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

TL;DR

This paper investigates the large-$n$ quantum $O(n)$ model with long-range interactions, revealing how parametric resonances influence finite-size dynamics, entanglement growth, and correlations, especially in the strong long-range regime.

Contribution

It derives resonance conditions and constructs a reduced multi-mode Hamiltonian to explain finite-size effects and deviations from mean-field behavior in the large-$n$ limit.

Findings

Resonance phase diagram elucidates conditions for resonance activation.

Multiple resonant modes cause logarithmic entanglement growth.

Spatially modulated correlations emerge due to resonances.

Abstract

We study the the large-$n$ dynamics of the long-range quantum $O(n)$ model, focusing on the strong long-range regime $\alpha<d$. 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.