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arXiv:2512.14868

Large-$n$ $O(n)$ with long-range interactions: integrability and resonance dynamics

Guido Giachetti, Nicolo Defenu

cond-mat.stat-mechquant-phSubmitted Dec 16, 2025• Announced Dec 18, 2025• 0 citations
v1 · Dec 16, 2025v2 · Jul 2, 2026
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Abstract

We study the the large-$n$ dynamics of the long-range quantum $O(n)$ model, focusing on the strong long-range regime $α<d$. The dynamics of the model exhibits non-trivial features on mesoscopic timescales $t\sim\ln N$, due to the activation of parametric resonances of the nearly degenerate quantum modes. By using recent results establishing the integrability of the large-$n$ limit, we derive the resonance conditions, and construct the reduced multi-mode Hamiltonian that captures the finite-size dynamics. This framework yields the resonance phase diagram and clarifies when and how deviations from mean-field behavior arise. In particular, the presence of multiple resonant modes enhances the logarithmic growth of entanglement and leads to spatially modulated correlations.

Comments
12 pages, 5 figures

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Understanding quantum many-body systems with long-range or infinite-range interactions is of relevance across a broad set of physical disciplines, including quantum optics, nuclear magnetic resonance, and nuclear physics. From a theoretical viewpoint, these systems are appealing since they can be efficiently studied with numerics, and in the thermodynamic limit are expected to be governed by mean-field equations of motion. Over the past years the capabilities to experimentally create long-range interacting systems have dramatically improved permitting their control in space and time. This allows us to induce and explore a plethora of nonequilibrium dynamical phases, including time crystals and even chaotic regimes. However, establishing the emergence of these phases from numerical simulations turns out to be surprisingly challenging. This difficulty led to the assertion that mean-field theory may not be applicable to time-dependent infinite-range interacting systems. Here, we rigorously prove that mean-field theory in fact exactly captures their dynamics, in the thermodynamic limit. We further provide bounds for finite-size effects and their dependence on the evolution time.