The quantum $p$-spin renormalization group in the large $N$ limit as a benchmark for functional renormalization group

TL;DR

This paper applies both perturbative and nonperturbative renormalization group methods to the quantum p-spin model, providing a benchmark for functional RG approaches in understanding glassy phases and fixed points.

Contribution

It develops a comprehensive RG framework for the quantum p-spin model, including perturbative calculations and nonperturbative approximations, to analyze its fixed points and phase structure.

Findings

Computed the one-loop beta function for the model

Identified fixed points and analyzed their structure

Assessed the reliability of approximation schemes for p=2 and p=3

Abstract

To gain a deeper understanding of the glassy phase in $p$-spin quantum models, this paper examines the dynamics of the $N$-vector $\bm{x} \in \mathbb{R}^N$ through the framework of renormalization group theory. First, we focus on perturbation theory, which is more suitable than nonperturbative techniques due to the specific temporal non-locality of the model after disorder integration. We compute the one-loop $\beta$-function and explore the structure of its fixed points. Next, we develop the nonperturbative renormalization group approach based on the standard Wetterich-Morris formalism, using two approximation schemes to address the model's non-locality. We investigate the vertex expansion in the symmetric phase and assess the reliability of the approximations for the fixed-point solutions. Finally, we extend our analysis beyond the symmetric phase by using an expansion around the vacuum of the local potential. Our numerical investigations particularly focus on the cases $p = 2$ and $p=3$.