Critical Behavior of the Bethe Lattice Spin Glass

J. M Carlson, J. T Chayes, L Chayes, J. P Sethna, D. J Thouless
1988 Europhysics letters  
We present results of a rigorous analysis of the k J king spin glass on the Bethe lattice with uncorrelated boundary conditions. We derive phase diagrams as functions of temperature 'us. percentage of ferromagnetic bonds, and, when half of the bonds are ferromagnetic, temperature 'us. external field. Critical exponents are also determined. Using bifurcation theory, we establish the existence of nontrivial distributions of single-site magnetizations within the low-temperaturehmall-field phase
more » ... mall-field phase boundaries; these solutions reflect spin glass, ferromagnetic and magnetized spin glass behavior. We have studied the 4 J Ising spin glass model on the Bethe lattice with varying fractions of ferromagnetic bonds [l-31. Here we present concise arguments for results which have been proven rigorously elsewhere [2, 31. Recursion relations for the distribution of single-site magnetizations are derived and studied as a dynamical system on an appropriate function space. Bulk properties are described by attractors of the recursion relations, and the phase transitions correspond to bifurcations in the dynamics. Using these methods, we derive the phase diagram as a function of temperature and fraction of ferromagnetic bonds, determine critical exponents at the phase boundaries, and prove the existence of a stable spin glass solution near the high-temperature paramagnetic phase boundary. At least in terms of single-site properties, the existence proof precludes chaos, an infinite hierarchy of transitions, and other bizarre possibilities. We also prove that (when half of the bonds are ferromagnetic) the spin glass transition persists in the presence of small external magnetic fields, and derive the asymptotics of the de Almeida-Thouless (AT) line[41. The Bethe lattice provides a useful alternative to the more commonly studied infiniterange Sherrington-Kirkpatrick (SK) model of spin glasses [5] ; here the interactions are short-range, the analysis is straightforward and the results have been made completely
doi:10.1209/0295-5075/5/4/013 fatcat:reegiohxcnb25lewpvck54glgu