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Free-Energy Bounds for Hierarchical Spin Models

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Abstract

In this paper we study two non-mean-field (NMF) 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 NMF 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 NMF 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.

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Acknowledgments

M. Castellana is grateful to S. Franz for useful discussions, to NSF for funding through Grants PHY-0957573 and CCF-0939370, to the Human Frontiers Science Program, to the Swartz Foundation, and to the W. M. Keck Foundation for financial support. A. Barra is grateful to MIUR for funding trough the grant FIRB RBFR08EKEV, and to Sapienza Università di Roma and to GNFM-INdAM for partial financial support. F. Guerra is grateful to Sapienza Università di Roma and to INFN for partial financial support.

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Authors and Affiliations

  1. Lewis-Sigler Institute for Integrative Genomics, Princeton University, Princeton, NJ, 08544, USA

    Michele Castellana

  2. Dipartimento di Fisica, Sapienza Università di Roma, Rome, Italy

    Adriano Barra & Francesco Guerra

  3. INFN Sezione di Roma, Rome, Italy

    Francesco Guerra

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  1. Michele Castellana
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  2. Adriano Barra
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  3. Francesco Guerra
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Correspondence to Michele Castellana.

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Castellana, M., Barra, A. & Guerra, F. Free-Energy Bounds for Hierarchical Spin Models. J Stat Phys 155, 211–222 (2014). https://doi.org/10.1007/s10955-014-0951-9

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