Operator dependence and robustness of spacetime-localized response in a quantum critical spin chain

Abstract

We investigate the phenomenon of spacetime-localized response in a quantum critical spin system, with particular attention to how it depends on the spatial profile and operator content of the applied perturbation, as well as its robustness against increase of amplitude and temporal discretization. Motivated by recent theoretical proposals linking such response patterns to the anti-de Sitter/conformal field theory correspondence, we numerically analyze the real-time dynamics of the one-dimensional transverse-field Ising model at criticality using the time-evolving block decimation algorithm. We find that sharply localized and periodically recurring responses emerge only for specific types of perturbations, namely those that correspond to local density fields in the continuum limit. In contrast, perturbations involving other spin components produce conventional propagating excitations without localization. Furthermore, we demonstrate that the response remains qualitatively robust when the time-dependent perturbation is approximated by a piecewise-linear function, highlighting the practical relevance of our findings for quantum simulation platforms with limited temporal resolution. Our results clarify the operator dependence of emergent bulk-like dynamics in critical spin chains and offer guidance for probing holographic physics in experimental settings.

Figures (9)

• Figure 1: (a) Schematic illustration of wavepacket propagation along null geodesics in AdS spacetime and its dual manifestation as a spacetime localized response on the CFT ring, which corresponds to the AdS boundary. (b) Representative spacetime points where the localized response appears. Here $\phi$ denotes the angular position on the CFT ring.
• Figure 2: Central positions of spatially localized perturbations: (a) single source centered at $\phi = 0$; (b) two sources centered at $\phi = 0$ and $\phi = -\pi/2$.
• Figure 3: Response profiles $|\delta\langle \sigma^x_j(t) \rangle|$ in the transverse-field Ising model on a one-dimensional ring, subject to a perturbation acting on $O_j=\sigma^x_j$, spatially localized and centered at $\phi = 0$. The simulations are performed with parameters $J = h = L / 4\pi$, $\Omega = 5$, $M = 0$, and $\sigma_t = \sigma_\phi = 0.4$, with $L = 32$. (a) Response to a weak perturbation with amplitude $A = 0.1 J \sqrt{2/(\sigma_t \sigma_\phi L)}$. (b) Response to a stronger perturbation with amplitude $A =J \sqrt{2/(\sigma_t \sigma_\phi L)}$. In both (a) and (b), the left panels show the time evolution of $|\delta\langle \sigma^x_j(t) \rangle|$ at three representative angular positions: $\phi = 0$, $\pi/2$, and $\pi$. The right panels display the corresponding spatiotemporal structure of the response $|\delta\langle \sigma^x_j(t) \rangle|$, visualized as intensity plots.
• Figure 4: Response to a weak perturbation acting on $O_j=\sigma^x_j$, spatially localized and centered at two angular positions $\phi = 0$ and $\phi = -\pi/2$, with the same system parameters as those used in Fig. \ref{['fig:fig1']}(a). Compared to the single-source case in Fig. \ref{['fig:fig1']}(a), multiple localized peaks emerge, corresponding to each excitation and its antipodal counterpart.
• Figure 5: Response to a spatially uniform perturbation acting on $O_j=\sigma^x_j$, corresponding to the limiting case of infinitely many localized sources, with the same system parameters as those used in Fig. \ref{['fig:fig1']}(a). In the left panel, responses from all sites are plotted on top of each other; they coincide within numerical accuracy because the perturbation is uniform.