and relaxation in classical magnets. Experimental evidence supports the predicted scaling collapses and exponent equalities.

6. Relevance to Experimental and Numerical Investigations

The broad applicability of the JSS scaling relation, as extended in (Henkel, 23 Apr 2025), enables its use in experimental and numerical analysis of ageing phenomena. Measured two-time correlators and response functions, when plotted against $y = t/s$, show the anticipated power-law decay. The fluctuation-dissipation ratio and its limiting value are consistent with the scaling predictions.

The theoretical framework provides a criterion for the relevance of nonlinearities in stochastic dynamical equations, determined by the dimensionality of the couplings. This is applicable for all $T \leq T_c$, relevant to glassy relaxation, coarsening, and domain growth. The generalized invariance and scaling also yield predictions for finite-size scaling phenomena, such as the appearance of plateaus in auto-correlation or response functions.

7. Summary of Key Relations and Table of Main Exponents

The core relations underpinning the JSS framework are summarized below.

Exponent/Scaling Law	Definition or Formula	Context
Initial slip exponent	$\Theta$	Growth of $m(t)$ after quench
Autocorrelation exp.	$\lambda$	Asymptotic decay in $F_C(y)$
Dynamic exponent	$z$	$t\sim \xi^z$ scaling
JSS relation	$\Theta = (d-\lambda)/z$	Universal in ageing regime
Two-time correlation	$C(t,s) \sim s^{-b} F_C(t/s)$	Large $y$: $F_C(y)\sim y^{-\lambda/z}$

These universal relations, together with generalized time-translation invariance, provide the foundation for a unified scaling description of ageing at, and below, the critical temperature.