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Equations of state and stability condition of mixed -spin glass model
Phys. Rev. E 113, 054133 – Published 20 May, 2026
DOI: https://doi.org/10.1103/fcsl-93pd
Abstract
The Sherrington-Kirkpatrick (SK) model is a foundational model for understanding spin glass systems. It is based on the pairwise interaction between each two spins in a fully connected lattice with quenched disordered interactions. The SK model, as an infinite-range model, simplifies the study of this system by eliminating fluctuations. An advanced SK model, known as the -spin model, introduces higher-order interactions that involve the interaction of P spins. This study focuses on the general Hamiltonian of the spin glass model with infinite-range interactions, referred to as the mixed -spin glass model, which consists of adding all -spin interaction terms. This study aims to derive the equations of state for this Hamiltonian, formulate the equations of state within the framework of the first replica symmetry breaking, and determine both the stability condition of the replica symmetric solution and the stability of the replicas belonging to the same group in the first step of replica symmetry breaking. Additionally, this study considers two special cases. The first case is a mixed -spin glass model in which all interactions are normally distributed around zero. The second case focuses on the SK model compound with perturbative higher-order interactions to investigate how these interactions influence the phase transition.
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