Random Field Ising-like effective theory of the glass transition II: Finite Dimensional Models

Abstract

As in the preceding paper we aim at identifying the effective theory that describes the fluctuations of the local overlap with an equilibrium reference configuration close to a putative thermodynamic glass transition. We focus here on the case of finite-dimensional glass-forming systems, in particular supercooled liquids. The main difficulty for going beyond the mean-field treatment comes from the presence of diverging point-to-set spatial correlations. We introduce a variational low-temperature approximation scheme that allows us to account, at least in part, for the effect of these correlations. The outcome is an effective theory for the overlap fluctuations in terms of a random-field + random-bond Ising model with additional, power-law decaying, pair and multi-body interactions generated by the point-to-set correlations. This theory is much more tractable than the original problem. We check the robustness of the approximation scheme by applying it to a fully connected model already studied in the companion paper. We discuss the physical implications of this mapping for glass-forming liquids and the possibility it offers to determine the presence or not of a finite-temperature thermodynamic glass transition.

26 pages (+11 pages of appendices), 8 figures. Sequel of paper I

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• Zero-temperature glass transition in two dimensions
• Attractive versus truncated repulsive supercooled liquids: The dynamics is encoded in the pair correlation function
• Statistical mechanics of coupled supercooled liquids in finite dimensions
• Random-Field Ising like effective theory of the glass transition: I Mean-Field Models
• On the overlap between configurations in glassy liquids
• Mean-field description for the architecture of low-energy excitations in glasses
• Large deviations of glassy effective potentials
• Possible instability of one-step replica symmetry breaking in p-spin Ising models outside mean-field theory
• Heterogeneous nucleation in the random field Ising model
• Inverse renormalization group of spin glasses
• Theory of the $\beta$-Relaxation Beyond Mode-Coupling Theory: A Microscopic Treatment
• Replica Field Theory for a Generalized Franz--Parisi Potential of Inhomogeneous Glassy Systems: New Closure and the Associated Self-Consistent Equation
• Overlap renormalization group transformations for disordered systems
• Dynamical phase transition in the activity-biased fully-connected random field Ising model: connection with glass-forming systems
• Static self-induced heterogeneity in glass-forming liquids: Overlap as a microscope