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Physica B Condensed Matter

Lieb–Robinson bound in one-dimensional inhomogeneous quantum systems

Hao Fu, Mingqiu Luo, Peiqing Tong

Published Apr 26, 2022• 0 citations
Publisher

Abstract

Lieb-Robinson bound (LRB) in one-dimensional noninteracting manyelectron systems with the disordered and quasiperiodic on-site potentials is studied numerically. For the short-range hopping system, a logarithmic light cone is found in the system with the disordered on-site potential for small time. Coefficient β decreases with the increasing strength of disordered. When time is large, the bound doesn't change with time. For the generalized Fibonacci quasiperiodic (GFQ) system, the on-site potential is taken as V or − V according to two kinds of GFQ sequences. It is found that the system has a power-law light cone. The exponent γ decreases with the increasing V . We also find that γ for the first class of GFQ system is larger than that for the second class of GFQ system with same V . Finally, the effects of the long-range hopping which decays like r − α with distance r on LRB are discussed.

DOI
10.1016/j.physb.2022.413958

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Lieb-Robinson bound (LRB) in one-dimensional noninteracting manyelectron systems with the disordered and quasiperiodic on-site potentials is studied numerically. For the short-range hopping system, a logarithmic light cone is found in the system with the disordered on-site potential for small time. Coefficient β decreases with the increasing strength of disordered. When time is large, the bound doesn't change with time. For the generalized Fibonacci quasiperiodic (GFQ) system, the on-site potential is taken as V or − V according to two kinds of GFQ sequences. It is found that the system has a power-law light cone. The exponent γ decreases with the increasing V . We also find that γ for the first class of GFQ system is larger than that for the second class of GFQ system with same V . Finally, the effects of the long-range hopping which decays like r − α with distance r on LRB are discussed.
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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.
We announce and sketch the rigorous proof of a new kind of anomalous (or sub-ballistic) Lieb-Robinson (LR) bound for an isotropic XYXYXY chain in a quasiperiodic transversal magnetic field. Instead of the usual effective light cone $|x|\ensuremath{\le}v|t|$, we obtain $|x|\ensuremath{\le}v|t{|}^{\ensuremath{\alpha}}$ for some $0<\ensuremath{\alpha}<1$. We can characterize the allowed values of $\ensuremath{\alpha}$ exactly as those exceeding the upper transport exponent ${\ensuremath{\alpha}}_{u}^{+}$ of a one-body Schr\"odinger operator. To our knowledge, this is the first rigorous derivation of anomalous quantum many-body transport. We also discuss anomalous LR bounds with power-law tails for a random dimer field.
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).