Dynamical Localization in Disordered Quantum Spin Systems

Hamza, Eman; Sims, Robert; Stolz, Günter

doi:10.1007/s00220-012-1544-6

Cited by 81 publications

(169 citation statements)

References 46 publications

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“…Let I ⊆ R and ρ(s, t) be a family of uniformly bounded operators on ℓ 2 (Z) with ρ(s, t) ≤ 1 for all s, t ∈ I. If there is some C ∈ (0, ∞), µ ∈ (µ 0 , ∞) such that for all x, y ∈ Z: 14) then for any n ∈ N and any pair of fermionic configurations x = (x 1 , . .…”

Section: Determinant Boundmentioning

confidence: 99%

Decay of Determinantal and Pfaffian Correlation Functionals in One-Dimensional Lattices

Sims

Warzel

2016

Commun. Math. Phys.

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We establish bounds on the decay of time-dependent multipoint correlation functionals of one-dimensional quasi-free fermions in terms of the decay properties of their two-point function. At a technical level, this is done with the help of bounds on certain bordered determinants and pfaffians. These bounds, which we prove, go beyond the well-known Hadamard estimates. Our main application of these results is a proof of strong (exponential) dynamical localization of spin-correlation functions in disordered XY -spin chains.

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Section: Determinant Boundmentioning

confidence: 99%

Decay of Determinantal and Pfaffian Correlation Functionals in One-Dimensional Lattices

Sims

Warzel

2016

Commun. Math. Phys.

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“…At this stage, H N can be diagonalized by a standard Bogoliubov transformation. One finds the following formula [8] for the Heisenberg dynamics (3) of the fermion operators: holds for all observables B ∈ O x 0 . As we will see, (10) allows us to prove LR fermi ðαÞ by controlling the one-body transport created by h. This is not surprising, because (10) is an expression of the fact that we are now describing free particles.…”

Section: Prl 113 127202 (2014) P H Y S I C a L R E V I E W L E T T Ementioning

confidence: 99%

“…The point is that, as originally realized in [8] and adapted here to our purposes, inverting the nonlocal Jordan-Wigner transformation essentially just requires summing up fermionic LR bounds: By an iteration argument, which is based only on ðABÞðtÞ ¼ AðtÞBðtÞ and the usual commutator rules, one can show…”

Section: Prl 113 127202 (2014) P H Y S I C a L R E V I E W L E T T Ementioning

confidence: 99%

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New Anomalous Lieb-Robinson Bounds in QuasiperiodicXYChains

et al. 2014

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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 XY chain in a quasiperiodic transversal magnetic field. Instead of the usual effective light cone jxj ≤ vjtj, we obtain jxj ≤ vjtj α for some 0 < α < 1. We can characterize the allowed values of α exactly as those exceeding the upper transport exponent α þ u of a one-body Schrödinger 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. Introduction.-Relativistic systems are local in the sense that information propagates at most at the speed of light. In their seminal paper [1], Lieb and Robinson found that nonrelativistic quantum spin systems described by local Hamiltonians satisfy a similar "quasilocality" under the Heisenberg dynamics. Their Lieb-Robinson (LR) bound and its recent generalizations [2,3] implies the existence of a "light cone" jxj ≤ vjtj in space-time, outside of which quantum correlations (concretely: commutators of local observables) are exponentially small. In other words, the LR bound shows that, to a good approximation, quantum correlations propagate at most ballistically, with a systemdependent "Lieb-Robinson velocity" v.About ten years ago, the general interest in LR bounds resurged when Hastings and coworkers realized that they are the key tool for deriving exponential clustering, a higher-dimensional Lieb-Schultz-Mattis theorem, and the celebrated area law for the entanglement entropy in onedimensional systems with a spectral gap [2,4,5]. These results highlight the role of entanglement in constraining the structure of ground states in gapped systems and yield many applications to quantum information theory, e.g., in developing algorithms to simulate quantum systems on a classical computer [6,7].In this Letter, we announce and sketch the rigorous proof of a new kind of anomalous (or sub-ballistic) Lieb-Robinson bound for an isotropic XY chain in a quasiperiodic transversal magnetic field. The LR bound is anomalous in the sense that the forward half of the ordinary light cone is changed to the region jxj ≤ vjtj α for some 0 < α < 1.Previous study has focused on the dependence of the Lieb-Robinson velocity v on the system details [3], with particular interest in the case v ¼ 0, since it may be interpreted as dynamical localization [8]. In a very recent paper [9], a logarithmic light cone was obtained for long-range, i.e., power-law decaying, interactions. The

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“…Interestingly even in the nonergodic phase, where transport is completely frozen, information continues to spread logarithmically in time via dephasing [22], as was initially established using the growth of the EE [23,24] and later using OTOCs [18,[25][26][27][28][29][30][31]. In contrast, the spreading of information in noninteracting Anderson insulators is completely frozen [32,33]. The ergodic phase of this model, occurring for W < W c , exhibits anomalous subdiffusive spin transport characterized by a dynamical exponent varying continuously with disorder strength [34][35][36][37][38][39][40][41][42][43][44], but also a sublinear EE growth [36] (cf.…”

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confidence: 99%

Information propagation in isolated quantum systems

2017

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Entanglement growth and out-of-time-order correlators (OTOC) are used to assess the propagation of information in isolated quantum systems. In this work, using large scale exact time-evolution we show that for weakly disordered nonintegrable systems information propagates behind a ballistically moving front, and the entanglement entropy growths linearly in time. For stronger disorder the motion of the information front is algebraic and sub-ballistic and is characterized by an exponent which depends on the strength of the disorder, similarly to the sublinear growth of the entanglement entropy. We show that the dynamical exponent associated with the information front coincides with the exponent of the growth of the entanglement entropy for both weak and strong disorder. We also demonstrate that the temporal dependence of the OTOC is characterized by a fast nonexponential growth, followed by a slow saturation after the passage of the information front. Finally,we discuss the implications of this behavioral change on the growth of the entanglement entropy.Introduction. -While the speed of light is the absolute upper limit of information propagation in both classical and quantum relativistic systems, surprisingly a velocity which plays a similar role exists also for shortrange interacting nonrelativistic quantum systems. This velocity, known as the Lieb-Robinson velocity, bounds the spreading of correlations in the system and implies that information about local initial excitations propagates within a causal "light-cone", similarly to the lightcone encountered in the theory of special relativity [1]. The shape of the light-cone can be obtained from the correlation function

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Dynamical Localization in Disordered Quantum Spin Systems

Cited by 81 publications

References 46 publications

Decay of Determinantal and Pfaffian Correlation Functionals in One-Dimensional Lattices

Decay of Determinantal and Pfaffian Correlation Functionals in One-Dimensional Lattices

New Anomalous Lieb-Robinson Bounds in QuasiperiodicXYChains

Information propagation in isolated quantum systems

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