Statistical mechanics of the spherical hierarchical model with random fields

TL;DR

This paper analytically investigates the equilibrium properties and phase transitions of the spherical hierarchical model with random fields, confirming the mapping to short-range models and ruling out a spin-glass phase.

Contribution

It provides exact analytical expressions for the critical line and exponents, and demonstrates the validity of the dimensional reduction and the absence of a spin-glass phase in this model.

Findings

Derived the critical line separating paramagnetic and ferromagnetic phases.

Computed critical exponents and confirmed the mapping to D-dimensional short-range models.

Established the absence of a spin-glass phase through stability and free-energy analysis.

Abstract

We study analytically the equilibrium properties of the spherical hierarchical model in the presence of random fields. The expression for the critical line separating a paramagnetic from a ferromagnetic phase is derived. The critical exponents characterising this phase transition are computed analytically and compared with those of the corresponding $D$-dimensional short-range model, leading to conclude that the usual mapping between one dimensional long-range models and $D$-dimensional short-range models holds exactly for this system, in contrast to models with Ising spins. Moreover, the critical exponents of the pure model and those of the random field model satisfy a relationship that mimics the dimensional reduction rule. The absence of a spin-glass phase is strongly supported by the local stability analysis of the replica symmetric saddle-point as well as by an independent computation of the free-energy using a renormalization-like approach. This latter result enlarges the class of random field models for which the spin-glass phase has been recently ruled out.