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arXiv:2005.10257 · Physical Review Research

Lieb-Robinson bounds and out-of-time order correlators in a long-range spin chain

Luis Colmenárez, David J. Luitz

Published Oct 8, 2020• 43 citations
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Abstract

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.

DOI
10.1103/physrevresearch.2.043047

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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.
Recent works have revealed the intricate effect of long-range interactions on information transport in quantum many-body systems: In D spatial dimensions, interactions decaying as a power-law r−α with α > 2D + 1 exhibit a Lieb-Robinson bound (LRB) with a linear light cone and the threshold 2D + 1 is sharp in general. Here, we observe that mutually commuting, long-range interactions satisfy an enhanced LRB of the form t r−α for any α > 0, and this scaling is sharp. In particular, the linear light cone occurs at α = 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 principle for which we adapt a recent subharmonicity argument of Wang-Hazzard. Similar enhancements hold for commuting interactions with stretched exponential decay.
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.
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.
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.
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.
The propagation of information in nonrelativistic quantum systems obeys a speed limit known as a Lieb-Robinson bound.We derive a new Lieb-Robinson bound for systems with interactions that decay with distance r as a power law, 1=r α .The bound implies an effective light cone tighter than all previous bounds.Our approach is based on a technique for approximating the time evolution of a system, which was first introduced as part of a quantum simulation algorithm by Haah et al., FOCS'18.To bound the error of the approximation, we use a known Lieb-Robinson bound that is weaker than the bound we establish.This result brings the analysis full circle, suggesting a deep connection between Lieb-Robinson bounds and digital quantum simulation.In addition to the new Lieb-Robinson bound, our analysis also gives an error bound for the Haah et al. quantum simulation algorithm when used to simulate power-law decaying interactions.In particular, we show that the gate count of the algorithm scales with the system size better than existing algorithms when α > 3D (where D is the number of dimensions).