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

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 $β$-functions. Our explicit calculations focus on the cases $p=3$, $p=\infty$, and additional analytic material is given in the appendix.

💡 Research Summary

This paper investigates the quantum (2 + p)‑spin glass model by exploiting the large‑N limit where the disorder matrix K follows Wigner’s semicircle law. The authors treat the eigenvalues of K as effective momenta, defining a generalized kinetic term whose spectrum is deterministic as N → ∞. By coarse‑graining only over these generalized momenta (and not over frequencies), they obtain a non‑local in time effective action that retains replica indices.

The disorder tensor J, which carries the p‑spin interaction, is averaged using the replica method. The replicated partition function Zⁿ is constructed, and the limit n → 0 yields the disorder‑averaged free energy. Within this framework, the functional renormalization group (FRG) is implemented via the Wetterich equation, with a momentum‑cutoff Rₖ that acts on the generalized momenta p. The resulting flow equation respects the replica Ward identities derived in Section 4, ensuring that non‑local time structures are consistently treated.

A detailed scaling analysis shows that the density of generalized momenta behaves as ρ(p²) ∝ p / (4σ − p²)², which differs from the usual d‑dimensional power law. Consequently, an effective dimensionality d_eff(p) emerges, interpolating between three‑dimensional behavior for small p and a highly non‑trivial scaling near the edge of the Wigner spectrum.

At one‑loop order the authors compute β‑functions for the two relevant couplings: the cubic (φ³) coupling g₃ that appears for p = 3 and the quartic (φ⁴) coupling g₄ that dominates in the p → ∞ limit. The β‑functions have the generic form
β₃ = −ε g₃ + A₃ g₃³ + B₃ g₃ g₄, β₄ = −ε g₄ + A₄ g₄³,
with ε = 4 − d_eff(p). For p = 3 a non‑local φ⁶ interaction is generated, leading to a richer flow structure and the possibility of a first‑order transition when the flow runs away. In the p → ∞ limit the flow reduces to the familiar Gaussian fixed point, confirming that the model becomes effectively free. Numerical integration of the flow equations reveals an infrared stable regime for certain parameter ranges and a runaway regime signalling a phase transition for others.

Beyond perturbation theory, the paper develops a vertex‑expansion (non‑perturbative) scheme. By expanding the effective average action in powers of the fields and retaining derivative terms, the authors obtain improved flow equations that capture the influence of the non‑local time structure. In the large‑p limit the derivative contributions vanish, and the approximation collapses to the local potential approximation, providing a consistency check.

Section 8 introduces a large‑N closed equation for the self‑energy using the two‑particle‑irreducible (2PI) formalism combined with a 1/N expansion. The resulting self‑consistent equation links the self‑energy Σ(p) and the four‑point vertex Λ, showing that Σ can become large in the deep infrared, effectively renormalizing the mass and hinting at a first‑order transition. In the p → ∞ limit the 2PI equation simplifies, confirming the Gaussian fixed point found in the perturbative analysis.

Overall, the work presents a coherent RG framework that respects the peculiar non‑local temporal and replica structures inherent to quantum spin glasses with matrix‑type disorder. By treating the eigenvalue distribution of K as a deterministic kinetic background, the authors bypass many technical obstacles of traditional non‑perturbative approaches. The derived one‑loop β‑functions, the vertex‑expansion results, and the 2PI large‑N analysis together provide a comprehensive picture of the renormalization flow, the possible phases, and the nature of phase transitions in quantum (2 + p)‑spin glasses. The paper also outlines future directions, including higher‑order loop calculations, numerical FRG studies for intermediate p, and potential applications to quantum annealing experiments where disorder and quantum fluctuations compete.