Delocalization in Random Polymer Models

Jitomirskaya, Svetlana; Schulz-Baldes, Hermann; Stolz, Günter

doi:10.1007/s00220-002-0757-5

Cited by 91 publications

(176 citation statements)

References 23 publications

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“…However, in the random dimer model, this probability is polynomially small for some β < 1. In fact, there are similar transport exponentsβ þ ðpÞ, related to timeaveraged pth moments of the position operator, which characterize when this is the case and which were determined explicitly in [39,40].…”

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: 87%

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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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: 87%

New Anomalous Lieb-Robinson Bounds in QuasiperiodicXYChains

et al. 2014

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“…In the zero mass case (m = 0) the operators ID ω (0, c), ω ∈ Ω, presents critical energies E V = ±V for V ∈ (0, c] , V = c/ √ 2, as defined in [18], since either…”

Section: Toolsmentioning

confidence: 99%

“…It was also numerically found in [13] and rigorously shown in [10,18] the existence of critical energies (in the sense of [18]; see ahead) at which the Lyapunov exponent vanishes; dynamical localization was obtained in [10] only after projecting onto closed energy intervals not containing such critical energies. Despite the similarity between the transfer matrices of the two models, it is not immediate the adaptation of the localization (delocalization) results to the BernoulliDirac model and each step needs to be verified; here, many points will not be detailed when they follow exactly the same lines of their Schrödinger counterpart.…”

Section: Introductionmentioning

confidence: 98%

“…With respect to nontrivial quantum transport, probed via dynamical delocalization (unbounded moments of the position operator), it was found in random polymer models [18] and in random palindrome models [7] (both including the important random dimer model), due to existence of critical energies [18]. Recently, for 1D discrete Schrödinger operators, Damanik, Sütő and Tcheremchantsev [9] have developed a general method which allows one to derive quantum dynamical lower bounds from upper bounds on the growth of norms of transfer matrices, and they applied this method to some substitution, Sturmian and prime models, among others.…”

Section: Introductionmentioning

confidence: 99%

“…By adapting the ideas of [18] (see also [9]) to ID(0, c) it will follow (see Theorem 5) that for an initial spinor Ψ well localized in space, there is 0 < C q < ∞ such that…”

Section: Introductionmentioning

confidence: 99%

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Spectral and localization properties for the one-dimensional Bernoulli discrete Dirac operator

Oliveira

Prado

2005

Journal of Mathematical Physics

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Abstract. A 1D Dirac tight-binding model is considered and it is shown that its nonrelativistic limit is the 1D discrete Schrödinger model. For random Bernoulli potentials taking two values (without correlations), for typical realizations and for all values of the mass, it is shown that its spectrum is pure point, whereas the zero mass case presents dynamical delocalization for specific values of the energy. The massive case presents dynamical localization (excluding some particular values of the energy). Finally, for general potentials the dynamical moments for distinct masses are compared, especially the massless and massive Bernoulli cases.

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Fine structure of the zeros of orthogonal polynomials IV: A priori bounds and clock behavior

Simon

2007

Comm Pure Appl Math

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Abstract. We prove locally uniform spacing for the zeros of orthogonal polynomials on the real line under weak conditions (Jacobi parameters approach the free ones and are of bounded variation). We prove that for ergodic discrete Schrödinger operators, Poisson behavior implies positive Lyapunov exponent. Both results depend on a priori bounds on eigenvalue spacings for which we provide several proofs.

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Delocalization in Random Polymer Models

Cited by 91 publications

References 23 publications

New Anomalous Lieb-Robinson Bounds in QuasiperiodicXYChains

New Anomalous Lieb-Robinson Bounds in QuasiperiodicXYChains

Spectral and localization properties for the one-dimensional Bernoulli discrete Dirac operator

Fine structure of the zeros of orthogonal polynomials IV: A priori bounds and clock behavior

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