Spin Glass on a Bethe Lattice: Order Parameter and Critical Susceptibility

Abstract

Since de Almeida and Thouless (AT) /1/ have shown the existence of the phase transition in the symmetric Sherrington-Kirkpatrick (SK) model 121 at finite external field H manifesting itself by the divergence at H = HAT(T) of the spin glass susceptibility where <. . . > T denotes a Gibbs average and <. . . > J is an average over random exchange interactions J.. , it has become clear that the Edwards-Anderson (EA) parameter 13 I 11 could not be the order parameter of this transition because q f 0 at H f 0 for any temperature T. The search for the adequate order parameter of the AT transition has lead to the replica symmetry breaking scheme by Parisi / 4 / in which the spin glass phase at H < HAT is described by the order parameter function q ( x ) , 0 < x < 1. The physical interpretation of q ( x ) in terms of the probability distribution function for the overlaps of the magnetizations in various "pure states" / 5 / leaves still open the question: why just x i G (1) diverges at the AT line? The clear physical arguments on this point are quite necessary as the mathematical rigorousness of the results obtained in 11 to 5 / using the so-called "replica trick" (the continuation from the integer replicas number n to the real n + 0) is rather doubtful. Also the attempts to consider the S K model without resorting to the "replica trick" 16 to 8 / have not elucidated the problem of the order parameter of the AT transition and the cause of the x '~~ divergence. The same is true f o r the results of the Bethe spin glass treatments / 9 to 151 in which the existence of the AT transition in finite field has been established using only two replicas / 9 , 13/. Thus the order parameter definitions considered in /14, 15/ rely essentially on the specific features of the Bethe lattice geometry which does ) Prospekt Stachki 194, 344104 Rostov-on-Don, Russia.