Renormalization-group computation of the critical exponents of hierarchical spin glasses: large-scale behavior and divergence of the correlation length.

In a recent work [M. Castellana and G. Parisi, Phys. Rev. E 82, 040105(R) (2010)], the large-scale behavior of the simplest non-mean-field spin-glass system has been analyzed, and the critical exponent related to the divergence of the correlation length has been computed at two loops within the $\ensuremath{\epsilon}$-expansion technique by two independent methods. By performing the explicit calculation of the critical exponents at two loops, one obtains that the two methods yield the same result. This shows that the underlying renormalization group ideas apply consistently in this disordered model, in such a way that an $\ensuremath{\epsilon}$-expansion can be set up. The question of the extension to high orders of this $\ensuremath{\epsilon}$-expansion is particularly interesting from the physical point of view. Indeed, once high orders of the series in $\ensuremath{\epsilon}$ for the critical exponents are known, one could check the convergence properties of the series, and find out if the ordinary series resummation techniques, yielding very accurate predictions for the Ising model, work also for this model. If this is the case, a consistent and predictive non-mean-field theory for such a disordered system could be established. In that regard, in this work we expose the underlying techniques of such a two-loop computation. We show with an explicit example that such a computation could be quite easily automatized, i.e., performed by a computer program, in order to compute high orders of the $\ensuremath{\epsilon}$-expansion, and so eventually make this theory physically predictive. Moreover, all the underlying renormalization group ideas implemented in such a computation are widely discussed and exposed.