Random-field Ising-like effective theory of the glass transition. I. Mean-field models

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Random-field Ising-like effective theory of the glass transition. I. Mean-field models

Giulio Biroli, Chiara Cammarota, Gilles Tarjus, and Marco Tarzia

Phys. Rev. B 98, 174205 – Published 15 November 2018

Abstract

In this paper, we address the problem of identifying the effective theory that describes the statistics of the fluctuations of what is thought to be the relevant order parameter for glassy systems—the overlap field with an equilibrium reference configuration—close to the putative thermodynamic glass transition. Our starting point is the mean-field theory of glass formation, which relies on the existence of a complex free-energy landscape with a multitude of metastable states. In this paper, we focus on archetypal mean-field models possessing this type of free-energy landscape and set up the framework to determine the exact effective theory. We show that the effective theory at the mean-field level is generically of the random-field + random-bond Ising type. We also discuss the main issues concerning the extension of our result to finite-dimensional systems. This extension is addressed in detail in the companion paper.

Received 18 July 2018

DOI:https://doi.org/10.1103/PhysRevB.98.174205

Physics Subject Headings (PhySH)

Glass transition

Disordered systems Glasses

Replica methods

Statistical Physics & Thermodynamics

Authors & Affiliations

Giulio Biroli^1,*, Chiara Cammarota^2,†, Gilles Tarjus^3,‡, and Marco Tarzia^3,§

¹IPhT, CEA/DSM-CNRS/URA 2306, CEA Saclay, F-91191 Gif-sur-Yvette Cedex, France Laboratoire de Physique Statistique, École Normale Supérieure, CNRS, France
²Department of Mathematics, King's College London, Strand, London WC2R 2LS, United Kingdom
³LPTMC, CNRS-UMR 7600, Sorbonne Université, 4 Pl. Jussieu, F-75005 Paris, France

^*giulio.biroli@cea.fr
^†chiara.cammarota@kcl.ac.uk
^‡tarjus@lptmc.jussieu.fr
^§tarzia@lptmc.jussieu.fr

Random field Ising-like effective theory of the glass transition. II. Finite-dimensional models

Giulio Biroli, Chiara Cammarota, Gilles Tarjus, and Marco Tarzia

Phys. Rev. B 98, 174206 (2018)

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Vol. 98, Iss. 17 — 1 November 2018

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Images

Figure 1
Sketch of the phase diagram of constrained glass-forming systems predicted by the mean-field/RFOT theory in the $(T − ε)$ plane (left) and in the $(T − c)$ plane (right), showing the regions where RFIM-like universality classes for the critical points in finite dimensions were identified (see text for a more detailed description). The dynamical (MCT) glass transitions are in the same universality class as the spinodal point of the RFIM [28] (green). The terminal critical points in the $(T − ε)$ plane and in the $(T − c)$ plane belong to the universality class of the RFIM [29, 31, 51] (violet). The first-order transition line in the $(T − ε)$ plane is described by a first-order transition in the presence of a random field [31] (brown). The situation in the absence of coupling $(ε = 0, c = 0)$ and close to the putative RFOT $T_{K}$ (yellow) has been investigated for the first time numerically in Ref. [35] and is the focus of the present paper. Note that one expects that the putative line of RFOT in the $(T − c)$ plane is described by a similar effective theory as that near $T_{K}$ .
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Figure 2
Sketch of the distribution of local Franz-Parisi potentials when the system is divided in cubic boxes larger than the size of the particles but smaller than the point-to-set correlation length. This illustrates the local fluctuations of the configurational entropy and of the surface tension.
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Figure 3
1-replica part $S_{1} (c)$ of the replicated effective action as a function of the overlap $c$ with a reference equilibrium configuration for the fully connected $2^{M}$ -KREM with $M = 3$ [Eqs. (14) and (15)]. Several values of the temperature, $T \geq T_{K} \approx 0.4$ (i.e., $β \leq β_{K} \approx 2.5)$ , are shown.
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Figure 4
Phase diagram of the $2^{M}$ -KREM in the $T − ε$ plane (left) and of the effective disordered theory in the plane $\sqrt{Δ_{h}} / J_{2} - H$ (right). For simplicity, the results are illustrated in the limit $M \to \infty$ . Upon decreasing the temperature in the original model (green lines) $\sqrt{Δ_{h}} / J_{2}$ increases as $\sqrt{8 / M} T$ and $H$ increases as $H = (Δ_{h}^{2} / J_{2} - M ln 2) / 2$ . The thermodynamic glass transition (RFOT) is met when $H$ goes through 0, i.e., $T_{K} = {(8 ln 2)}^{- 1 / 2}$ (yellow circles). The first-order transition line (red) in the $T − ε$ becomes the vertical line at $H = 0$ in the right panel, which terminates at the critical point of the effective disordered Ising model where $\sqrt{Δ_{h}} / J_{2} = 1$ (violet circles).
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Figure 5
Comparison between the (truncated) effective theory and the exact result for the fully connected $2^{M}$ -KREM with $M = 3$ : mean overlap $〈 p 〉$ with a reference equilibrium configuration versus the inverse temperature $β$ . The black curve (circles) represents the exact solution [Eqs. (A4) and (A6)], whereas the red curve (squares) corresponds to the prediction of the effective random-field + random-bond Ising Hamiltonian, with the coupling constants of the multibody interactions (up to four-body) numerically extracted from Fig. 3 as discussed in the text and the covariances of the random variables given in Eq. (22). The counterpart of the overlaps in the disordered Ising model is the magnetization $(1 + m) / 2$ , where $m = \bar{〈 σ_{i} 〉}$ [see Appendix pp2 and Eqs. (B2)]. The effective disordered theory has been derived by using the annealed approximation to average over the random energies, which is in principle justified only above $T_{K}$ and at $T_{K}$ (which includes the jump of $〈 p 〉)$ . The blue curve (diamonds) shows the prediction of the effective theory below $T_{K}$ with an additional approximation, i.e., when the coupling constants and the covariances are obtained by continuing the results obtained via the annealed approximation in the low-temperature phase.
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Figure 6
Local part of the variance of the effective random field, $Δ_{2} (p_{1}, p_{2}) = \partial_{p_{1}} \partial_{p_{2}} S_{2} (p_{1}, p_{2})$ , and of the 2-replica action, $S_{2} (p_{1}, p_{2}) / N$ , for equal arguments $p_{1} = p_{2}$ as a function of $p_{1}$ (blue curve, circles) for the $p$ -spin model in the Kac limit with $p = 3$ and $β = 1.7$ (such that $β_{d} < β < β_{K})$ . $Δ_{2}$ is zero in the liquid minimum $(p_{1} = 0)$ , grows as $p_{1}$ is increased, so that it is strictly positive at the metastable glassy minimum. $S_{2} (p_{1}, p_{1})$ behaves similarly. We also display the local part of the first cumulant $S_{1} (p_{1}) / N$ given in Eq. (29), to show the position of the secondary minimum around $p_{★} \approx 0.65$ : grey curve and diamonds (note that the region around the barrier is not properly described by the present RS solution but this is irrelevant for our illustrative purpose). $Δ_{2}$ has been multiplied by 50 to plot the three curves on the same scale.
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