Phase diagram for the Bethe lattice spin glass (abstract)

Abstract

We present the results of rigorous calculations for the ±J Ising spin-glass model on the Bethe lattice. The phase diagram for varying temperature and fraction of ferromagnetic bonds is derived near the paramagnetic phase boundary. In addition to the spin-glass and paramagnetic phases, we find a nontrivial ferromagnetic phase and a magnetized spin-glass phase, characterized by diverging Edwards–Anderson susceptibility. The recursion relation for the distribution of single-site magnetizations is studied as a dynamical system on an appropriate function space, the bulk thermodynamics is described by the attractors of the recursion relation, and the phase transitions correspond to bifurcations in the dynamics. Using bifurcation theory, we establish the existence of a stable distribution of single-site magnetizations near the paramagnetic phase boundary. At least in single-site properties, the existence proof precludes chaos, and infinite hierarchy of transitions, and other conceivable bizarre possibilities. While our phase diagram is very similar to the phase diagram for the Sherrington–Kirkpatrick model, the Bethe lattice provides a useful description of the mean-field behavior of spin glasses because the interactions are short range, the analysis is much more straightforward, and the results have been made completely rigorous.