Equilibrium properties of disordered spin models with two-scale interactions

Jack Raymond, David Saad

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Methods for understanding classical disordered spin systems with interactions conforming to some idealized graphical structure are well developed. The equilibrium properties of the Sherrington-Kirkpatrick model, which has a densely connected structure, have become well understood. Many features generalize to sparse Erdös- Rényi graph structures above the percolation threshold and to Bethe lattices when appropriate boundary conditions apply. In this paper, we consider spin states subject to a combination of sparse strong interactions with weak dense interactions, which we term a composite model. The equilibrium properties are examined through the replica method, with exact analysis of the high-temperature paramagnetic, spin-glass, and ferromagnetic phases by perturbative schemes. We present results of replica symmetric variational approximations, where perturbative approaches fail at lower temperature. Results demonstrate re-entrant behaviors from spin glass to ferromagnetic phases as temperature is lowered, including transitions from replica symmetry broken to replica symmetric phases. The nature of high-temperature transitions is found to be sensitive to the connectivity profile in the sparse subgraph, with regular connectivity a discontinuous transition from the paramagnetic to ferromagnetic phases is apparent.
Original languageEnglish
Article number031138
Pages (from-to)031138
Number of pages1
JournalPhysical Review E
Issue number3
Publication statusPublished - 24 Sept 2009

Bibliographical note

© 2009 The American Physical Society.


  • classical disordered spin systems
  • equilibrium properties
  • Sherrington-Kirkpatrick model
  • sparse Erdös-Rényi graph structures
  • percolation threshold
  • Bethe lattices
  • sparse strong interactions
  • weak dense interactions


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