Two-dimensional Bloch electrons in perpendicular magnetic fields: An exact calculation of the Hofstadter butterfly spectrum

Janecek, S.; Aichinger, Michael; Hernández, Eduardo

doi:10.1103/physrevb.87.235429

Cited by 12 publications

(13 citation statements)

References 42 publications

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“…The additional β-dependence of the parameters of the model can be interpreted as a dependence of the potential and hopping terms on the effective magnetic field, inducing a deformation of the spectrum with respect to the original Hofstadter's model [3]. Similar spectral deformations have also been found for realistic condensed-matter systems, due to, for instance, diamagnetic effects [51].…”

Section: The Spectrum Of Bichromatic Structuresmentioning

confidence: 53%

Topological edge states in bichromatic photonic crystals

Alpeggiani¹,

Kuipers²

2019

Optica

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Bichromatic photonic crystal structures are based on the coexistence of two different periodicities in the dielectric constant profile. They are realized starting from a photonic crystal waveguide and modifying the lattice constant only in the waveguide region. In this work, we numerically investigate the spectral and topological properties of bichromatic structures. Our calculations demonstrate that they provide a photonic analog of the integer quantum Hall state, a well known example of a topological insulator. The nontrivial topology of the bandstructure is illustrated by the formation of strongly localized, topologically protected boundary modes when finite-sized bichromatic structures are embedded in a larger photonic crystal.

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Section: The Spectrum Of Bichromatic Structuresmentioning

confidence: 53%

Topological edge states in bichromatic photonic crystals

Alpeggiani¹,

Kuipers²

2019

Optica

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“…6a), similar to graphene, the inter-Landau level coupling becomes important only for large amplitudes of the periodic potential, V 0 > 100 meV. This is true except for two degenerate LLs of type (29) and (30), for which the inter-level coupling becomes strong even for small amplitudes V 0 because of the degeneracy of the levels. In this case, the structure of the energy spectrum near zero energy becomes complicated because of the mixture of two degenerate butterfly structures.…”

Section: Bilayer Graphenementioning

confidence: 95%

“…The butterfly structure is no longer described by the simple Harper equation. In [29], a detailed analysis of the Hofstadter butterfly spectrum was done for strong and intermediate periodic potential strength. The magnetic field splits the Bloch bands and introduces coupling of the states of different Bloch bands.…”

Section: Introductionmentioning

confidence: 99%

Fractal butterflies of Dirac fermions in monolayer and bilayer graphene

Chakraborty¹,

Apalkov²

2015

IET Circuits, Devices & Systems

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We present an overview of the theoretical understanding of Hofstadter butterflies in monolayer and bilayer graphene. After a brief introduction on the past work in conventional semiconductor systems, we discuss the novel electronic properties of monolayer and bilayer graphene that helped to detect experimentally the fractal nature of the energy spectrum. We have discussed the theoretical background on the Moiré pattern in graphene. This pattern was crucial in determining the butterfly structure. We have also touched upon the role of electron-electron interaction in the butterfly pattern in graphene. We conclude by discussing the future prospects of butterfly search, especially for interacting Dirac fermions.

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“…by means of the Peierls substitution [4]. Our approach would also apply to the Hofstadter butterfly spectrum resulting from the exact (numerical) calculation, since the topological Wannier diagram is preserved [66].…”

Section: D Hofstadter Modelmentioning

confidence: 99%

Linking topological features of the Hofstadter model to optical diffraction figures

et al. 2022

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In two, three and even four spatial dimensions, the transverse responses experienced by a charged particle on a lattice in a uniform magnetic field are fully controlled by topological invariants called Chern numbers, which characterize the energy bands of the underlying Hofstadter Hamiltonian. These remarkable features, solely arising from the magnetic translational symmetry, are captured by Diophantine equations which relate the fraction of occupied states, the magnetic flux and the Chern numbers of the system bands. Here we investigate the close analogy between the topological properties of Hofstadter Hamiltonians and the diffraction figures resulting from optical gratings. In particular, we show that there is a one-to-one relation between the above mentioned Diophantine equation and the Bragg condition determining the far-field positions of the optical diffraction peaks. As an interesting consequence of this mapping, we discuss how the robustness of diffraction figures to structural disorder in the grating is a direct analogy of the robustness of transverse conductance in the Quantum Hall effect.

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Two-dimensional Bloch electrons in perpendicular magnetic fields: An exact calculation of the Hofstadter butterfly spectrum

Cited by 12 publications

References 42 publications

Topological edge states in bichromatic photonic crystals

Topological edge states in bichromatic photonic crystals

Fractal butterflies of Dirac fermions in monolayer and bilayer graphene

Linking topological features of the Hofstadter model to optical diffraction figures

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