Nature of perturbation theory in spin glasses

The high-order behaviour 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 high-order behaviour 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 behaviour of the terms in the expansion of the critical exponents in a power series in ε = 6 - d. We have also corrected errors in the existing ε expansions at third order