Anomalous Minimum and Scaling Behavior of Localization Length Near an Isolated Flat Band

Li, Ge

doi:10.1002/andp.201600182

Cited by 15 publications

(25 citation statements)

References 37 publications

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“…All the characteristic values mentioned above depend on the shape of the disorder distribution (see table 1). The numbers for uniform and Gaussian disorders agree with those to be read off from the figures of [16]. Finally, for the distributions we have considered, all these quantities are monotonic functions of the kurtosis κ.…”

Section: )supporting

confidence: 74%

“…This is the value of the inverse localization length for w ≪ |E| ≪ 1, i.e., |x| ≫ 1. This result can be recovered by more elementary means, along the lines of [16].…”

Section: )mentioning

confidence: 85%

“…First, the tight-binding model with Cauchy disorder, known as the Lloyd model [29], is solvable on any regular lattice in any dimension, in the sense that the average one-particle Green's function can be evaluated exactly. Second, an effective Cauchy disorder emerges in several instances of 1D structures possessing flat bands [12,13,14,15,16], including the pyrochlore ladder and the diamond chain, investigated in sections 4 and 5.…”

Section: Cauchy Disorder (The Lloyd Model)mentioning

confidence: 99%

“…For a usual dispersive band, ξ is known to diverge with exponent 2 inside the band, and with exponent 2/3 in the vicinity of band edges. More exotic exponents have been recently reported for various weakly disordered 1D structures having flat bands, including 0 for the stub chain, 1/2 for the cross-stitch and pyrochlore ladders, and 4/3 for the diamond chain and Lieb ladder [12,13,14,15,16].…”

Section: Introductionmentioning

confidence: 98%

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Scaling laws for weakly disordered 1D flat bands

Luck¹

2019

J. Phys. A: Math. Theor.

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We investigate Anderson localization on various 1D structures having flat bands. The main focus is on the scaling laws obeyed by the localization length at weak disorder in the vicinity of flat-band energies. A careful distinction is made between situations where the scaling functions are universal (i.e., depend on the disorder distribution only through its width) and where they keep depending on the full shape of the disorder distribution, even in the weak-disorder scaling regime. Three examples are analyzed in detail. On the stub chain, one central flat band is isolated from two lateral dispersive ones. The localization length remains microscopic at weak disorder and exhibits disorder-specific features. On the pyrochlore ladder, the two flat bands are tangent to a dispersive one. The localization length diverges with exponent 1/2 and a non-universal scaling law, whose dependence on the disorder distribution is predicted analytically. On the diamond chain, a central flat band intersects two symmetric dispersive ones. The localization length exhibits two successive scaling regimes, diverging first with exponent 4/3 and a universal law, and then (i.e., further away from the pristine flat band) with exponent 1 and a non-universal law. Both scaling functions are also derived by analytical means.

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Section: )supporting

confidence: 74%

“…This is the value of the inverse localization length for w ≪ |E| ≪ 1, i.e., |x| ≫ 1. This result can be recovered by more elementary means, along the lines of [16].…”

Section: )mentioning

confidence: 85%

Section: Cauchy Disorder (The Lloyd Model)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

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Scaling laws for weakly disordered 1D flat bands

Luck¹

2019

J. Phys. A: Math. Theor.

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“…5 depicts the relation between the asymptotic values of the averaged IPR and the asymmetric coupling; the system size is chosen N = 40; χ c is approximately proportional to 1 − α/β. The localization length of the eigenstates is [ln( α/β)] −1 [94].…”

Section: Nonuniform Systemmentioning

confidence: 99%

Non-Hermitian phase transition and eigenstate localization induced by asymmetric coupling

2019

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We investigate a uniformly coupled non-Hermitian system with asymmetric coupling amplitude. The asymmetric coupling equals to a symmetric coupling threaded by an imaginary gauge field. In a closed configuration, the imaginary gauge field leads to an imaginary magnetic flux, which induces a non-Hermitian phase transition. For an open boundary, the imaginary gauge field results in an eigenstate localization. The eigenstates under Dirac and biorthogonal norms and the scaling laws are quantitatively investigated to show the affect of asymmetric coupling induced one-way amplification. However, the imaginary magnetic flux does not inevitably induce the non-Hermitian phase transition for systems without translation invariance, this is elucidated from the non-Hermitian phase transition in the non-Hermitian ring with a single coupling defect. Our findings provide insights into the non-Hermitian phase transition and one-way localization.

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Artificial flat band systems: from lattice models to experiments

Leykam

Andreanov

Flach

2018

Advances in Physics: X

503

400

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Certain lattice wave systems in translationally invariant settings have one or more spectral bands that are strictly flat or independent of momentum in the tight binding approximation, arising from either internal symmetries or fine-tuned coupling. These flat bands display remarkable stronglyinteracting phases of matter. Originally considered as a theoretical convenience useful for obtaining exact analytical solutions of ferromagnetism, flat bands have now been observed in a variety of settings, ranging from electronic systems to ultracold atomic gases and photonic devices. Here we review the design and implementation of flat bands and chart future directions of this exciting field.

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Anomalous Minimum and Scaling Behavior of Localization Length Near an Isolated Flat Band

Cited by 15 publications

References 37 publications

Scaling laws for weakly disordered 1D flat bands

Scaling laws for weakly disordered 1D flat bands

Non-Hermitian phase transition and eigenstate localization induced by asymmetric coupling

Artificial flat band systems: from lattice models to experiments

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