2013/12/31 by Michele Castellana, Adriano Barra, Francesco Guerra · 1 citation
Mathematics · Physics and Astronomy · #Advanced Condensed Matter Physics #Ferromagnetism #Hamiltonian (control theory) #Ising model #Limit (mathematics) #Quantum many-body systems #Spin (aerodynamics) #Spin glass #Spins #Theoretical and Computational Physics #Thermodynamic limit #cond-mat.dis-nn #math-ph #math.MP
paper · pdf · doi:10.1007/s10955-014-0951-9
published as Journal of Statistical Physics 155(2), 211 (2014)
openalex publication_date 2014/02/24 · arxiv created 2014/03/17 · arxiv updated 2014/09/09 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
In this paper we study two non-mean-field spin models built on a hierarchical lattice: The hierarchical Edward-Anderson model (HEA) of a spin glass, and Dyson's hierarchical model (DHM) of a ferromagnet. For the HEA, we prove the existence of the thermodynamic limit of the free energy and the replica-symmetry-breaking (RSB) free-energy bounds previously derived for the Sherrington-Kirkpatrick model of a spin glass. These RSB mean-field bounds are exact only if the order-parameter fluctuations (OPF) vanish: Given that such fluctuations are not negligible in non-mean-field models, we develop a novel strategy to tackle part of OPF in hierarchical models. The method is based on absorbing part of OPF of a block of spins into an effective Hamiltonian of the underlying spin blocks. We illustrate this method for DHM and show that, compared to the mean-field bound for the free energy, it provides a tighter non-mean-field bound, with a critical temperature closer to the exact one. To extend this method to the HEA model, a suitable generalization of Griffith's correlation inequalities for Ising ferromagnets is needed: Since correlation inequalities for spin glasses are still an open topic, we leave the extension of this method to hierarchical spin glasses as a future perspective.