Slow dynamics and magnon bound states in the 2D long-range quantum Ising model

Abstract

The dynamics of long-range quantum Ising models represents a current frontier in experimental physics, notably in trapped ions or Rydberg atomic systems. However, a theoretical description of these dynamics beyond 1D remains a significant challenge for conventional methods. Here, we address this challenge by means of neural quantum states to simulate global quenches from the fully polarized ferromagnetic state in the 2D quantum Ising model with power-law decaying interactions. From these numerically exact simulations, we find that the dynamics exhibit slow relaxation with long-lived oscillations. We explain this behavior through a theory for the formation of magnon bound states, which are generated, as we show, through effective attractive interactions between magnons that persist over several lattice sites due to the power-law nature of the interactions. Our results are readily observable in current quantum simulation platforms realizing long-range interacting models such as in Rydberg atomic systems.

Figures (5)

• Figure 1: Slow dynamics and magnon binding. (A) Spatiotemporal normalized connected correlator $\tilde{C}(d,t)$ after a quench from $|\!\downarrow\cdots\downarrow\rangle$ on an $11\times11$ lattice ($\alpha=3$, $g/J=0.2$), showing slow dynamics and oscillations. (B) Schematic illustration of two magnons at separation $d$. The lower panel sketches the effective attractive potential $U(d)$ for $\alpha=3$ from Eq. \ref{['eq:U_d']}. Inset: the corresponding two-magnon bound-state probability peaked at $\tilde{r}=\sqrt{5}$ on an $101\times101$ lattice.
• Figure 2: Fourier spectroscopy for the post-quench dynamics ($9\times9$, $\alpha = 3$ and $g/J = 0.5$). (A) Spatiotemporal connected correlator $\tilde{C}(d,t)$, together with the site-averaged longitudinal magnetization $\langle \hat{S}_i^z(t)\rangle$. (B) FFTs $\mathcal{F}[\,C(d,t)\,]$ for $d=1,2,3$, and $\mathcal{F}[\,\langle\hat{S}_i^z(t)\rangle\,]$, plotted versus frequency $\omega/J$. Vertical dashed lines mark energy gaps $\Delta^{(\nu\leftrightarrow \nu')}_{i,j}=E^\nu_i-E^{\nu'}_j$ from effective Hamiltonians restricted to the $\nu$ magnon sectors. The matching between spectral peaks and energy gaps quantitatively validates the few-body effective description. Inset: Fourier spectra of exact dynamics for a $5\times 5$ system (evolved up to $Jt = 200$).
• Figure 3: Eigenstate structure in the two-magnon sector ($101 \times 101$, $g/J=0.2$). Inverse participation ratio (IPR) versus mean pair separation $\bar{d}$ for all eigenstates. (A) $\alpha=2$: a continuous tail to large $\bar{d}$ indicates bound states and quasilocalized states. (B) $\alpha=3$: intermediate regime with bound states at intermediate separations. (C) $\alpha=6$: sharp drop in IPR with $\bar{d}$ signals short-range binding only. Insets: real-space distributions of representative eigenstates in the $(\mathbf{r}_1-\mathbf{r}_2)$ plane.
• Figure 4: $\tilde{C}(d,t)$ for a $9\times9$ lattice with $g/J=0.2$. The panels compare interaction exponents $\alpha = 2,3$ with $\alpha = 6$ and the nearest-neighbor limit $\alpha = \infty$.
• Figure S1: Single-magnon dispersion for different power-law exponents $\alpha$. Left: dispersion $E_1(k_x,k_y{=}0)-E_0$ along the $k_x$ direction. Right: dispersion along the high-symmetry path $X\to M\to\Gamma\to X\to S$. Long-range interactions induce a nonanalytic cusp at $\Gamma$, which gradually disappears as $\alpha$ becomes large.