Genericity of the absence of multiplicative logarithmic corrections in two-time ageing scaling at the upper critical dimension

The kinetic spherical model — the $n\to\infty$ limit of the $O(n)$ ferromagnet — quenched to its upper critical dimension $d^{*}=4$ shows simple ageing: the leading scaling of the two-time autocorrelation and autoresponse in the limit $s,t\to\infty$ with $y=t/s>1$ fixed contains no multiplicative logarithmic correction factors, and the growing length scales as $L(t)\sim t^{1/2}$; logarithms at $T=T_c$ enter only additively, as corrections to leading scaling. Ebbinghaus, Grandclaude and Henkel (2008) explicitly left open to what extent this compensation of logarithms is generic. The question remains open: the compensation has since been reproduced only within the spherical/large-$n$ family (including an irreversible asymmetric dynamics and a conserved order parameter), while for finite-$n$ $O(n)$ models at $d=4$, for tricritical points with $d^{*}=3$, and for other model classes, no leading two-time scaling form at $d=d^{*}$ is known. The problem is to settle this: determine the leading two-time ageing scaling at the upper critical dimension in cases where it is currently unknown — including the tricritical test explicitly suggested by the source — and decide whether multiplicative logarithmic factors reappear or the compensation persists.

Background

Ageing is the generic long-time behaviour of a many-body system rapidly quenched from a disordered equilibrium state to a temperature at or below its critical temperature $T_c$: relaxation is slow and non-exponential, time-translation invariance is broken (the response depends on the system's age), and the dynamics exhibits scaling. Ageing is probed through two-time observables, chiefly the autocorrelation $C(t,s)=\langle\phi(t,\mathbf r)\,\phi(s,\mathbf r)\rangle$ and the linear autoresponse $R(t,s)=\delta\langle\phi(t,\mathbf r)\rangle/\delta h(s,\mathbf r)$ of the order-parameter field $\phi$, where $s$ is the waiting time (time elapsed since the quench), $t>s$ the observation time, and $h$ a field conjugate to $\phi$. Systems with so-called simple ageing obey, in the scaling limit $s,t\to\infty$ with fixed ratio $y=t/s>1$, the forms $C(t,s)=s^{-b}f_C(y)$ and $R(t,s)=s^{-1-a}f_R(y)$, with a typical domain size growing algebraically, $L(t)\sim t^{1/z}$.

The spherical model of a ferromagnet — the $n\to\infty$ limit of the $O(n)$ vector models, in which the spin length is fixed only by a global constraint enforced through a time-dependent field — is exactly solvable: the constraint field $g(t)$ obeys a Volterra-type integral equation, from which Godrèche and Luck (J. Phys. A 33, 9141 (2000)) derived the two-time correlation and response of the kinetic model in all dimensions except $d=4$.

Four is the upper critical dimension $d^{*}$ of the model: the dimension at which the renormalization-group fixed point controlling the critical point becomes Gaussian and at which mean-field exponents begin to hold. At $d=d^{*}$, equilibrium critical scaling is modified by multiplicative logarithmic correction factors — textbook knowledge for the 4D spherical model — and a systematic theory of such equilibrium logarithms exists. By analogy, one would expect non-equilibrium ageing scaling to acquire multiplicative logarithmic factors at $d=d^{*}$ as well.

Ebbinghaus, Grandclaude and Henkel tested this for the kinetic spherical model with non-conserved order-parameter dynamics (Model A type: purely relaxational Langevin dynamics of a non-conserved field) quenched from a fully disordered state, and found the opposite. At $d=4$ the two-time scaling functions contain no multiplicative logarithmic factors — they follow from the $d\neq4$ expressions by analytic continuation $d\to4$ — and $L(t)\sim t^{1/2}$ without logarithms. Logarithms do occur, but only additively, i.e. in corrections to the leading scaling, for quenches onto $T=T_c$. The mechanism is elementary but sharp: in the scaling limit $\ln t/\ln s=\ln(ys)/\ln s\simeq 1+\ln y/(2\ln s)\to1$, so powers of $\ln t$ and $\ln s$ compensate; and at criticality the singular constraint-field asymptotic $g(t)\sim(\ln t)^{-1}$ — which follows from a $p\ln p$ term in the small-$p$ behaviour of the Laplace transform of the model's dispersion integral — enters the observables only through ratios in which the (not fully controlled) amplitudes cancel.

Not every system behaves this way. Where topological defects such as vortices drive the coarsening, $L(t)\sim(t/\ln t)^{1/2}$, and logarithmic corrections in ageing have been observed numerically in frustrated magnets. Conversely, at a tricritical point — where a line of first-order transitions meets a second-order one and the upper critical dimension drops to $d^{*}=3$ — one-loop field theory (Janssen and Oerding, 1994) finds the habitually expected multiplicative logarithms already in single-time observables such as $M(t)\sim M_0(\ln t)^{-a(n)}$, with the exponent $a(n)$ diverging as $n\to\infty$; but only single-time quantities were computed there, so the two-time compensation mechanism could not be tested.

The open question is thus whether the compensation of logarithmic factors in the leading two-time ageing scaling at an upper critical dimension is a generic property of non-equilibrium critical dynamics or a peculiarity of the $n\to\infty$ limit, in which loop corrections are absent. The suspicion that $n=\infty$ is special is fed precisely by the divergence of $a(n)$ as $n\to\infty$. The answer determines whether the standard simple-ageing analysis of two-time data remains valid for experiments and simulations at marginal dimensions — four-dimensional ferromagnets, tricritical systems in $d=3$ — and whether exactly solved spherical-model results are faithful guides there.

Problem Statement

Determine whether the absence of multiplicative logarithmic corrections in the leading two-time ageing scaling at the upper critical dimension, established for the spherical model ($n=\infty$) at $d^{*}=4$, is generic, or whether it is specific to the exactly solvable large-$n$ limit.

For a system quenched from a disordered state to its critical point at its upper critical dimension $d^{*}$, derive (or measure) the leading long-time behaviour of the two-time autocorrelation $C(t,s)$ and autoresponse $R(t,s)$ in the ageing scaling limit $s,t\to\infty$ with $y=t/s>1$ fixed, and decide whether multiplicative logarithmic factors — powers of $\ln s$, factors beyond the compensating ratio $\ln t/\ln s$, or $(\ln t)^{-a}$-type prefactors as found at one loop for single-time tricritical observables — appear in the leading scaling, or whether, as in the 4D spherical model, logarithms enter only additively, as corrections to the leading scaling. The scientifically linked open cases include: (a) finite-$n$ $O(n)$ models with non-conserved dynamics at $d=4$, where loop corrections are present and the $n\to\infty$ limit may be singular; (b) dynamics at a tricritical point, $d^{*}=3$ — for example the spherical metamagnet (a spherical model with a competing interaction, placed in an external field so as to sit at its tricritical point), whose study the source explicitly suggests as a test of the two-time compensation, or tricritical $O(n)$ dynamics — where the analysis must be redone with the tricritical singularity structure of the constraint or mass; and (c) other classes of dynamics at their $d^{*}$, with different conserved quantities, symmetry groups, or fluctuation-dissipation properties.

The objective is discharged when the presence or absence of multiplicative logarithmic factors in the leading two-time ageing scaling at $d=d^{*}$ is settled, with explicit scaling forms, for at least one case whose leading two-time behaviour at the upper critical dimension is currently unknown — the tricritical test explicitly suggested by the source (the spherical metamagnet at $d^{*}=3$) and finite-$n$ $O(n)$ models at $d=4$ being prominent instances — or by a general argument fixing the answer for a broad class of dynamics. Establishing that loop corrections reintroduce multiplicative logarithms at finite $n$ or at a tricritical point answers the question just as definitively as establishing that the compensation persists.

References

1. M. Ebbinghaus, H. Grandclaude, M. Henkel, Absence of logarithmic scaling in the ageing behaviour of the 4D spherical model, Eur. Phys. J. B 63, 85–91 (2008), DOI 10.1140/epjb/e2008-00202-2, arXiv:0709.3220, https://arxiv.org/abs/0709.3220
2. C. Godrèche, J.-M. Luck, Response of non-equilibrium systems at criticality: ferromagnetic models in dimension two and above, J. Phys. A: Math. Gen. 33, 9141–9164 (2000), DOI 10.1088/0305-4470/33/50/302, arXiv:cond-mat/0001264, https://arxiv.org/abs/cond-mat/0001264
3. C. Godrèche, J.-M. Luck, Asymmetric Langevin dynamics for the ferromagnetic spherical model, J. Stat. Mech. (2013) P05006, DOI 10.1088/1742-5468/2013/05/P05006, arXiv:1302.4658, https://arxiv.org/abs/1302.4658
4. H. K. Janssen, K. Oerding, Non-equilibrium relaxation at a tricritical point, J. Phys. A: Math. Gen. 27, 715–723 (1994), DOI 10.1088/0305-4470/27/3/016, https://doi.org/10.1088/0305-4470/27/3/016
5. P. Calabrese, A. Gambassi, Aging in ferromagnetic systems at criticality near four dimensions, Phys. Rev. E 65, 066120 (2002), DOI 10.1103/PhysRevE.65.066120, arXiv:cond-mat/0203096, https://arxiv.org/abs/cond-mat/0203096
6. M. Vodret, A. Chiocchetta, A. Gambassi, Universal amplitudes ratios for critical aging via functional renormalization group, J. Phys. A: Math. Theor. 55, 264008 (2022), DOI 10.1088/1751-8121/ac7182, arXiv:2112.03015, https://arxiv.org/abs/2112.03015
7. F. Baumann, M. Henkel, Kinetics of phase-separation in the critical spherical model and local scale-invariance, J. Stat. Mech. (2007) P01012, DOI 10.1088/1742-5468/2007/01/P01012, arXiv:cond-mat/0611652, https://arxiv.org/abs/cond-mat/0611652
8. J.-C. Walter, C. Chatelain, Logarithmic corrections in the ageing of the fully frustrated Ising model, J. Stat. Mech. (2008) P07005, DOI 10.1088/1742-5468/2008/07/P07005, arXiv:0707.1959, https://arxiv.org/abs/0707.1959
9. X. Durang, M. Henkel, Exactly solvable models of growing interfaces and lattice gases: the Arcetri models, ageing and logarithmic sub-ageing, J. Stat. Mech. (2017) P123206, DOI 10.1088/1742-5468/aa9a53, arXiv:1708.08237, https://arxiv.org/abs/1708.08237
10. S. Wald, G. T. Landi, M. Henkel, Lindblad dynamics of the quantum spherical model, J. Stat. Mech. (2018) 013103, DOI 10.1088/1742-5468/aa9f44, arXiv:1707.06273, https://arxiv.org/abs/1707.06273
11. J. C. Halimeh, M. F. Maghrebi, Quantum aging and dynamical universality in the long-range $O(N\to\infty)$ model, Phys. Rev. E 103, 052142 (2021), DOI 10.1103/PhysRevE.103.052142, arXiv:2008.08583, https://arxiv.org/abs/2008.08583
12. M. Henkel, Non-equilibrium relaxations: ageing and finite-size effects, Condens. Matter Phys. 26, 13501 (2023), DOI 10.5488/cmp.26.13501, arXiv:2211.03657, https://arxiv.org/abs/2211.03657
13. M. Henkel, Physical ageing from generalised time-translation-invariance, Nucl. Phys. B 1017, 116968 (2025), DOI 10.1016/j.nuclphysb.2025.116968, arXiv:2504.16857, https://arxiv.org/abs/2504.16857
14. L. Moueddene, M. Henkel, Short-time dynamics in phase-ordering kinetics, Int. J. Theor. Phys. 65, 92 (2026), DOI 10.1007/s10773-025-06205-0, arXiv:2511.00498, https://arxiv.org/abs/2511.00498
15. Yuan Chen, Hong-Dong Liang, Critical aging properties of the two-dimensional spherical model with the long-range interaction $r^{-3}$, Physica A 391, 4661–4674 (2012), DOI 10.1016/j.physa.2012.05.012, https://doi.org/10.1016/j.physa.2012.05.012