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

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.

💡 Research Summary

The authors investigate the dynamics of the quantum O(n) model with power‑law decaying interactions J(r) ∝ r^{‑α} in the strong long‑range regime (α < d, with d = 1). In the large‑n limit the model becomes exactly solvable: the collective mean‑field mode (\bar\eta(t)) obeys a classical Hamiltonian describing a particle in a quartic central potential, while all other Fourier modes experience a periodic drive generated by (\bar\eta(t)). For finite but large system size N, the equations for each mode reduce to a Floquet‑Hill problem. Using the recently proved integrability of the finite‑N system (Neumann‑Uhlenbeck integrals), the authors derive explicit expressions for the dressed single‑mode energies (\epsilon_\nu) and show that the stability of a mode is governed by the sign of the determinant (\Delta_\nu). When (\Delta_\nu<0) a parametric resonance occurs, leading to exponential growth of the corresponding mode on a mesoscopic timescale (t\sim\ln N).

The resonance condition translates into a set of boundaries (r^*\nu(\alpha,\lambda)) in the (r, α) plane, defining a resonance phase diagram. Because the dispersion (\omega\nu) collapses toward 1 as α decreases, many modes can become resonant simultaneously (multi‑resonant regime). In this case the authors construct a reduced multi‑mode Hamiltonian that includes a collective quartic interaction among the resonant modes. This Hamiltonian can be diagonalized exactly, yielding a spectrum of (|\mathcal{R}|) independent Floquet frequencies, where (\mathcal{R}) is the set of resonant modes.

Entanglement dynamics is dramatically affected: a single resonant mode produces a logarithmic growth (\mathcal{S}(t)\sim\frac12\ln t), whereas M resonant modes give (\mathcal{S}(t)\sim\frac{M}{2}\ln t). The enhanced growth is linked to the breaking of an effective permutation symmetry when multiple modes are amplified. Spatial correlations acquire a modulated structure, (\langle\Phi_j\Phi_{j’}\rangle\propto\sum_{\nu\in\mathcal{R}}|\eta_\nu(t)|^2 e^{ik_\nu (j-j’)}), reflecting the interference of the resonant wavevectors.

Overall, the paper provides a unified analytical framework that combines mean‑field dynamics, exact integrability, and Floquet theory to describe finite‑size effects in strongly long‑range quantum systems. The resonance phase diagram and the multi‑mode effective Hamiltonian offer new tools for understanding prethermalization, persistent oscillations, and anomalous entanglement spreading in a broad class of experimentally relevant platforms such as trapped ions and Rydberg atom arrays.