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

TL;DR

This work investigates the far-from-equilibrium dynamics of the quantum $O(n)$ model with strong long-range interactions ($α<d$) in the large-$n$ limit. By exploiting integrability, it identifies resonance conditions among nearly degenerate quantum modes and constructs a reduced multi-mode Hamiltonian that captures finite-size dynamics on mesoscopic timescales $t_{Ehr}\sim\ln N$. The analysis yields a resonance phase diagram and shows how the presence of multiple resonances enhances entanglement spreading and induces spatially modulated correlations, signaling a departure from mean-field behavior. The results provide a unified, semi-analytic framework for understanding how long-range interactions shape dynamical criticality, entanglement production, and the breakdown of simple mean-field pictures in large-$n$ quantum systems.

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.

Figures (5)

• Figure 1: Structure of the $p_\nu$, $\eta_\nu$, $|\bar{\eta}| = \xi$ section of the phase space in the linear regime $p_\nu,\eta_\nu = O(1)$ in the off-resonant $(a)$ and resonant case $(b)$ respectively. The presence of the integrals of motion constraints the trajectories $(p_\nu, \eta_\nu, |\bar{\eta}|)$ (blue) on the manifold $\epsilon_\nu = \rm const$ (yellow), while the trajectory of the $(p_{|\bar{\eta}|}, |\bar{\eta}|)$ is shown in the inset. While in the off-resonant case the $|\bar{\eta}| = \rm const$ sections of the manifolds are ellipses, preventing the indefinite growth of the $\eta_\nu$, $p_\nu$; if a resonance occurs $\eta_\nu$, $p_\nu$ are no longer bounded as these sections are hyperbolas.
• Figure 2: Numerical estimate (diamonds) vs the theoretical estimate (Eq. \ref{['eq:FloEig']}) of the Floquet eigenvalues $\lambda^F_{\nu}$ corresponding to the Fourier modes $\nu = 0, \dots, 3$ in the off-resonant (left) and resonant (right) regimes respectively, for a ground state quench ($m_{\rm gs} = 6$) as a function of the final bare mass $r$. The vertical dotted lines represent the theoretical estimate of the boundaries $r^*_\nu$ of the resonant regions (see Eq. \ref{['eq:r*nu']}). For the off-resonant regime ($r > r^*_\nu$, left) the Floquet frequencies are $\Omega^F_\nu = - \text{i} \ \lambda^F_{\nu}$ are shown.
• Figure 3: Phase diagram of the number of resonances $\mathcal{N}$, for a ground state quench $r_- \rightarrow r$ from the massive phase as a function (left) of the ground-state mass $m_{\rm gs}$ and of the post-quench bare mass $r$ for $\lambda = 1$, $\alpha = 0.5$; (right) of the range $\alpha$ of the interaction and of the post-quench bare mass $r$ for $\lambda = 1$, $m_{\rm gs} = 8$. The unbroken lines separating the boundaries corresponds to the values of $r_\nu^{*}$ in Eq. \ref{['eq:r*nu']}. We see that the boundary $r^{*}_0$ of the off-resonant region is always below the thermal critical value $r_c^{\rm th}$ (dotted blue line), with $r^{*} \rightarrow r_c^{\rm th}$ as $m_{\rm gs} \rightarrow 0$. Further decreasing $r$ more and more resonances are triggered, with the system becoming less and less stable as $m_{\rm gs} \rightarrow 0$ and $\alpha \rightarrow 1$.
• Figure 4: Sketch of the dynamics of the Jacobi quasiparticles of the long-range $O(n)$ model in the case $\mathcal{N} = 2$. Here the dotted lines represents the dispersion $\omega_\nu^2$ of the modes, the activated ones, $\nu = 0,1$, with $\Delta_\nu < 0$ being in yellow. Each resonant mode corresponds to a Jacobi quasiparticle, which oscillates within the corresponding accessible region, while the last quasiparticle ($u_\mathcal{N} > \omega_\mathcal{N}^2 \equiv 1$) corresponds instead to the classical degree of freedom. The poles corresponding to the off-resonant modes, absent from Eq. \ref{['eq:V(u)']}, are here shown to clarify their irrelevance in the thermodynamic limit.
• Figure 5: Spectral density of $\mathcal{A}(t) = m^2(t) - \overline{m^2}$ ($\overline{\cdot}$ representing the time average), for the single-resonance ($\mathcal{N}=1$) phase. The location of the peaks is compared with the expected quasi-periodic comb-structure $p_0 \ \Omega_0 + p_1 \ \Omega_1$, $p_0, p_1 \in \mathbb{Z}$ (dashed vertical lines), finding an excellent agreement. Here $\Omega_{1}$, corresponding to the most prominent peak, coincides with the frequency of the classical motion in Eq. \ref{['eq:mathcaHetab']}, while $\Omega_0 \sim (\ln N)^{-1}$ is the resonant frequency in Eq. \ref{['eq:Omeganu']}. Peaks close to the different harmonics $p_1 \Omega_1$ of the classical frequency are highlighted in different colors. We considered a quench from $m_{\rm gs} = 6$ to $r=-2$ with $\lambda = 1$, $N = 10^{14}$. Due to the logarithmic scaling in $\Omega_0$, such a large value of $N$ is needed both to have a clear scale separation and to get rid of finite-size effects.