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arXiv · 2504.08925

Universality of $SU(\infty)$ relaxation dynamics for $SU(n_f)$-symmetric spin-models

Duff Neill·Hanqing Liu
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Abstract

Spin-models, where the $N$ spins interact pairwise with a $SU(n_f)$ symmetry preserving hamiltonian, famously simplify in the large $n_f$, $N$ limits, as derived by Sachdev and Ye when exploring mean-field behavior of spin-glasses. We present numerical evidence that for a large class of models, the large $n_f$ limit is not necessary: the same dynamical equations can describe the relaxation processes at high temperatures for a set of classical models inspired from mean-field treatments of interacting dense neutrino gases, up to times set by the radius of convergence of the perturbation series for the correlation function. After a simple rescaling of time, the dynamics display a surprising universality, being identical for any value of $n_f$ as long as the rank of the coupling matrix is small. As a corollary of our results, we find that the direct interaction approximation originating from the study of stochastic flows in fluid turbulence should be thought of as only a short-time approximation for generic random coupling systems.

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Duff Neill, Hanqing Liu. 2025-04-11. Universality of $SU(\infty)$ relaxation dynamics for $SU(n_f)$-symmetric spin-models. https://arxiv.org/abs/2504.08925

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