Large time effective kinetics β-functions for quantum (2+p)-spin glass

This paper examines the quantum $(2+p)$-spin dynamics of a N-vector x∈ RN through the lens of renormalization group (RG) theory. The RG is based on a coarse-graining over the eigenvalues of matrix-like disorder, viewed as an effective kinetic whose eigenvalue distribution undergoes a deterministic law in the large N limit. We focus our investigation on perturbation theory and vertex expansion for effective average action, which proves more amenable than standard nonperturbative approaches due to the distinct non-local temporal and replicative structures that emerge in the effective interactions following disorder integration. Our work entails the formulation of rules to address these non-localities within the framework of perturbation theory, culminating in the derivation of one-loop β-functions. Our explicit calculations focus on the cases $p=3$, p=∞, and additional analytic material is given in the appendix.