Localization for Random Block Operators Related to the XY Spin Chain

Chapman, Jacob W.; Stolz, Günter

doi:10.1007/s00023-014-0328-2

Cited by 27 publications

(47 citation statements)

References 33 publications

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“…A result in [12] covers the regime of large disorder in case {ν ξ } are iid with absolutely continuous distribution with a compact support. In [10] strong dynamical localization (2.43) is established for strong enough spin coupling |µ|.…”

Section: The Eigenfunction Correlator Of H N Exhibits Complete Strongmentioning

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: The Eigenfunction Correlator Of H N Exhibits Complete Strongmentioning

confidence: 99%

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

Sims

Warzel

2016

Commun. Math. Phys.

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“…A more detailed review of such results-including another new quantitative statement-follows. 4 We note that the case of an absolutely continuous probability distribution was already studied by E. Le Page.…”

mentioning

confidence: 89%

“…This corresponds to a GL 2 (R)-valued random multiplicative process whose probability distribution is a finite sum of point masses. 4 As a consequence of our general theory, we derive a local modulus of continuity (namely weak-Hölder) for the Lyapunov exponent regarded as a function of the support of the measure. Furthermore, this implies the same local modulus of continuity for the integrated density of states (IDS) of a random Jacobi operator, near every energy level with positive Lyapunov exponent.…”

Section: Introduction and Statementsmentioning

confidence: 99%

Large Deviations for Products of Random Two Dimensional Matrices

Duarte

Klein

2019

Commun. Math. Phys.

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We establish large deviation type estimates for i.i.d. products of two dimensional random matrices with finitely supported probability distribution. The estimates are stable under perturbations and require no irreducibility assumptions. In consequence, we obtain a uniform local modulus of continuity for the corresponding Lyapunov exponent regarded as a function of the support of the distribution. This in turn has consequences on the modulus of continuity of the integrated density of states and on the localization properties of random Jacobi operators. 1 arXiv:1810.08194v1 [math.DS] 18 Oct 2018 1 Irreducibility refers to the non existence of proper subspaces invariant under the closed semigroup T µ generated by the support of the measure µ. There are different versions of this notion.2Contractivity refers to the existence in T µ of matrices with arbitrarily large gaps between consecutive singular values. 3A more detailed review of such results-including another new quantitative statement-follows. 4 We note that the case of an absolutely continuous probability distribution was already studied by E. Le Page.

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“…To produce a nontrivial situation in the spirit of Theorem 1, it is furthermore sufficient to just have randomness of say the even sites, which is achieved by choosing λ od = 0. This particular situation is of interest for the study of certain random quantum spin chains [4].…”

Section: Examples and Numerical Illustrationmentioning

confidence: 99%

Pseudo-gaps for random hopping models

Dorsch

Schulz-Baldes

2020

J. Phys. A: Math. Theor.

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For one-dimensional random Schrödinger operators, the integrated density of states is known to be given in terms of the (averaged) rotation number of the Prüfer phase dynamics. This paper develops a controlled perturbation theory for the rotation number around an energy, at which all the transfer matrices commute and are hyperbolic. Such a hyperbolic critical energy appears in random hopping models. The main result is a Hölder continuity of the rotation number at the critical energy that, under certain conditions on the randomness, implies the existence of a pseudo-gap. The proof uses renewal theory. The result is illustrated by numerics.

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Localization for Random Block Operators Related to the XY Spin Chain

Cited by 27 publications

References 33 publications

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

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

Large Deviations for Products of Random Two Dimensional Matrices

Pseudo-gaps for random hopping models

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