We have studied in detail the $M\text{−}p$ 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 nine quintic-order coefficients, and five 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.

• Received 27 August 2023
• Revised 1 November 2023
• Accepted 7 November 2023

DOI:https://doi.org/10.1103/PhysRevE.108.054134