Transport exponents of Sturmian Hamiltonians

Damanik, David; Gorodetski, Anton; Liu, Qinghui; Qu, Yan-Hui

doi:10.1016/j.jfa.2015.05.018

Cited by 8 publications

(14 citation statements)

References 36 publications

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“…In proving LR bounds however, we are naturally led to consider ℓ 1 -norms instead. Since this means we do not have orthogonality at our disposal, we need to develop an alternative approach 3 and we find that combining a resolution of the identity with Combes-Thomas estimates works. Though we do not state them explicitly, our approach yields pointwise (i.e.…”

Section: Proof Of the First Main Resultsmentioning

confidence: 99%

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On anomalous Lieb–Robinson bounds for the Fibonacci XY chain

Damanik¹,

Lemm²,

Lukić³

et al. 2016

J. Spectr. Theory

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Abstract. 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| α with 0 < α < 1. In fact, we can characterize the values of α for which such a bound holds as those exceeding α + 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, 4] 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.

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Section: Proof Of the First Main Resultsmentioning

confidence: 99%

“…The proof was based on methods from [25], which were since found to be flawed [3]. We believe that the extension can be proved by combining our methods with the ones in [3], but we leave this task to future work.…”

Section: Introductionmentioning

confidence: 98%

On anomalous Lieb–Robinson bounds for the Fibonacci XY chain

Damanik¹,

Lemm²,

Lukić³

et al. 2016

J. Spectr. Theory

Self Cite

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“…Remark 5.11. The proof is based on an inequality that is similar to 5.4, which is proved in [4,Proposition 4.8,b)] and is analogous to the proof of Proposition 5.1.…”

Section: Sturmian Potentialsmentioning

confidence: 94%

“…Such potentials are called Sturmian. The band structure of the spectrum analysis for this model was applied to estimating transport exponents for single-site initial states in [4]. The ideas described in the previous section can be used to extend some of their results to initial states with wide support.…”

Section: Sturmian Potentialsmentioning

confidence: 99%

Transport exponents of states with large support

Gerbuz

2019

Rev. Math. Phys.

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“…By the diffusion equation, the diffusion of particles is proportional to √ τ. If the measurements are not performed, generally the transport property of quantum many body systems should be influenced by the potential [5][6][7][8]. For example, if the potential is random, the system shows the localization (Anderson localization) [9][10][11] and if the potential is periodic the system shows the ballistic transport [12].…”

Section: Introductionmentioning

confidence: 99%

Stochastic Process Emerged from Lattice Fermion Systems by Repeated Measurements and Long-Time Limit

Yamaga

2020

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It is known that, in quantum theory, measurements may suppress Hamiltonian dynamics of a system. A famous example is the ‘Quantum Zeno Effect’. This is the phenomena that, if one performs the measurements M times asking whether the system is in the same state as the one at the initial time until the fixed measurement time t, then survival probability tends to 1 by taking the limit M→∞. This is the case for fixed measurement time t. It is known that, if one takes measurement time infinite at appropriate scaling, the ‘Quantum Zeno Effect’ does not occur and the effect of Hamiltonian dynamics emerges. In the present paper, we consider the long time repeated measurements and the dynamics of quantum many body systems in the scaling where the effect of measurements and dynamics are balanced. We show that the stochastic process, called the symmetric simple exclusion process (SSEP), is obtained from the repeated and long time measurements of configuration of particles in finite lattice fermion systems. The emerging stochastic process is independent of potential and interaction of the underlying Hamiltonian of the system.

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Transport exponents of Sturmian Hamiltonians

Cited by 8 publications

References 36 publications

On anomalous Lieb–Robinson bounds for the Fibonacci XY chain

On anomalous Lieb–Robinson bounds for the Fibonacci XY chain

Transport exponents of states with large support

Stochastic Process Emerged from Lattice Fermion Systems by Repeated Measurements and Long-Time Limit

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