Revealing effects of local dimension on variable-range interacting model by connecting Lieb-Robinson bounds and multipartite entanglement

TL;DR

The paper investigates how the local Hilbert space dimension $d$ shapes information spreading in spin-$s$ Ising models with variable-range interactions decaying as $1/|r|^{alpha}$. It develops a dynamical framework based on multiqudit weighted graph states (WGS) and analyzes mutual information, Lieb-Robinson bounds, and genuine multipartite entanglement (GME). Key findings include a dimension-dependent transition in mutual information spreading near $alpha \,\sim\, log_2 d$, a dimension-sensitive light-cone structure with $t_{|r|} \propto |r|^{\alpha}/d$ (for $p=2$) and $t_{|r|} \propto |r|^{\alpha}/d^2$ (for $p=\infty$), and a direct correspondence between maximal LR propagation and maximal GME (edge effects under open boundaries). These results link dynamic nonlocality, information flow, and entanglement in LR qudit systems and offer a framework for probing LR physics and potential MBQC resources.

Abstract

A spin-$s$ variable-range interacting Ising model may display qualitatively different behaviors depending on the fall-off rate of the interactions, as already seen in equilibrium studies of spin-1/2 systems. We propose a dynamical method using weighted graph states, generated through time evolution that confirms the existence of the transition point in the fall-off rate for the spin-$s$ Ising model. Moreover, the dependence of local dimension on information spreading and multipartite entanglement profile in this model remains unclear, which we establish here. In particular, our analysis shows that the maximum of genuine multipartite entanglement (GME) with the fall-off rate serves as a clear indicator of the information spreading, which aligns with changes in the profile of the Lieb-Robinson bound. Further, in the case of an open chain, the spread of information is related to the divergence in the first derivative of GME. Additionally, we validate this signature by performing a scaling analysis of the time-averaged mutual information.

Figures (11)

• Figure 1: (a) Time-averaged mutual information, $\langle I\rangle_{t_0}(\alpha, r)$(ordinate) for qutrit (spin-$1$, $d=3$) WGS in $1$D with OBC, against the distance $r$ (abscissa) between two sites, in logarithmic scales, for various fall-off rates $\alpha$. The average is taken from $0$ to $t_0=15\pi$. Although, $\log\langle I\rangle_{t_0}$ decreases linearly with $\log r$ at large $r$ for all $\alpha$, its behavior changes from linear to quadratic on increasing $\alpha$ at small $r$. The quadratic fitting is done from $r=1$ to $15$ which shows good agreement in inset (a)(ii). (b) Quadratic coefficient $\tilde{\beta}_2$ (ordinate) of $\log\langle I\rangle_{t_0}(\alpha, r)$ scaling with $\log r$, against fall-off rate $\alpha$ (absissca), showing $\alpha\sim\log_2d$ (vertical bars) in the change of $\langle I\rangle_{t_0}(\alpha, r)$ profile from linear to quadratic (in log scales). Here, $N=4000$, and all the axes are dimensionless.
• Figure 2: Information spread for $d=3$ in the LR Ising model on general lattice, quantified by the commutator $C^d_2$ (colorbar) [Eq. (\ref{['eq:c2']})] for fall-off rates (a) $\alpha=0.8$ (b) $\alpha=1.3$ in the time $t$ (ordinate)--distance $|\vec{r}|$ (abscissa) plane. The dashed (black) line represents the $t_{|\vec{r}|}d=2\pi|\vec{r}|^\alpha$ line [Eq. (\ref{['eq:lrb_main']})] showing maximum $C^d_2$. All the axes are dimensionless.
• Figure 3: Non-analyticity of GGM, $\mathcal{G}^o$, for $d=4$ in the open chain. (a) $\delta\mathcal{G}^o_d=\mathcal{G}^{\max}_d-\mathcal{G}^o_d$ with respect to the fall-off rate $\alpha$ (ordinate) and time $t/\pi$ (abscissa). At time, $t_{n,x}=2\pi nx^\alpha/d, \delta\mathcal{G}^o_d=0$ ($<10^{-3}$ with $\Delta\alpha=\Delta t/\pi=10^{-2}$ spacings) shown in logarithmic scales. Various $(n,x)$-pairs are marked by arrows. (b) $\frac{\mathrm{d}\mathcal{G}_d}{\mathrm{d}\alpha}$ (ordinate) vs the fall-off rate $\alpha$ (abscissa) for PBC and OBC at $t=2.8\pi$. $\frac{\mathrm{d}\mathcal{G}_d^p}{\mathrm{d}\alpha}$ is continuous, while $\frac{\mathrm{d}\mathcal{G}_d^o}{\mathrm{d}\alpha}$ is discontinuous for $n=1$ and different $x$ with different vertical lines, and $n=2, x=2$ shown by an arrow. The ordinate is in symmetric-logarithmic scale ($10^{-5}$ linear threshold). Here $N=4000$, and all axes are dimensionless.
• Figure 4: Mutual information, $I(\alpha,r,t)$ (ordinate) against time $t$ (abscissa) for the various fall-off rates $\alpha=0.7,1.5,2.5$ (a)--(c) for a spin-$1/2$ Ising Hamiltonian on a $1$D lattice used for evolution. Here $N=10^3$. Different lines represent different values of separation $r$ between two qubits in the lattice. Both the axes are dimensionless.
• Figure 5: All specifications remain the same as in Fig. \ref{['fig:i_rt_sig_d2']}, except that qubits are replaced with qutrits ($d = 3$).