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arXiv:1410.3466 · Physical Review Letters

Nearly Linear Light Cones in Long-Range Interacting Quantum Systems

Michael Foss‐Feig, Zhexuan Gong, Charles W. Clark, Alexey Vyacheslavovich Gorshkov

Published Apr 13, 2015• 211 citations
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Abstract

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.

DOI
10.1103/physrevlett.114.157201

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In nonrelativistic quantum theories with short-range Hamiltonians, a velocity vvv can be chosen such that the influence of any local perturbation is approximately confined to within a distance rrr until a time t\ensuremath∼r/vt\ensuremath{\sim}r/vt\ensuremath∼r/v, thereby defining a linear light cone and giving rise to an emergent notion of locality. In systems with power-law (1/r\ensuremathα1/{r}^{\ensuremath{\alpha}}1/r\ensuremathα) interactions, when \ensuremathα\ensuremath{\alpha}\ensuremathα exceeds the dimension DDD, an analogous bound confines influences to within a distance rrr only until a time t\ensuremath∼(\ensuremathα/v)logrt\ensuremath{\sim}(\ensuremath{\alpha}/v)\mathrm{log}rt\ensuremath∼(\ensuremathα/v)logr, 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 \ensuremathα>2D\ensuremath{\alpha}>2D\ensuremathα>2D and become linear as \ensuremathα\ensuremath→\ensuremath∞\ensuremath{\alpha}\ensuremath{\rightarrow}\ensuremath{\infty}\ensuremathα\ensuremath→\ensuremath∞. 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.
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 rrr 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 ppp-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.
In this paper, we prove a family of Lieb-Robinson bounds for discrete spin systems with long-range interactions. Our results apply for arbitrary kkk-body interactions, so long as they decay with a power law greater than kdkdkd, where ddd is the dimension of the system. More precisely, we require that the sum of the norm of terms with diameter greater than or equal to RRR, acting on any one site, decays as a power law $1/{R}^{\ensuremath{\alpha}}$, with $\ensuremath{\alpha}>d$. These bounds allow us to prove that, at any fixed time, the spatial decay of a time evolved operator follows arbitrarily closely to $1/{r}^{\ensuremath{\alpha}}$. Moreover, we introduce an alternative light cone definition for power-law interacting quantum systems which captures the region of the system where changing the Hamiltonian can affect the evolution of a local operator. In short-range interacting systems, this light cone agrees with the conventional definition. However, in long-range interacting systems, our definition yields a stricter light cone, which is more relevant in most physical contexts.
Recent works have revealed the intricate effect of long-range interactions on information transport in quantum many-body systems: In DDD spatial dimensions, interactions decaying as a power-law r−αr^{-α}r−α with α>2D+1α> 2 D+1α>2D+1 exhibit a Lieb-Robinson bound (LRB) with a linear light cone and the threshold 2D+12D +12D+1 is sharp in general. Here, we observe that mutually commuting, long-range interactions satisfy an enhanced LRB of the form t r−αt \, r^{-α}tr−α for any α>0α>0α>0, and this scaling is sharp. In particular, the linear light cone occurs at α=1α= 1α=1 in any dimension. Part of our motivation stems from quantum error-correcting codes. As applications, we derive enhanced bounds on ground state correlations and an enhanced local perturbations perturb locally (LPPL) principle for which we adapt a recent subharmonicity argument of Wang-Hazzard. Similar enhancements hold for commuting interactions with stretched exponential decay.
Lieb-Robinson bounds quantify the maximal speed of information spreading in nonrelativistic quantum systems.We discuss the relation of Lieb-Robinson bounds to out-of-time order correlators, which correspond to different norms of commutators C(r, t ) = [A i (t ), B i+r ] of local operators.Using an exact Krylov space-time evolution technique, we calculate these two different norms of such commutators for the spin-1/2 Heisenberg chain with interactions decaying as a power law 1/r α with distance r.Our numerical analysis shows that both norms (operator norm and normalized Frobenius norm) exhibit the same asymptotic behavior, namely, a linear growth in time at short times and a power-law decay in space at long distance, leading asymptotically to power-law light cones for α < 1 and to linear light cones for α > 1.The asymptotic form of the tails of C(r, t ) ∝ t/r α is described by short-time perturbation theory, which is valid at short times and long distances.