Large time effective kinetics $\beta$-functions for quantum (2+p)-spin glass

TL;DR

This paper develops a renormalization group approach to analyze the quantum $(2+p)$-spin glass model, deriving one-loop $eta$-functions for specific cases, advancing understanding of its effective kinetics.

Contribution

It introduces a novel RG framework for quantum $(2+p)$-spin glasses, deriving explicit one-loop $eta$-functions using perturbation theory and addressing non-local temporal interactions.

Findings

Derived one-loop $eta$-functions for $p=3$ and $p=\infty$ cases.

Formulated rules to handle non-localities in effective interactions.

Provided analytic results and detailed calculations in the appendix.

Abstract

This paper examines the quantum $(2+p)$-spin dynamics of a $N$-vector $\textbf{x}\in \mathbb{R}^N$ 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 $\beta$-functions. Our explicit calculations focus on the cases $p=3$, $p=\infty$, and additional analytic material is given in the appendix.