Scaling and Renormalization Group in Replica-Symmetry-Breaking Space: Evidence for a Simple Analytical Solution of the Sherrington-Kirkpatrick Model at Zero Temperature

Abstract

Using numerical self-consistent solutions of a sequence of finite replica symmetry breakings (RSB) and Wilson's renormalization group but with the number of RSB steps playing a role of decimation scales, we report evidence for a nontrivial T-->0 limit of the Parisi order function q(x) for the Sherrington-Kirkpatrick spin glass. Supported by scaling in RSB space, the fixed point order function is conjectured to be q*(a)=sqrt[pi]/2 a/xi erf(xi/a) on 0<or=a<or=infinity, where x/T --> a at T =0 and xi approximately 1.13+/-0.01. Xi plays the role of a correlation length in a-space. q*(a) may be viewed as the solution of an effective 1D field theory.

Time dependent local field distribution and metastable states in the SK-spin-glass

“…Recently Oppermann and coworkers [24] have investigated the replica symmetry breaking solution for T = 0 in high order. They find = −0.763 · · · and again a linear behavior of the local field distribution with slope 0.3.…”

“…The local field distribution obtained from the Edwards measure [16,19,20] is discussed in Section 3 and Appendix A. The local field distribution has also been computed in high orders of replica symmetry breaking by Oppermann et.al [24]. This is discussed for comparison in Section 4.…”

“…For the above bias = −0.76, κ = 1.35 and P (0 + ) = 0.175. The bias is choosen such that the energy is close to the ground state energy obtained in replica theory [24].…”

Local field distributions in spin glasses

“…Through accurate studies of a large sequence of finite-replica-symmetry breakings Refs. [9] and [10] have demonstrated that within replica theory the correct linear behaviour of P (h) for the SK model requires the full limit of infinite replicasymmetry breaking order; any finite-order truncation yielding a fictitious (if decreasing with RSB-order) hole in P (h) near h = 0.…”

“…Since then there have been several further studies of P (h) for the SK model (see, for example, Refs. [7], [8], [9], and [10]) and the nature of the local field distributions is well understood. On the other hand, there has been little work on the study of P (h) for other spin-glass models.…”