Shaping Causality: Emergence of Nonlocal Light Cones in Long-Range Quantum Systems

TL;DR

The paper shows that long-range spin chains can exhibit emergent locality or programmable nonlocal light cones depending on initial conditions, through a controlled mapping to a hard-core boson model with a key nonlocal interaction $W(i,j) \\propto 1/|i-j|^{\\alpha}$. By projecting onto excitation bands and applying Kac rescaling, the authors isolate intraband and interband nonlocal contributions and reveal that locality emerges when $W$ is suppressed or in the thermodynamic limit, while nonlocality can be engineered by arranging initial excitations and tuning $J_{\\mathrm{long}}$ and $\\alpha$. The work identifies two nonlocal pathways—intraband $W$-driven correlations within a band and interband couplings arising from band transitions—each with distinct scaling and controllability, and demonstrates that nonlocal light cones can be selectively programmed across independent propagation channels. These insights offer a microscopic framework for robust quantum memories, error correction, and programmable information transport in long-range quantum simulators. The analysis leverages Holstein–Primakoff/Boson mappings, band projections, and fidelity diagnostics to rigorously connect microscopic interactions to emergent causal structures.

Abstract

While for non-relativistic short-range interactions, the spread of information is local, remaining confined in an effective light cone, long-range interactions can generate either nonlocal (faster-than-ballistic) or local (ballistic) spread of correlations depending on the initial conditions. This makes long-range interactions a rich platform for controlling the spread of information. Here, we derive an effective Hamiltonian analytically and identify the specific interaction term that drives nonlocality in a wide class of long-range spin chains. This allows us to understand the conditions for the emergence of local behavior in the presence of nonlocal interactions and to identify a regime where the causal space-time landscape can be precisely designed. Indeed, we show that for large long-range interaction strength or large system size, initial conditions can be chosen in a way that allows a local perturbation to generate nonlocal signals at programmable distant positions, which then propagate within effective light cones. The possibility of engineering the emergence of nonlocal Lieb-Robinson-like light cones allows one to shape the causal landscape of long-range interacting systems, with direct applications to quantum information processing devices, quantum memories, error correction, and information transport in programmable quantum simulators.

Figures (19)

• Figure 1: Density plots of the Frobenius norm showing light-cone dynamics under the full Hamiltonian \ref{['LFIM']} with initial states prepared in band $b=1$: (2; 4) (notation defined in Eq. \ref{['pairnotation']}). At weaker long-range coupling ($J_{\mathrm{long}}=1/3$, left), the nonlocal signal outside the light cone is strong, while at stronger longe-range coupling ($J_{\mathrm{long}}=20$, right), locality emerges. The threshold for first arrival times is 0.01. In both cases we fix $N=15$ and $\alpha=0$. Here and throughout we set $J_z=1$. The insets show outside-the-cone profiles at time $t=1$.
• Figure 2: Density plots of the Frobenius norm showing light-cone dynamics under the full Hamiltonian \ref{['LFIM']} with $|\uparrow_z, \downarrow_z, \uparrow_z, \ldots \rangle$ and $|\uparrow_z, \uparrow_z, \uparrow_z, \ldots \rangle$ as the two inititial states being compared. Here $N=13$, $J_{\mathrm{long}}=2$, and $\alpha=0$. Instantaneous propagation of the perturbation to large distances occurs for the full Hamiltonian (left), while removing $W$ restores a strict Lieb-Robinson-local light cone (right). Nonlocality originates from $W$. The inset shows the outside-the-cone profile at time $t=1$.
• Figure 3: Density plots of the Frobenius norm showing light-cone dynamics under the projected Hamiltonian ${\hat{H}}_{\mathrm{eff}}^{b=2}$ with initial states prepared in band $b=2$ for $\alpha=0.5$ and $J_{\mathrm{long}}/J_z=2$. In the close configuration $(2,4;\,3,4)$ (left) (notation defined in Eq. \ref{['pairnotation']}), the additional excitation at site 4 lies within the light cone (except at very short times), and no detectable nonlocal signal is observed outside. In the far configuration $(2,28;\,3,28)$ (right), the additional excitation at site 28 lies outside the light cone of the excitation, and a relatively weak but clear nonlocal signal appears showing conditional nonlocality from $W$ interactions. The insets show outside-the-cone profiles at $t=1$.
• Figure 4: Time-averaged Frobenius norm for dynamics under the full Hamiltonian for ${t \in [0,2]}$ at site 13 for the close configuration, probing the interband nonlocal signal at $\alpha=0$. The expected scaling is $\|\Delta \rho_n(t)\|_F^{\rm interband} \lesssim J_{\mathrm{long}}^{-2} N^{-2}$. Left: dependence on $J_{\mathrm{long}}$ for fixed $N=15$ and initial states in different bands (notation defined in Eq. \ref{['pairnotation']}): $b=1\colon (2;\,3)$, $b=2\colon (2,5;\,3,5)$, $b=3\colon (2,5,7;\,3,5,7)$, $b=5\colon (2,5,7,9,11;\,3,5,7,9,11)$, and $b=7\colon (2,5,7,9,11,13,15;\,3,5,7,9,11,13,15)$. Right: dependence on system size $N$ at fixed $J_{\mathrm{long}}=20$. Without Kac rescaling, the nonlocal signal would decay even faster with $N$. Data points show simulation results; dashed lines are fits.
• Figure 5: Time-averaged Frobenius norm for dynamics under the full Hamiltonian, for ${t \in [0,2]}$ at site 13 as a function of $J_{\mathrm{long}}$ shows competing scaling of intra- and interband nonlocalities at $\alpha=0.5$ and $N=15$. Left: Close configurations, where intraband nonlocality is suppressed and the residual signal isolates interband effects (bands $b=1\colon (2;\,3)$, $b=2\colon (2,5;\,3,5)$, $b=3\colon (2,5,7;\,3,5,7)$). Right: Far configurations at the same parameters (bands $b=2\colon (2,13;\,3,13)$ and $b=3\colon (2,11,13;\,3,11,13)$), where intraband nonlocality asymptotically dominates over interband effects. For reference, the intraband signal of band $b=2$ is taken from Fig. \ref{['WintraScaling']} in the Supplemental Material. Intraband $N$-scaling only emerges for large $N$; for small chains, the competition between the two terms masks any clear scaling.