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

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