Real Space Migdal-Kadanoff Renormalisation of Glassy Systems: Recent Results and a Critical Assessment

Abstract

In this manuscript, in honour of L. Kadanoff, we present recent progress obtained in the description of finite dimensional glassy systems thanks to the Migdal-Kadanoff renormalisation group (MK-RG). We provide a critical assessment of the method, in particular discuss its limitation in describing situations in which an infinite number of pure states might be present, and analyse the MK-RG flow in the limit of infinite dimensions. MK-RG predicts that the spin-glass transition in a field and the glass transition are governed by zero-temperature fixed points of the renormalization group flow. This implies a typical energy scale that grows, approaching the transition, as a power of the correlation length, thus leading to enormously large time-scales as expected from experiments and simulations. These fixed points exist only in dimensions larger than $d_L>3$ but they nevertheless influence the RG flow below it, in particular in three dimensions. MK-RG thus predicts a similar behavior for spin-glasses in a field and models of glasses and relates it to the presence of avoided critical points.

Journal of Statistical Physics, Special Issue: Dedicated to the Memory of Leo Kadanoff

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Cited by in corpus (12)

• Perspective: Configurational entropy of glass-forming liquids
• Zero-temperature glass transition in two dimensions
• Configurational entropy of polydisperse supercooled liquids
• Unexpected upper critical dimension for spin glass models in a field predicted by the loop expansion around the Bethe solution at zero temperature
• Multicritical point on the de Almeida-Thouless line in spin glasses in $d>6$ dimensions
• Dimensional reduction breakdown and correction to scaling in the random-field Ising model
• The Ising spin glass on random graphs at zero temperature: not all spins are glassy in the glassy phase
• Why replica symmetry breaking does not occur below six dimensions in Ising spin glasses
• Multiscaling in the 3D critical site-diluted Ising ferromagnet
• Nature of spin glass order in physical dimensions
• Spin-s Spin-Glass Phases in the d=3 Ising Model
• Overlap renormalization group transformations for disordered systems