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.
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.
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 1/2a 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.
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.
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 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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