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arXiv:1408.1796 · Physical Review Letters

New Anomalous Lieb-Robinson Bounds in Quasiperiodic XY Chains

David Damanik, Marius Lemm, Milivoje Lukić, William N. Yessen

Published Sep 18, 2014• 23 citations
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Abstract

We announce and sketch the rigorous proof of a new kind of anomalous (or sub-ballistic) Lieb-Robinson (LR) bound for an isotropic $XY$ 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.

DOI
10.1103/physrevlett.113.127202

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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≤v∣t∣|x|\ensuremath{\le}v|t|∣x∣\ensuremath≤v∣t∣, we obtain ∣x∣\ensuremath≤v∣t∣\ensuremathα|x|\ensuremath{\le}v|t{|}^{\ensuremath{\alpha}}∣x∣\ensuremath≤v∣t∣\ensuremathα for some 0<\ensuremathα<10<\ensuremath{\alpha}<10<\ensuremathα<1. We can characterize the allowed values of \ensuremathα\ensuremath{\alpha}\ensuremathα exactly as those exceeding the upper transport exponent \ensuremathαu+{\ensuremath{\alpha}}_{u}^{+}\ensuremathαu+​ 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.
We rigorously prove a new kind of anomalous (or sub-ballistic) Lieb–Robinson bound for the isotropic XY chain with Fibonacci external magnetic field at arbitrary coupling. It is anomalous in that the usual exponential decay in |x|-v|t| is replaced by exponential decay in |x|-v|t|^\alpha with 0<\alpha<1 . In fact, we can characterize the values of \alpha for which such a bound holds as those exceeding \alpha_u^+ , the upper transport exponent of the one-body Fibonacci Hamiltonian. Following the approach of [14], we relate Lieb–Robinson bounds to dynamical bounds for the one-body Hamiltonian corresponding to the XY chain via the Jordan–Wigner transformation; in our case the one-body Hamiltonian with Fibonacci potential. We can bound its dynamics by adapting techniques developed in [8, 9, 2, 3] to our purposes. To our knowledge, this is the first rigorous derivation of anomalous quantum many-body transport. Along the way, we prove a new result about the one-body Fibonacci Hamiltonian: the upper transport exponent agrees with the time-averaged upper transport exponent, see Corollary 2.9. We also explain why our method does not extend to yield anomalous Lieb–Robinson bounds of power-law type for the random dimer model.
By tightening the conventional Lieb-Robinson bounds to better handle systems that lack translation invariance, we determine the extent to which "weak links" suppress operator growth in disordered onedimensional spin chains.In particular, we prove that ballistic growth is impossible when the distribution of coupling strengths μ(J ) has a sufficiently heavy tail at small J and we identify the correct dynamical exponent to use instead.Furthermore, through a detailed analysis of the special case in which the couplings are genuinely random and independent, we find that the standard formulation of Lieb-Robinson bounds is insufficient to capture the complexity of the dynamics-we must distinguish between bounds that hold for all sites of the chain and bounds that hold for a subsequence of sites and we show by explicit example that these two can have dramatically different behaviors.All the same, our result for the dynamical exponent is tight, in that we prove by counterexample that there cannot exist any Lieb-Robinson bound with a smaller exponent.We close by discussing the implications of our results, both major and minor, for numerous applications ranging from quench dynamics to the structure of ground states.
An active area of research in quantum many-body physics is the extent to which quenched disorder impedes ``natural'' behaviors, such as transport and thermalization. This is an inherently challenging problem to study, and thus it is particularly valuable to have rigorous results that unambiguously prove the absence of transport in certain situations. This work provides such a result: the author uses here the machinery of Lieb-Robinson bounds to prove that ballistic and even diffusive motion is indeed impossible when the distribution of on-site disorder is sufficiently heavy tailed, regardless of most other details of the Hamiltonian.
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 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 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.