^{1}and Mauro Sellitto

^{2}

### Abstract

We provide extended evidence that mode-coupling theory (MCT) of supercooled liquids for the schematic model admits a microscopic realization based on facilitated spin models with tunable facilitation. Depending on the facilitation strength, one observes two distinct dynamical glass transition lines—continuous and discontinuous—merging at a dynamical tricritical-like point with critical decay exponents consistently related by MCT predictions. The mechanisms of dynamical arrest can be naturally interpreted in geometrical terms: the discontinuous and continuous transitions correspond to bootstrap and standard percolation processes, in which the incipient spanning cluster of frozen spins forms either a compact or a fractalstructure, respectively. Our cooperative dynamical facilitation picture of glassy behavior is complementary to the one based on disordered systems and can account for higher-order singularity scenarios in the absence of a finite temperature thermodynamic glass transition. We briefly comment on the relevance of our results to finite spatial dimensions and to the schematic model.

We thank D. de Martino and F. Caccioli for their participation in early stages of this work, and W. Götze for clarifications about MCT. J.J.A. is a member of the INCT-Sistemas Complexos and is partially supported by the Brazilian agencies Capes, CNPq, and FAPERGS.

I. INTRODUCTION

II. MODE-COUPLING THEORY: THE SCENARIO

III. FACILITATED SPIN SYSTEMS AND BOOTSTRAP PERCOLATION ON BETHE LATTICE

A. Noncooperative dynamics

B. Cooperative dynamics

1. Continuous transition

2. Discontinuous transition

IV. FACILITATED SPIN MIXTURES WITH TUNABLE FACILITATION: EXACT RESULTS

V. NUMERICAL SIMULATIONS OF A SPIN MIXTURE WITH DISCONTINUOUS AND CONTINUOUS GLASS TRANSITIONS

A. Persistence

B. Relaxation times and critical decay exponents

VI. DISCUSSION AND CONCLUSIONS

### Key Topics

- Glass transitions
- 39.0
- Percolation
- 10.0
- Relaxation times
- 8.0
- Classical spin models
- 7.0
- Mean field theory
- 7.0

## Figures

Phase diagram for *z* = 4 and facilitation as in Eq. (20). The dark region is the glassy phase. The dotted/solid line is the hybrid/continuous transition. Inset: Fraction of frozen spins, Eq. (A8), vs temperature for several values of *q* and *r* = 10^{−3}. Below *q* = 1/2, Φ jumps to a finite value at the transition which is represented by the dotted line.

Phase diagram for *z* = 4 and facilitation as in Eq. (20). The dark region is the glassy phase. The dotted/solid line is the hybrid/continuous transition. Inset: Fraction of frozen spins, Eq. (A8), vs temperature for several values of *q* and *r* = 10^{−3}. Below *q* = 1/2, Φ jumps to a finite value at the transition which is represented by the dotted line.

Persistence vs time for several values of *q* both in the discontinuous (*q* = 0, 0.2, and 0.4) and the continuous region (0.52) of the phase diagram, Fig. 1, for *r* = 10^{−3} and several temperatures, either above and below . The solid black lines show the theoretical prediction, Eq. (A8), for the plateau heights for while the dotted ones signal the critical plateau at , *q* < 1/2. For temperatures above the transition, the long time value of the persistence is zero.

Persistence vs time for several values of *q* both in the discontinuous (*q* = 0, 0.2, and 0.4) and the continuous region (0.52) of the phase diagram, Fig. 1, for *r* = 10^{−3} and several temperatures, either above and below . The solid black lines show the theoretical prediction, Eq. (A8), for the plateau heights for while the dotted ones signal the critical plateau at , *q* < 1/2. For temperatures above the transition, the long time value of the persistence is zero.

Asymptotic fraction of blocked spins Φ (plateau height) as the temperature approaches for several values of *r* and *q* = 0.52 in the continuous region of the phase diagram. Points are the result of simulations while the lines are from Eq. (A8). The relevant temperature interval, in which Φ ∼ ε^{−1}, is not accessible in our numerical simulation when *r* ≠ 0. Inset: The same for *q* = 0.2 for which the transition is discontinuous (notice the different labels in the *y*-axis).

Asymptotic fraction of blocked spins Φ (plateau height) as the temperature approaches for several values of *r* and *q* = 0.52 in the continuous region of the phase diagram. Points are the result of simulations while the lines are from Eq. (A8). The relevant temperature interval, in which Φ ∼ ε^{−1}, is not accessible in our numerical simulation when *r* ≠ 0. Inset: The same for *q* = 0.2 for which the transition is discontinuous (notice the different labels in the *y*-axis).

Data collapse of persistence data for several values of *q* versus the rescaled time, *t*/τ, where τ is the characteristic time associated with the α relaxation. We obtain a good collapse for data such that ϕ(*t*) < Φ_{ c } and, as Φ_{ c } → 0 when *q* → 1/2, the collapse region decreases as well, disappearing above the dynamical tricritical point. The collapsed region is well described by a KWW stretched exponential, shown as solid lines for *q* = 0 and 0.2. The stretched exponent is 0.5 for *q* = 0 and 0.35 for *q* = 0.2.

Data collapse of persistence data for several values of *q* versus the rescaled time, *t*/τ, where τ is the characteristic time associated with the α relaxation. We obtain a good collapse for data such that ϕ(*t*) < Φ_{ c } and, as Φ_{ c } → 0 when *q* → 1/2, the collapse region decreases as well, disappearing above the dynamical tricritical point. The collapsed region is well described by a KWW stretched exponential, shown as solid lines for *q* = 0 and 0.2. The stretched exponent is 0.5 for *q* = 0 and 0.35 for *q* = 0.2.

Equilibrium relaxation at criticality for discontinuous () and continuous () ergodic-nonergodic transition: solid lines in the main panels are power-law fits with exponents *a* ≃ 0.23 and 0.31, respectively. Inset: β-relaxation time vs temperature difference . The solid lines are power-law functions with exponents 1/2*a* and 1/*a*, where the value of *a* is obtained by the fits in the main panel.

Equilibrium relaxation at criticality for discontinuous () and continuous () ergodic-nonergodic transition: solid lines in the main panels are power-law fits with exponents *a* ≃ 0.23 and 0.31, respectively. Inset: β-relaxation time vs temperature difference . The solid lines are power-law functions with exponents 1/2*a* and 1/*a*, where the value of *a* is obtained by the fits in the main panel.

Symmetric departure from the critical power law behavior for curves at temperatures equidistant from , for *q* = 0.52. The closer the temperature gets to , the more symmetrical the curves become and the longer it takes to depart from the critical power law. An analogous behavior is observed for *q* = 0.2 as well.

Symmetric departure from the critical power law behavior for curves at temperatures equidistant from , for *q* = 0.52. The closer the temperature gets to , the more symmetrical the curves become and the longer it takes to depart from the critical power law. An analogous behavior is observed for *q* = 0.2 as well.

Exponent *a* obtained from the critical relaxation (circles) and from the characteristic time as the critical line is approached from above (squares). The solid lines are power-law fits that vanish as *q* → 1/2, consistently with a logarithmic relaxation at that point.

Exponent *a* obtained from the critical relaxation (circles) and from the characteristic time as the critical line is approached from above (squares). The solid lines are power-law fits that vanish as *q* → 1/2, consistently with a logarithmic relaxation at that point.

Estimate of the exponent *b* for *q* = 0.2 and *r* = 10^{−3}. For this value of *q*, *T* _{ c } ≃ 0.58 and one obtains γ ≃ 2.81, *a* ≃ 0.25 and, using Eq. (7), *b* ≃ 0.61. This predicted value is in very good agreement with the simulation, as shown by the solid line whose slope is 0.61.

Estimate of the exponent *b* for *q* = 0.2 and *r* = 10^{−3}. For this value of *q*, *T* _{ c } ≃ 0.58 and one obtains γ ≃ 2.81, *a* ≃ 0.25 and, using Eq. (7), *b* ≃ 0.61. This predicted value is in very good agreement with the simulation, as shown by the solid line whose slope is 0.61.

Phase diagram, temperature *T* vs fraction of constrained spins *q*, for the ternary mixture 0+2+3 (and 0+2+4) on a Bethe lattice with *z* = 4 and facilitation as in Eq. (A10). The dotted/full lines are discontinuous/continuous liquid glass transitions for various *r*. The line passing through the circles is the tricritical line.

Phase diagram, temperature *T* vs fraction of constrained spins *q*, for the ternary mixture 0+2+3 (and 0+2+4) on a Bethe lattice with *z* = 4 and facilitation as in Eq. (A10). The dotted/full lines are discontinuous/continuous liquid glass transitions for various *r*. The line passing through the circles is the tricritical line.

Phase diagram for the ternary mixtures 0+1+2 on a Bethe lattice with *z* = 4 and facilitation as in Eq. (A18). The glassy region is located below the lines which represent discontinuous liquid-glass transitions for various *r*.

Phase diagram for the ternary mixtures 0+1+2 on a Bethe lattice with *z* = 4 and facilitation as in Eq. (A18). The glassy region is located below the lines which represent discontinuous liquid-glass transitions for various *r*.

Phase diagram for the ternary mixture 0+1+3 (and 0+1+4) on a Bethe lattice with *z* = 4 and facilitation as in Eq. (A20). The glassy region is located below the lines which represent continuous liquid-glass transitions for various *r*.

Phase diagram for the ternary mixture 0+1+3 (and 0+1+4) on a Bethe lattice with *z* = 4 and facilitation as in Eq. (A20). The glassy region is located below the lines which represent continuous liquid-glass transitions for various *r*.

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