Fig. 3: Linear stability analysis.

a, b Share the same x-axis and show the real and imaginary parts of the first two eigenvalues λ₁ (triangles) and λ₂ (circles) of the Jacobian \({{\mathcal{M}}}\) with fixed J_x = 5.9, J_z = γ = 1. The nonzero conjugate pairs of the imaginary parts indicate oscillatory dynamics. This oscillation decays to a stationary spin-density-wave (SDW) phase when the real parts are negative, and stabilizes to a nonstationary oscillatory (OSC) phase when the real parts are positive. c, d Share the same y-axis and show the robustness of dynamics of \(\langle {\hat{\sigma }}^{y}\rangle \) for different initial states \(| {\psi }_{n}(t=0)\left.\right\rangle =(| \uparrow \left.\right\rangle +\sqrt{99}{e}^{i({r}_{n}+{c}_{n})\pi /k}| \downarrow \left.\right\rangle )/10\) with k = 3 (black solid line), 6 (blue short-dashed line), 9 (red dotted line), 12 (green dot-dashed line), 15 (magenta solid line). The zoom-in plot of (d) and the Fourier spectrum of the selected dynamical regions (Γt ∈ [400, 1000]) of (d) are respectively shown in (d1 and d2). The dashed lines in (a, b) highlight zero values.