Statistical mechanics of the spherical hierarchical model with random fields

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.

arXiv comment23 pages, 2 figures

References in corpus (17)

• Relations between Short Range and Long Range Ising models
• Study of the de Almeida-Thouless line using power-law diluted one-dimensional Ising spin glasses
• Diluted one-dimensional spin glasses with power law decaying interactions
• Elusive Glassy Phase in the Random Field Ising Model
• The correspondence between long-range and short-range spin glasses
• The Hierarchical Random Energy Model
• Ensemble renormalization group for disordered systems
• Real space Renormalization Group analysis of a non-mean field spin-glass
• Overlap Interfaces in Hierarchical Spin-Glass models
• Random field Ising model: dimensional reduction or spin-glass phase?
• No spin glass phase in ferromagnetic random-field random-temperature scalar Ginzburg-Landau model
• Imry-Ma criterion for long-range random field Ising model: short-/long-range equivalence in a field
• Critical scaling in random-field systems: 2 or 3 independent exponents?
• Transition between localized and extended states in the hierarchical Anderson model
• Ensemble renormalization group for the random field hierarchical model
• Critical Exponents of the Random Field Hierarchical Model
• The renormalization flow of the hierarchical Anderson model at weak disorder