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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 $p=3$., Comment: 74 pages, 32 figures

In this paper, we consider a renormalization group perspective on the quantum dynamics of a particle moving in the Euclidean $\mathbb{R}^N$ space through the complex landscape provided by a disordered Hamiltonian of type $2+p$. We focus on the large $N$ limit, where the coarse-graining procedure is unconventional: it is based on the Wigner spectrum of the rank-2 disorder. The main consequence of this choice is that canonical dimensions depend on the scale, and the flow equations fail to become autonomous, preventing the existence of global fixed points. One of the main features of the underlying renormalization group flow is the existence of finite-scale singularities for initial conditions sufficiently close to the Gaussian region and for rank-$p$ disorder intensity large enough. Using the Luttinger-Ward formalism, we show that these finite-scale singularities hide (and should be resolved by) a phase transition that breaks time-translation invariance., Comment: 15 pages, 11 figures

This paper aims to study the functional renormalization group for quantum $(2+p)$-spin dynamics of a $N$-vector $\textbf{x}\in \mathbb{R}^N$. By fixing the gauge symmetry in the construction of the FRG, that breaks the $O(N)$-symmetry and deriving the corresponding non-trivial Ward identity we can: In the first time coarse grain and focus on this study using a more attractive method such as the effective vertex expansion, and in the second time explore this model beyond the symmetry phase. We show finite scale singularities due to the disorder, interpreted as the signal in the perturbation theory. The unconventional renormalization group approach is based on coarse-graining over the eigenvalues of matrix-like disorder, viewed as an effective kinetic term, with an eigenvalue distribution following a deterministic law in the large $N$ limit. As an illustration, the case where $p=3$ is scrutinized., Comment: 65 pages, 19 figures. A few typing errors have been corrected, as well as spelling mistakes

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., Comment: 32 pages, 33 figures