Ward identities and local potential approximation for large time quantum (2+p)-spin glass dynamics

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.