Anomalous dynamical scaling and bifractality in the one-dimensional Anderson model

Arias, S. de Toro; Luck, J. M.

doi:10.1088/0305-4470/31/38/007

Cited by 19 publications

(21 citation statements)

References 20 publications

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“…When n 0 is closer to the middle of the chain [ Fig. 3 (l)], only the t −1 decay is seen [59,60], which is the asymptotic behavior of [J 0 (2t)] 2 in Eq. (26).…”

Section: A Survival Probabilitymentioning

confidence: 94%

Level repulsion and dynamics in the finite one-dimensional Anderson model

2019

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This work shows that dynamical features typical of full random matrices can be observed also in the simple finite one-dimensional (1D) noninteracting Anderson model with nearest neighbor couplings. In the thermodynamic limit, all eigenstates of this model are exponentially localized in configuration space for any infinitesimal onsite disorder strength W . But this is not the case when the model is finite and the localization length is larger than the system size L, which is a picture that can be experimentally investigated. We analyze the degree of energy-level repulsion, the structure of the eigenstates, and the time evolution of the finite 1D Anderson model as a function of the parameter ξ ∝ (W 2 L) −1 . As ξ increases, all energy-level statistics typical of random matrix theory are observed. The statistics are reflected in the corresponding eigenstates and also in the dynamics. We show that the probability in time to find a particle initially placed on the first site of an open chain decays as fast as in full random matrices and much faster that when the particle is initially placed far from the edges. We also see that at long times, the presence of energy-level repulsion manifests in the form of the correlation hole. In addition, our results demonstrate that the hole is not exclusive to random matrix statistics, but emerges also for W = 0, when it is in fact deeper. arXiv:1904.11989v2 [cond-mat.dis-nn]

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“…When n 0 is closer to the middle of the chain [ Fig. 3 (l)], only the t −1 decay is seen [59,60], which is the asymptotic behavior of [J 0 (2t)] 2 in Eq. (26).…”

Section: A Survival Probabilitymentioning

confidence: 94%

Level repulsion and dynamics in the finite one-dimensional Anderson model

2019

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“…The Fourier integral in Eq. 5 can be then performed exactly and the wavefunction at the n-th site obtained in terms of the n-th order Bessel functions as [20,21,36]: ψ(n, t) = i −n J n (v e t). The argument of the Bessel functions is proportional to the time t, with proportionality factor v e , the maximum of the group velocity, v e = max |v g | = | dω(q) dq | = 2g 1 .…”

Section: Extended Quantum Walk Model: Long-time Dynamicsmentioning

confidence: 99%

“…e iϕ(n,q)t ; ϕ(n, q) ≡ n t q − w(q) (6) We note that under site reflection symmetry, ϕ(−n, q) = ϕ(n, −q). The saddle point approximation assumes that the dominant contribution to the integral comes from a small region of q around the saddle point solutions q * satisfying the equation [20,22]:…”

Section: Extended Quantum Walk Model: Long-time Dynamicsmentioning

confidence: 99%

“…Hence near ordinary fronts, p(n, t) ≈ 1 t while it exhibits sub-diffusive behaviour (smaller than 1/t) at extremal fronts. In particular, near the maximal fronts, the probability density scales as 1/t 2/3 [20,22].…”

Section: Extended Quantum Walk Model: Long-time Dynamicsmentioning

confidence: 99%

“…The spread of the wave packet is bounded by the existence of a maximal group velocity with which the fronts can propagate. The local probability exhibits an anomalous sub-diffusive scaling in the front region [20,22]. Existence of a maximal velocity at which information can spread or correlations can develop has long been known in the context of non-relativistic spin systems [23].…”

Section: Introductionmentioning

confidence: 99%

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Light-cone and local front dynamics of a single-particle extended quantum walk

Bhandari

Durganandini

2019

Phys. Rev. A

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We study the light-cone and front dynamics of a single particle continuous time extended quantum walk on a one dimensional lattice with finite range hopping. We show that, in general, for an initially localized state, propagating wave fronts can be characterized as ordinary or extremal fronts with the latter exhibiting an anomalous sub-diffusive scaling behaviour in the front region. We investigate the dynamical global and local scaling properties of the cumulative probability distribution function for the extended walk with nearest and next-nearest neighbour hopping using analytical and numerical methods. The global scaling shows the existence of a 'causal light-cone' corresponding to excitations travelling with a velocity smaller than a maximal 'light velocity'. Maximal fronts moving with fixed 'light velocity' bound the causal cone. The front regions spread with time sub-diffusively exhibiting a local Airy scaling which leads to an internal staircase structure. At a certain critical next-nearest neighbour hopping strength, there is a transition from a phase with one 'causal cone' to a phase with two nested 'causal cones' and the existence of an internal staircase structure in the corresponding cumulative distribution profiles. We also connect the study to that in spin chain systems and indicate that a single particle quantum walk on the one dimensional lattice already captures the many body physics of a spin chain system. In particular, we suggest that the time evolution of a single particle quantum walk on the one dimensional lattice with an initially localized state is equivalent to the time evolution of a domain wall initial state in a corresponding spin chain system. probability density p(n, t) = |ψ(n, t)| 2 scales as

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Quantum diffusion and scaling of localized interacting electrons

Halfpap

2001

Annalen der Physik

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A new numerical method is introduced that enables a reliable study of disorder‐induced localization of interacting particles. It is based on a quantum mechanical time evolution calculation combined with a finite size scaling analysis. The time evolution of up to four particles in one dimension is studied and localization lengths are defined via the long‐time saturation values of the mean radius, the inverse participation ratio and the center of mass extension. A systematic study of finite size effects using the finite size scaling method is performed in order to extract the localization lengths in the limit of an infinite system size. For a single particle, the well‐known scaling of the localization length λ1 with disorder strength W is observed, λ1 ∝ W—2. For two particles, an interaction‐induced delocalization is found, confirming previous results obtained by numerically calculating matrix elements of the two‐particle Green's function: in the limit of small disorder, the localization length increases with decreasing disorder as λ2 ∝ W—4 and can be much larger than <$>\mitlambda λ1. For three and four particles, delocalization is even stronger. Based on analytical arguments, an upper bound for the n‐particle localization length λn is derived and shown to be in agreement with the numerical data, λn ∝ λ1. Although the localization length increases superexponentially with particle number and can become arbitrarily large for small disorder, it does not diverge for finite λ1 and n. Hence, no extendedstates exist in one dimension, at least for spinless fermions.

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Anomalous dynamical scaling and bifractality in the one-dimensional Anderson model

Cited by 19 publications

References 20 publications

Level repulsion and dynamics in the finite one-dimensional Anderson model

Level repulsion and dynamics in the finite one-dimensional Anderson model

Light-cone and local front dynamics of a single-particle extended quantum walk

Quantum diffusion and scaling of localized interacting electrons

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