Nature of perturbation theory in spin glasses

Abstract

The high-order behavior of the perturbation expansion in the cubic replica field theory of spin glasses in the paramagnetic phase has been investigated. The study starts with the zero-dimensional version of the replica field theory and this is shown to be equivalent to the problem of finding finite size corrections in a modified spherical spin glass near the critical temperature. We find that the highorder behavior of the perturbation series is described, to leading order, by coefficients of alternating signs (suggesting that the cubic field theory is well-defined) but that there are also subdominant terms with a complicated dependence of their sign on the order. Our results are then extended to the d-dimensional field theory and in particular used to determine the high-order behavior of the terms in the expansion of the critical exponents in a power series in ǫ = 6 − d. We have also corrected errors in the existing ǫ expansions at third order.

Nontrivial Critical Fixed Point for Replica-Symmetry-Breaking Transitions

“…As is standard [53] and proven in some case [52], we assume that the saddle-point solution governing the large-order behavior takes the separable and spherically-symmetric form…”

“…Consequently, as has been observed for the Abelian gauge theory with background fields [50], Borel-summability depends on the ratio of two couplings, as encoded in the saddle-point solution to the classical equations of motion for replicons [51]. Among nontrivial saddles, we assume [52,53] that the saddle of the form…”

Finite-size critical scaling in Ising spin glasses in the mean-field regime

“…However, an approach equivalent to this was used by three of us in Ref. 45 and it results in studying the finite-size scaling function for the spherical SK spin-glass model, which according to the arguments in the aforementioned reference should have an identical scaling function f (x). However, this approach is hard to extend to the behavior in a field, so instead we present an approach which does permit, in principle, an extension to finite fields.…”

An exact relation between free energy fluctuations and bond chaos in the Sherrington–Kirkpatrick model

“…The variable x = τ N 1/3 is the correct scaling combination in the critical region [18,19]. Keeping x fixed and letting N tend to infinity in − 1 2 log(xN −1/3 ) results in Eq.…”