We argue that chaotic power-law interacting systems have emergent limits on information propagation, analogous to relativistic light cones, which depend on the spatial dimension d and the exponent α governing the decay of interactions. Using the dephasing nature of quantum chaos, we map the problem to a stochastic model with a known phase diagram. A linear light cone results for α≥d+1/2. We also provide a Lévy flight (long-range random walk) interpretation of the results and show consistent numerical data for 1D long-range spin models with 200 sites.
We argue that chaotic power-law interacting systems have emergent limits on information propagation, analogous to relativistic light cones, which depend on the spatial dimension d and the exponent α governing the decay of interactions. Using the dephasing nature of quantum chaos, we map the problem to a stochastic model with a known phase diagram. A linear light cone results for α≥d+1/2. We also provide a Lévy flight (long-range random walk) interpretation of the results and show consistent numerical data for 1D long-range spin models with 200 sites.
We use an out-of-time-order commutator (OTOC) to diagnose the propagation of chaos in one-dimensional long-range power law interaction system. We map the evolution of OTOC to a classical stochastic dynamics problem and use a Brownian quantum circuit to exactly derive the master equation. We vary two parameters: The number of qubits N on each site (the on-site Hilbert space dimension) and the power law exponent $\ensuremath{\alpha}$. Three light cone structures of OTOC appear at N=1: (1) logarithmic when $0.5<\ensuremath{\alpha}\ensuremath{\lesssim}0.8$, (2) sublinear power law when $0.8\ensuremath{\lesssim}\ensuremath{\alpha}\ensuremath{\lesssim}1.5$, and (3) linear when $\ensuremath{\alpha}\ensuremath{\gtrsim}1.5$. The OTOC scales as $exp(\ensuremath{\lambda}t)/{x}^{2\ensuremath{\alpha}}$ and ${t}^{2\ensuremath{\alpha}/\ensuremath{\zeta}}/{x}^{2\ensuremath{\alpha}}$, respectively, beyond the light cones in the first two cases. When $\ensuremath{\alpha}\ensuremath{\ge}2$, the OTOC has essentially the same diffusive broadening as systems with short-range interactions, suggesting a complete recovery of locality. In the large N limit, it is always a logarithmic light cone asymptotically, although a linear light cone can appear before the transition time for $\ensuremath{\alpha}\ensuremath{\gtrsim}1.5$. This implies the locality is never fully recovered for finite $\ensuremath{\alpha}$. Our result provides a unified physical picture for the chaos dynamics in a long-range power law interaction system.
Locality imposes stringent constraints on the spreading of information in nonrelativistic quantum systems, which is reminiscent of a ``light cone,'' a causal structure arising in their relativistic counterparts. Long-range interactions can potentially soften such constraints, allowing almost instantaneous long jumps of particles, into the ``spacelike'' region, thus defying causality. Since interactions decaying as a power law with distance ${r}^{\ensuremath{-}\ensuremath{\alpha}}$ are ubiquitous in nature, it is pertinent to understand what is the fate of causality and information spreading in such systems. Using a numerically exact technique we address these questions by studying the out-of-time-order correlation function of a representative generic system in one dimension. We show that while the interactions are long range, their effect on information spreading is asymptotically negligible as long as $\ensuremath{\alpha}>1$. In this range we find a complex compound behavior, where, after a short transient, a fully local behavior emerges, yielding asymptotic light cones virtually indistinguishable from light cones in corresponding local models. The long-range nature of the interaction is only expressed in the power-law leaking of information from the light cone, with the same exponent as the exponent of the interaction $\ensuremath{\alpha}$. Our results directly apply also to corresponding response functions and suggest that previously obtained rigorous bounds on information spreading in long-range interacting systems are not tight, and could be potentially improved.
In locally interacting quantum many-body systems, the velocity of information propagation is finitely bounded, and a linear light cone can be defined.Outside the light cone, the amount of information rapidly decays with distance.When systems have long-range interactions, it is highly nontrivial whether such a linear light cone exists.Herein, we consider generic long-range interacting systems with decaying interactions, such as R -α with distance R. We prove the existence of the linear light cone for α > 2D þ 1 (D, the spatial dimension), where we obtain the Lieb-Robinson bound as k½O i ðtÞ; O j k ≲ t 2Dþ1 ðR -vtÞ -α with v ¼ Oð1Þ for two arbitrary operators O i and O j separated by a distance R.Moreover, we provide an explicit quantum-state transfer protocol that achieves the above bound up to a constant coefficient and violates the linear light cone for α < 2D þ 1.In the regime of α > 2D þ 1, our result characterizes the best general constraints on the information spreading.
In many-body quantum systems with spatially local interactions, quantum information propagates with a finite velocity, reminiscent of the ``light cone'' of relativity. In systems with long-range interactions which decay with distance r as $1/{r}^{\ensuremath{\alpha}}$, however, there are multiple light cones which control different information theoretic tasks. We show an optimal (up to logarithms) ``Frobenius light cone'' obeying $t\ensuremath{\sim}{r}^{min(\ensuremath{\alpha}\ensuremath{-}1,1)}$ for $\ensuremath{\alpha}>1$ in one-dimensional power-law interacting systems with finite local dimension: this controls, among other physical properties, the butterfly velocity characterizing many-body chaos and operator growth. We construct an explicit random Hamiltonian protocol that saturates the bound and settles the optimal Frobenius light cone in one dimension. We partially extend our constraints on the Frobenius light cone to a several operator p-norms, and show that Lieb-Robinson bounds can be saturated in at most an exponentially small ${e}^{\ensuremath{-}\mathrm{\ensuremath{\Omega}}(r)}$ fraction of the many-body Hilbert space.
The Lieb-Robinson theorem states that information propagates with a finite velocity in quantum systems on a lattice with nearest-neighbor interactions. What are the speed limits on information propagation in quantum systems with power-law interactions, which decay as 1/r^{α} at distance r? Here, we present a definitive answer to this question for all exponents α>2d and all spatial dimensions d. Schematically, information takes time at least r^{min{1,α-2d}} to propagate a distance r. As recent state transfer protocols saturate this bound, our work closes a decades-long hunt for optimal Lieb-Robinson bounds on quantum information dynamics with power-law interactions.
In nonrelativistic quantum theories with short-range Hamiltonians, a velocity v can be chosen such that the influence of any local perturbation is approximately confined to within a distance r until a time $t\ensuremath{\sim}r/v$, thereby defining a linear light cone and giving rise to an emergent notion of locality. In systems with power-law ($1/{r}^{\ensuremath{\alpha}}$) interactions, when $\ensuremath{\alpha}$ exceeds the dimension D, an analogous bound confines influences to within a distance r only until a time $t\ensuremath{\sim}(\ensuremath{\alpha}/v)\mathrm{log}r$, suggesting that the velocity, as calculated from the slope of the light cone, may grow exponentially in time. We rule out this possibility; light cones of power-law interacting systems are bounded by a polynomial for $\ensuremath{\alpha}>2D$ and become linear as $\ensuremath{\alpha}\ensuremath{\rightarrow}\ensuremath{\infty}$. Our results impose strong new constraints on the growth of correlations and the production of entangled states in a variety of rapidly emerging, long-range interacting atomic, molecular, and optical systems.