Statistical mechanics of the spherical hierarchical model with random fields

Fernando L. Metz (1, 2) , Jacopo Rocchi (2) , Pierfrancesco Urbani (3)

1 UFRGS - Universidade Federal do Rio Grande do Sul [Porto Alegre]
2 Dipartimento di Fisica [Roma La Sapienza]
3 IPHT - Institut de Physique Théorique - UMR CNRS 3681

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.

Domaines: Mécanique statistique [cond-mat.stat-mech]

Fernando L. Metz, Jacopo Rocchi, Pierfrancesco Urbani. Statistical mechanics of the spherical hierarchical model with random fields. Journal of Statistical Mechanics: Theory and Experiment, 2014, 2014, pp.09018. ⟨10.1088/1742-5468/2014/09/P09018⟩. ⟨cea-01463186⟩