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arXiv:2411.11089

Large time effective kinetics $β$-functions for quantum (2 + p)-spin glass II: Effective vertex expansion, local potential approximation and symmetries

Bêm-Biéri Barthélémy Natta, Vincent Lahoche, Dine Ousmane Samary, Parham Radpay

Published Nov 17, 2024• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2411.11089

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Large time effective kinetics βββ-functions for quantum (2 + p)-spin glass II: Effective vertex expansion, local potential approximation and symmetries
This paper aims to study the functional renormalization group for quantum (2+p)(2+p)(2+p)-spin dynamics of a NNN-vector x∈RN\textbf{x}\in \mathbb{R}^Nx∈RN. By fixing the gauge symmetry in the construction of the FRG, that breaks the O(N)O(N)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 NNN limit. As an illustration, the case where p=3p=3p=3 is scrutinized.
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Abstract To gain a deeper understanding of the glassy phase in p-spin quantum models, this paper examines the dynamics of the N-vector x ∈ R N through the framework of renormalization group (RG) 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 β-function and explore the structure of its fixed points. Next, we develop the nonperturbative RG 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.
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A renormalization group transformation suitable for spin glasses, and more generally for disordered systems, is presented. The approach is non-standard for two reasons. 1. The coarse graining transformation is performed on the distribution of site overlaps, which clearly is not a Boltzmann-Gibbs distribution. 2. The additional interactions are realized introducing a gauge-field disorder distribution, while the Hamiltonian is kept fixed. Universality classes are thus naturally defined on a large set of models, going from ℤ2 and Gaussian spin glasses to Ising and fully frustrated models, and others. The role of frustration and of the deconfinement transition can also be analysed in terms of fixed points. The proposed analysis is tested numerically on the ℤ2 Edwards-Anderson model in d=4. Good estimates of the critical index ν and of Tc are obtained, and an RG flow diagram is sketched for the first time.