Spin-Glass on a Bethe Lattice

The Ising spin-glass in a magnetic field is studied for the Bethe lattice. There is an instability that agrees with the replica-symmetry-breaking transition found for the infinite-range model. Correlation lengths are finite on both sides of the transition, but there is a correlation length that diverges at the transition. Some features are different from those of the infinite-range model, and in particular the magnetic susceptibility and internal energy vary smoothly through the transition. An analogy with the localization transition on the Bethe lattice is pointed out.

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