Edge states in the integer quantum Hall effect and the Riemann surface of the Bloch function

Hatsugai, Yasuhiro

doi:10.1103/physrevb.48.11851

Cited by 399 publications

(653 citation statements)

References 49 publications

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“…When one describes the Bloch electrons as graphene on the cylinder with the Landau gauge, similar decomposition of the two dimensional hamiltonian is possible. In this case, corresponding one dimensional system with parameter k y is not a simple harmonic oscillator but the Harper equation 14,19 . To describe the edge states ( bound states ) and the Bloch states ( scattering states ) on the same footing, we need to consider a complex energy surface.…”

Section: B Bulk -Edge Correspondencementioning

confidence: 99%

“…Also the number of edge modes, I j has also topological meaning. Here let us give rough idea of the topological meaning of the edge modes 14 . When one discusses usual electrons in a magnetic field B in a Landau gauge in x-direction, two dimensional hamiltonian is decomposed into the sum of the one dimensional hamiltonian of the harmonic oscillators (with parameter k y ).…”

Section: B Bulk -Edge Correspondencementioning

confidence: 99%

“…It is specified by considering edge states that is implicitly assumed in the Laghlin argument 14 . The Hall conductance is given by the number of edge modes in the energy gap that the Fermi energy lies.…”

Section: A Edge States Of the Quantum Hall Statesmentioning

confidence: 99%

“…Then the winding numbers of the loops around the holes ( gaps ) of the RS give the numbers of edge state, I j is defined. By using the Laughlin's argument, the winding number I j gives the Hall conductance when the chemical potential lies in the j-the gap 14 . In this way, the relation eq.…”

Section: B Bulk -Edge Correspondencementioning

confidence: 99%

“…Quantized value of the Hall conductance (in a suitable unit) is a number of electrons carried across the system when the AB flux is adiabatically increased by one flux quantum. Considering a system with edges, this quantized but ambiguous integer is uniquely specified by the number of edge modes I j in the j-th Landau gap where the Fermi energy lies 13,14 σ Edge xy = e 2 h I j . It implies that the edge states has essential importance for the physics of the quantum Hall effects.…”

Section: A Edge States Of the Quantum Hall Statesmentioning

confidence: 99%

See 4 more Smart Citations

Bulk-edge correspondence in graphene with/without magnetic field: Chiral symmetry, Dirac fermions and edge states

Hatsugai

2009

Solid State Communications

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There are two types of edge states in graphene with/without magnetic field. One is a quantum Hall edge state, which is topologically protected against small perturbation. The other is a chiral zero mode that is localized near the boundary with/without magnetic field. The latter is also topological but is guaranteed to be zero energy by the chiral symmetry, which is also responsible for massless Dirac like dispersion. Conceptual roles of the edge states are stressed and reviewed from a view point of the bulk-edge correspondence and the topological order.

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Section: B Bulk -Edge Correspondencementioning

confidence: 99%

Section: B Bulk -Edge Correspondencementioning

confidence: 99%

Section: A Edge States Of the Quantum Hall Statesmentioning

confidence: 99%

Section: B Bulk -Edge Correspondencementioning

confidence: 99%

Section: A Edge States Of the Quantum Hall Statesmentioning

confidence: 99%

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Bulk-edge correspondence in graphene with/without magnetic field: Chiral symmetry, Dirac fermions and edge states

Hatsugai

2009

Solid State Communications

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Topology and Symmetry in Quantum Materials

2022

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Interest in topological materials continues to grow unabated in view of their conceptual novelties as well as their potential as platforms for transformational new technologies. Electronic states in a topological material are robust against perturbations and support unconventional electromagnetic responses. The first‐principles band‐theory paradigm has been a key player in the field by providing successful prediction of many new classes of topological materials. This perspective presents a cross section through the recent work on understanding the role of geometry and topology in generating topological states and their responses to external stimuli, and as a basis for connecting theory and experiment within the band theory framework. In this work, effective strategies for topological materials discovery and impactful directions for future topological materials research are also commented.

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Topological phase transitions and Thouless pumping of light in photonic waveguide arrays

Qin

Mei

et al. 2016

Laser & Photonics Reviews

107

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Photonic waveguide arrays provide an excellent platform for simulating conventional topological systems, and they can also be employed for the study of novel topological phases in photonics systems. However, a direct measurement of bulk topological invariants remains a great challenge. Here we study topological features of generalized commensurate Aubry‐André‐Harper (AAH) photonic waveguide arrays and construct a topological phase diagram by calculating all bulk Chern numbers, and then explore the bulk‐edge correspondence by analyzing the topological edge states and their winding numbers. In contrast to incommensurate AAH models, diagonal and off‐diagonal commensurate AAH models are not topologically equivalent. In particular, there appear nontrivial topological phases with large Chern numbers and topological phase transitions. By implementing Thouless pumping of light in photonic waveguide arrays, we propose a simple scheme to measure the bulk Chern numbers.

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Edge states in the integer quantum Hall effect and the Riemann surface of the Bloch function

Cited by 399 publications

References 49 publications

Bulk-edge correspondence in graphene with/without magnetic field: Chiral symmetry, Dirac fermions and edge states

Bulk-edge correspondence in graphene with/without magnetic field: Chiral symmetry, Dirac fermions and edge states

Topology and Symmetry in Quantum Materials

Topological phase transitions and Thouless pumping of light in photonic waveguide arrays

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