Logarithmic critical slowing down in complex systems: from statics to dynamics

We consider second-order phase transitions in which the order parameter is a replicated overlap matrix. We focus on a tricritical point that occurs in a variety of mean-field models and that, more generically, describes higher-order liquid-liquid or liquid-glass transitions. We show that the static replicated theory implies slowing down with a logarithmic decay in time. The dynamical equations turn out to be those predicted by schematic Mode Coupling Theory for supercooled viscous liquids at an $A_3$ singularity, where the parameter exponent is $\lambda =1$. We obtain a quantitative expression for the parameter $\mu$ of the logarithmic decay in terms of cumulants of the overlap, which are physically observable in experiments or numerical simulations.

In the present work we study a peculiar kind of critical slowing down occurring in the dynamics of slowly relaxing complex glassy systems, in which the correlation function of the relevant dynamic variables decays logarithmically in time. This is different from the usual behavior of, e.g., the correlation function of density fluctuations in supercooled liquids next to the dynamic arrest occurring in mean-field theories for glasses, somehow describing the real-world (off-equilibrium) glass transition of liquid glass-formers. In that case, the correlator next to the transition displays a two-step behavior: towards a plateau at short times and from the plateau towards zero correlation at longer times, with the plateau becoming longer and longer as the external parameters bring the system nearer to the dynamic arrest line.

In Götze’s Mode-Coupling Theory (MCT), a dynamic arrest critical point is referred to as an $A_2$ singularity, according to the classification of Arnold’s catastrophe theory. The critical point corresponding to a logarithmic decay is, instead, an $A_3$ cusp singularity, a tricritical point signalling the end-point of a liquid-liquid (or glass-glass) dynamic transition. Such tricritical behavior has been investigated in attractive colloids, in hypernetted-chain approximations, and in liquid models with pinned particles. The $-\ln t$ behavior of the correlation function appears to be the correct fitting law for about a decade or two in most of the known experiments and numerical simulations of repulsive colloids.

In what follows we develop a derivation of the logarithmic decay starting from the static replicated free energy expansion. The key insight is that the third cumulants of the order-parameter distribution control the asymptotic dynamic exponent, and that these cumulants are physically observable. Together with the connection to the schematic MCT $A_3$ fixed point, this provides a quantitative bridge between equilibrium and dynamic descriptions of higher-order glass transitions.

We work in the limit of infinite dimensions, in which $Q_{ab}$ is naturally identified with the averaged density-density fluctuations in momentum space in a replicated system at wave vector $\mathbf{k}$:

$$Q_{ab}\equiv \frac{1}{V}⟨\delta {\rho }_a^{\ast }(\mathbf{k})\delta {\rho }_b(\mathbf{k})⟩,\text{\hspace{1em}(3)}$$

where ${\rho }_a(\mathbf{k})$ is the Fourier transform of the density of replica $a$. In the structural-glass setting we are interested in the time-correlation function

$$C(t)\equiv \frac{1}{V}⟨\delta {\rho }^{\ast }(\mathbf{k},0)\delta \rho (\mathbf{k},t)⟩.\text{\hspace{1em}(6)}$$

In the liquid / paramagnetic phase, $C(t)$ decays exponentially but the correlation time diverges at the critical point. As shown in [24], the structure of the replicated free energy near the tricritical point determines the leading dynamical behavior of $C(t)$ through a static-to-dynamic mapping that we make explicit in Sec. III.

In this paper we extend the standard analysis of the tricritical replica free energy by retaining the cubic and quartic invariants and tracking the contribution of each to the long-time dynamical equation. The $y_3$ term, actually, vanishes — as will be shown in Sec. III A — under the conditions that select the asymptotic dynamics. The remaining quartic vertices $y_1,y_2,y_4$ combine to give the central parameter $\mu$.

We will also consider the critical behavior of the physical susceptibilities. In particular, we show that close to the critical point, where $r$ vanishes linearly with the external parameters in mean-field models, the three-point susceptibilities ${\omega }_i(i=1,2)$ diverge as

$${\omega }_i=\frac{w_i}{r^3},\text{\hspace{1em}(23)}$$

while the four-point quartic susceptibilities ${\upsilon }_i(i=1,\dots ,4)$ diverge with a different exponent. The combination ${\upsilon }_1+{\upsilon }_2-{\upsilon }_4$ enters the main result for the logarithmic decay; this combination is what enables the cumulant-level interpretation of $\mu$ that we expand on in Sec. IV.

The main result is the logarithmic relaxation of the correlator close to the tricritical point. Writing the singular part of the relaxation as $g(y)$ with $y=\ln (t/t_1)$, we find the asymptotic expansion

$$C(t)-C(\infty )=\frac{2{\pi }^2}{3\mu {\ln }^2(t/t_1)}+\frac{24\zeta (3)}{\mu {\ln }^3(t/t_1)}\ln \ln (t/t_1)+\cdots ,\text{\hspace{1em}(16)}$$

where the parameter $\mu$ depends on the quartic coupling constants $y_1,y_2,y_4,w$ through

$$\mu =-\frac{y_1+y_2-y_4}{3w}.\text{\hspace{1em}(17)}$$

Equivalently, in terms of the third cumulants of the order-parameter distribution,

$$\mu =-r\frac{{\upsilon }_1+{\upsilon }_2-{\upsilon }_4}{3{\omega }_1}.\text{\hspace{1em}(18)}$$

Eqs. (17) and (18) connect $\mu$ to physically observable cumulants — the central claim advertised in the abstract. The remainder of this section traces the derivation from the static replicated free energy through the Laplace-transform structure that yields Eq. (16).

The static result is obtained in the so-called Fast Motion (FM) limit. The Laplace transform of a function $A(t)$ satisfies

$$\mathrm{LT}\left[\frac{dA(t)}{dt}\right]=-iz\hat{A}(z),\text{\hspace{1em}(39)}$$

and the contributions of the quadratic vertices $w_1,w_2$ to the dynamic equation become

$$w_1(\delta Q^2{)}_{ab}\to w_{1}z{\hat{G}}^2(z),w_2\delta Q_{ab}^2\to w_2\mathrm{LT}[G^2(t)].\text{\hspace{1em}(41-42)}$$

After Eq. (50), we shorten $\mu \equiv (y_1+y_2-y_4)/(3w_1)$. The Laplace-transform structure of Eq. (51) is then

$$0=z\left\{\frac{w_2}{w_1}\mathrm{LT}[G^2(t)]+z{\hat{G}}^2(z)\right\}-\mu z\mathrm{LT}[G^3(t)].\text{\hspace{1em}(51)}$$

In the equal-coupling limit $w_1=w_2$, the leading order combines Eq. (51) with the Taylor expansion in Eq. (53)–(57),

$$z\left\{\mathrm{LT}[G^2(t)]+z{\hat{G}}^2(z)\right\}=-\zeta (2)(g^{\prime }{)}^2+\cdots ,\text{\hspace{1em}(57)}$$

with $\zeta (2)={\pi }^2/6$, which gives, at leading order,

$$0=-\frac{{\pi }^2}{6}(g^{\prime }{)}^2+\mu g^3.\text{\hspace{1em}(58a)}$$

Note: the manuscript writes Eq. (58) with prefactor $-(2{\pi }^2/3)$, which is larger by a factor of four relative to the form implied by Eq. (57). Section IV and Eq. (125) of the appendix return to the negative-sign convention for $\mu$. We address these notation issues, and the corresponding effect on the leading coefficient of Eq. (16), at the end of the section.

To express $\mu$ in terms of physical observables we expand around the saddle $\delta {\tilde{Q}}_{ab}$:

$$\delta {\tilde{Q}}_{ab}=\frac{1}{N}\sum _i{s}_i^a{s}_i^b-q,q\equiv \frac{1}{N}\sum _i\overline{⟨s_i^a{s}_i^b{⟩}_J^2}.\text{\hspace{1em}(68-69)}$$

The cubic and quartic cumulants of the overlap distribution $W_i,{\upsilon }_i$ are related to the susceptibilities ${\omega }_i$ by identities established in [40]. The key combination entering Eq. (18) is

$${\upsilon }_4=\frac{6}{N}\sum _{ijk}⟨s_i{s}_j{s}_k{⟩}_c⟨s_i{s}_j{s}_{\ell }{⟩}_c⟨s_{\ell }{s}_k{⟩}_c.\text{\hspace{1em}(108)}$$

The summation indices and contraction structure of Eq. (108) — and the analogous Eq. (114) for the cubic free-energy expansion — are subtle and we revisit them in the equation-audit below.

We have shown that the static replicated theory of the tricritical point yields, through the Fast Motion limit, dynamical equations whose leading asymptotic behavior matches the schematic $A_3$ singularity of Mode Coupling Theory at parameter exponent $\lambda =1$. The leading coefficient $\mu$ is given in closed form by Eq. (17), and equivalently in cumulant language by Eq. (18), connecting the asymptotic decay to physical observables in experiments and numerical simulations.

Beyond the present static derivation, the framework suggests that static replicated theories can be connected to off-equilibrium dynamics as well [58?]. We discuss preasymptotic $-\ln t$ fits and their relation to the asymptotic $1/{\ln }^{2}t$ law in the discussion of Sec. IV. The two regimes are not equivalent and we point out where each is appropriate. The full off-equilibrium extension is left to future work.