Universality class of replica symmetry breaking, scaling behavior, and the low-temperature fixed-point order function of the Sherrington-Kirkpatrick model

A scaling theory of replica symmetry breaking (RSB) in the Sherrington-Kirkpatrick (SK) model is presented in the framework of critical phenomena for the scaling regime of large RSB orders κ, small temperatures T, and small (homogeneous) magnetic fields H. We employ the pseudodynamical picture [R. Oppermann, M. J. Schmidt, and D. Sherrington, Phys. Rev. Lett. 98, 127201 (2007)], where two critical points CP1 and CP2 are associated with the order function's pseudodynamical limits lim_a→∞0.2em0exq(a)=1 and lim_a→00.2em0exq(a)=0 at $(T=0$, $H=0$, 1∕κ=0). CP1- and CP2-dominated contributions to the free energy functional $F[q(a)]$ require an unconventional scaling hypothesis. We determine the scaling contributions in accordance with detailed numerical self-consistent solutions for up to 200 orders of RSB. Power laws, scaling functions, and crossover lines are obtained. CP1-dominated behavior is found for the nonequilibrium susceptibility, which decays like χ₁=κ^-5∕3f₁(T∕κ^-5∕3), for the entropy, which obeys S(T=0)~χ₁², and for the subclass of diverging parameters aᵢ=κ5∕3f_aᵢ(T∕κ^-5∕3) [describing Parisi box sizes mᵢ(T)≡aᵢ(T)T], with f₁(ζ)~ζ and f_aᵢ(ζ)~1∕ζ for ζ→∞, while $f(0)$ is finite. CP2-dominated behavior, controlled by the magnetic field H while temperature is irrelevant, is retrieved in the plateau height (or width) of the order function $q(a)$ according to qₚₗ(H)=κ^-1fₚₗ(H2∕3∕κ^-1) with fₚₗ(ζ)_ζ→∞~ζ and fₚₗ(0) finite. Divergent characteristic RSB orders κCP1(T)~T^-3∕5 and κCP2(H)~H^-2∕3, respectively, describe the crossover from mean field SK- to RSB-critical behavior with rational-valued exponents extracted with high precision from our RSB data. The order function $q(a)$ is obtained as a fixed-point function q*(a*) of RSB flow, in agreement with integrated fixed-point energy and susceptibility distributions.