Replica Symmetry Broken States of some Glass Models

Abstract

We have studied in detail the $M$-$p$ balanced spin glass model, especially the case $p=4$. These types of model have relevance to structural glasses. The models possess two kinds of broken replica states; those with one-step replica symmetry breaking (1RSB) and those with full replica symmetry breaking (FRSB). To determine which arises requires studying the Landau expansion to quintic order. There are 9 quintic order coefficients, and 5 quartic order coefficients, whose values we determine for this model. We show that it is only for $2\le M<2.4714\cdots$ that the transition at mean-field level is to a state with FRSB, while for larger $M$ values there is either a continuous transition to a state with 1RSB (when $M\le 3$) or a discontinuous transition for $M>3$. The Gardner transition from a 1RSB state at low temperatures to a state with FRSB also requires the Landau expansion to be taken to quintic order. Our result for the form of FRSB in the Gardner phase is similar to that found when $2\le M<2.4714\cdots$, but differs from that given in the early paper of Gross et al. [Phys. Rev. Lett. 55, 304 (1985)]. Finally we discuss the effects of fluctuations on our mean-field solutions using the scheme of Höller and Read [Phys. Rev. E 101, 042114 (2020)}] and argue that such fluctuations will remove both the continuous 1RSB transition and discontinuous 1RSB transitions when $8>d\ge 6$ leaving just the FRSB continuous transition. We suggest values for $M$ and $p$ which might be used in simulations to confirm whether fluctuation corrections do indeed remove the 1RSB transitions.

Two new figures; 21 pages, 10 figures

References in corpus (7)

• Supercooled Liquids for Pedestrians
• Finite size corrections in the Sherrington-Kirkpatrick model
• Spin Glass in a Field: a New Zero-Temperature Fixed Point in Finite Dimensions
• Origin of the Growing Length Scale in M-p-Spin Glass Models
• Interface free energies in p-spin glass models
• Study of the de Almeida-Thouless (AT) line in the one-dimensional diluted power-law XY spin glass
• Continuous replica-symmetry breaking in mean-field spin-glass models: Perturbation expansion without the replica trick