Solving the universal aging equations for critical off-equilibrium dynamics in glassy systems for general parameter exponent $\lambda$

Caltagirone, Parisi, and Rizzo (Phys. Rev. E 87, 032134 (2013)) showed that when a glassy system with a continuous mean-field transition is quenched to its critical temperature, the aging regime of the two-time correlation and response is governed by two universal nonlinear integral equations for scaling functions $C(a)$ and $R(a)$, depending on the single parameter exponent $\lambda=w_2/w_1$, the ratio of the cubic proper vertices of the equilibrium replicated Gibbs free energy. They exhibited an exact solution only for $\lambda=0$, a variational numerical scheme reliable up to $\lambda\approx0.55$ that misses the proven nonanalytic subleading corrections near $a=0$, and they restricted the derivation to a special cubic replica action, leaving the continuous transition in a field (replicon-only) and the discontinuous generic-$A_2$ transition unaddressed. No published work solves the aging equations for general $\lambda$, provides controlled large-$\lambda$ solutions, or derives the off-equilibrium equations for the other transition classes. The source-faithful open core therefore survives: exact or controlled solutions of the universal aging equations for general $\lambda$, and the extension of the replica derivation to the replicon-only and generic-$A_2$ classes.

Background

Glassy systems such as spin glasses and supercooled liquids relax ever more slowly on cooling, so that near a critical temperature the relaxation time exceeds accessible times and the system falls out of equilibrium. A satisfactory theory must therefore describe the off-equilibrium (aging) regime, in which observables depend on the age of the system since preparation, not only on time differences. In mean-field spin-glass models the order parameter is a two-point function: the spin-spin correlation $C(t,s)=N^{-1}\sum_i\overline{\langle s_i(t)s_i(s)\rangle}$ (disorder average overlined) and the response $R(t,s)=\sum_i\delta\langle s_i(t)\rangle/\delta h_i(s)$ to a small perturbing field. Aging means that for large times $t$ and $s$ with fixed ratio $a=s/t$ these functions decay as powers of $t$ rather than of $t-s$. The replica method describes disordered systems by introducing $n$ copies (replicas) of the system and an $n\times n$ overlap matrix order parameter $Q_{ab}$; the replicated Gibbs free energy is the corresponding Landau functional. Parisi and Rizzo (Phys. Rev. E 87, 012101 (2013)) showed that the equilibrium critical dynamics of a whole class of glassy models (the Sherrington-Kirkpatrick (SK) spin glass in zero field, spherical $p$-spin models, i.e. mean-field models with continuous spins under a global sphere constraint and random $p$-spin couplings, including the mixed $(p+q)$-spin variants in which $\lambda$ can be tuned, the three-state Potts glass, a disordered three-state generalization of the Ising spin glass, and schematic Mode-Coupling Theory (MCT) models, i.e. simplified few-mode versions of the liquid-glass dynamical equations, at their type-A transition where the order parameter grows continuously from zero, also called a degenerate $A_2$ singularity) follows from the cubic expansion of the replicated Gibbs free energy near the critical point, $G(Q)=a_T\frac{T-T_c}{2}\sum_{a,b}Q_{ab}^2-\frac{w_1}{6}\mathrm{Tr}Q^3-\frac{w_2}{6}\sum_{a,b}Q_{ab}^3$: the exponent $\nu$ of the equilibrium power-law decay $C_{\rm eq}(\tau)\propto\tau^{-\nu}$ is fixed by the single parameter exponent $\lambda=w_2/w_1$ through $\Gamma^2(1-\nu)/\Gamma(1-2\nu)=\lambda$, where $\Gamma$ is the gamma function and $w_1,w_2$ are the cubic proper vertices, i.e. the coefficients of the two cubic invariants of the overlap matrix in this expansion. Caltagirone, Parisi, and Rizzo (the source paper) extended this to off-equilibrium dynamics after a quench from $T=\infty$ to $T_c$. In the aging regime they derived the scaling ansatz $C(t,at)=c\,C(a)t^{-\nu}$ and $R(t,at)=c\,R(a)t^{-\nu-1}$, where $c$ is a model-dependent constant prefactor fixed by matching to the model's short-time equilibrium behavior, and the scaling functions $C(a)$ and $R(a)$ on $a\in[0,1]$ obey two quadratic singular integral equations (Eqs. (58)-(59) of the source) built from the equilibrium kernels $C_{\rm eq}(a)=1/(\nu(1-a)^{\nu})$ and $R_{\rm eq}(a)=1/(1-a)^{\nu+1}$ and depending on the model only through $\lambda$. The matching condition $a\to1$ reproduces equilibrium critical dynamics; the small-$a$ behavior $C(a)\propto a^{\alpha}$, $R(a)\propto a^{\alpha-1}$ defines a new exponent $\alpha$ whose determination requires the full solution of the equations. One-time quantities then follow: the energy approaches equilibrium as $E(t)-E_\infty\propto t^{-\Upsilon}$ with $\Upsilon=2\nu$, and the remanent magnetization decays as $m_R(t)\propto t^{-\delta}$ with $\delta=\alpha+\nu$. The exact solution is known only at $\lambda=0$ (the spherical 2-spin model): $C(a)=4a^{3/4}/((1+a)(1-a)^{1/2})$, $R(a)=a^{-1/4}/(1-a)^{3/2}$. For $\lambda\neq0$ only a variational Gauss-Newton scheme (trial functions equal to the $\lambda=0$ solution times a polynomial correction) is available, yielding consistent results only up to $\lambda\approx0.55$; the asymptotic analysis in the paper proves that for $\lambda\neq0$ the first subleading correction near $a=0$ is necessarily nonanalytic (a non-integer power of $a$), which the polynomial trial basis cannot represent. The derivation also assumes the special cubic structure above, in which the replicon, longitudinal, and anomalous eigenvalues of the replica stability matrix (the Hessian of the replicated Gibbs free energy at the replica-symmetric solution) vanish simultaneously. In an external field, on the de Almeida-Thouless line (the critical line in the temperature-field plane below which the replica-symmetric solution is unstable), only the replicon, the eigenvalue whose isolated vanishing signals a continuous transition to a replica-symmetry-broken phase, vanishes; the discontinuous generic-$A_2$ transition of MCT (the standard liquid-glass transition) is a further class. Both cases require extra quadratic replica-mixing terms $\sum m2_{abc}Q_{ac}Q_{ab}$, $\sum m3_{abcd}Q_{ab}Q_{cd}$ in the action, and both extensions were explicitly left for future work.

Problem Statement

The problem is to complete the determination of critical off-equilibrium (aging) dynamics from the equilibrium replicated Gibbs free energy for glassy systems, in the framework of Caltagirone, Parisi, and Rizzo (Phys. Rev. E 87, 032134 (2013)). It has two linked components serving this single objective. Component A (solve the universal equations): determine the scaling functions $C(a)$ and $R(a)$ solving the two universal quadratic singular integral equations (Eqs. (58) and (59) of the source, as printed in the published paper and in its preprint arXiv:1212.0475v2) for general parameter exponent $\lambda=w_2/w_1$ — the full open interval $(0,1)$ on which the equations and their equilibrium kernels are well defined ($\lambda=0$ is the exactly solved benchmark case; $\lambda=1$ is the tricritical $A_3$ endpoint where the power-law framework degenerates into logarithmic behavior, so the required control is the limit $\lambda\to1^-$) — either in closed analytical form or through a solution scheme with quantified error control valid for all $\lambda\in(0,1)$, including the large-$\lambda$ regime near $\lambda=1$; together with the exponent $\alpha(\lambda)$ governing the small-$a$ behavior $C(a)\propto a^{\alpha}$, $R(a)\propto a^{\alpha-1}$, and consequently the remanent-magnetization exponent $\delta=\alpha+\nu$; the solution must satisfy the matching to equilibrium critical behavior as $a\to1$ (with $\nu$ fixed by $\Gamma^2(1-\nu)/\Gamma(1-2\nu)=\lambda$), reproduce the known $\lambda=0$ exact solution, and capture the proven nonanalytic subleading corrections near $a=0$ for $\lambda\neq0$. Component B (extend the framework): extend the replica derivation beyond the special cubic replicated Gibbs free energy with vanishing quadratic replica-mixing coefficients, deriving the off-equilibrium scaling equations — and their solutions or exponent relations — for (i) the continuous spin-glass transition in an external field, where only the replicon eigenvalue vanishes and additional quadratic replica-mixing terms are present, and (ii) the discontinuous generic-$A_2$ transition of Mode-Coupling Theory (the standard liquid-glass transition).

References

1. F. Caltagirone, G. Parisi, T. Rizzo, "Critical off-equilibrium dynamics in glassy systems", Phys. Rev. E 87, 032134 (2013). DOI: 10.1103/PhysRevE.87.032134, https://doi.org/10.1103/PhysRevE.87.032134 (preprint arXiv:1212.0475, https://arxiv.org/abs/1212.0475)
2. G. Parisi, T. Rizzo, "Critical dynamics in glassy systems", Phys. Rev. E 87, 012101 (2013). DOI: 10.1103/PhysRevE.87.012101, https://doi.org/10.1103/PhysRevE.87.012101 (preprint arXiv:1205.3360, https://arxiv.org/abs/1205.3360)
3. F. Caltagirone, U. Ferrari, L. Leuzzi, G. Parisi, F. Ricci-Tersenghi, T. Rizzo, "Critical slowing down exponents of mode coupling theory", Phys. Rev. Lett. 108, 085702 (2012). DOI: 10.1103/PhysRevLett.108.085702, https://doi.org/10.1103/PhysRevLett.108.085702
4. T. Rizzo, "Replica-symmetry-breaking transitions and off-equilibrium dynamics", Phys. Rev. E 88, 032135 (2013). DOI: 10.1103/PhysRevE.88.032135, https://doi.org/10.1103/PhysRevE.88.032135 (preprint arXiv:1305.2070, https://arxiv.org/abs/1305.2070)
5. G. Parisi, F. Ricci-Tersenghi, T. Rizzo, "Diluted mean-field spin-glass models at criticality", J. Stat. Mech. (2014) P04013. DOI: 10.1088/1742-5468/2014/04/P04013, https://doi.org/10.1088/1742-5468/2014/04/P04013 (preprint arXiv:1401.1729, https://arxiv.org/abs/1401.1729)
6. A. Billoire, "Numerical study of the dynamics of some long range spin glass models", J. Stat. Mech. (2015) P07027. DOI: 10.1088/1742-5468/2015/07/P07027, https://doi.org/10.1088/1742-5468/2015/07/P07027 (preprint arXiv:1505.00676, https://arxiv.org/abs/1505.00676)
7. L. A. Fernandez, I. Gonzalez-Adalid Pemartin, V. Martin-Mayor, G. Parisi, F. Ricci-Tersenghi, T. Rizzo, J. J. Ruiz-Lorenzo, M. Veca, "Numerical test of the replica-symmetric Hamiltonian for correlations of the critical state of spin glasses in a field", Phys. Rev. E 105, 054106 (2022). DOI: 10.1103/PhysRevE.105.054106, https://doi.org/10.1103/PhysRevE.105.054106 (preprint arXiv:2107.06636, https://arxiv.org/abs/2107.06636)
8. G. Folena, F. Zamponi, "On weak ergodicity breaking in mean-field spin glasses", SciPost Phys. 15, 109 (2023). DOI: 10.21468/SciPostPhys.15.3.109, https://doi.org/10.21468/SciPostPhys.15.3.109 (preprint arXiv:2303.00026, https://arxiv.org/abs/2303.00026)
9. L. Leuzzi, T. Rizzo, "Logarithmic critical slowing down in complex systems: from statics to dynamics", Phys. Rev. B 109, 174211 (2024). DOI: 10.1103/PhysRevB.109.174211, https://doi.org/10.1103/PhysRevB.109.174211 (preprint arXiv:2403.07565, https://arxiv.org/abs/2403.07565)
10. G. Perrupato, T. Rizzo, "Solution of the critical dynamics of the mean-field Kob-Andersen model", Phys. Rev. E 112, 045402 (2025). DOI: 10.1103/yms3-km6k, https://doi.org/10.1103/yms3-km6k (preprint arXiv:2505.16451, https://arxiv.org/abs/2505.16451)
11. M. C. Angelini, S. Palazzi, G. Parisi, T. Rizzo, "Critical exponents of the spin-glass transition in a field at zero temperature", Proc. Natl. Acad. Sci. U.S.A. 122 (37), e2511882122 (2025). DOI: 10.1073/pnas.2511882122, https://doi.org/10.1073/pnas.2511882122 (preprint arXiv:2502.21089, https://arxiv.org/abs/2502.21089)