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Quantized Hall Conductance in a Two-Dimensional Periodic Potential
Open accessAbstract
The Hall conductance of a two-dimensional electron gas has been studied in a uniform magnetic field and a periodic substrate potential $U$. The Kubo formula is written in a form that makes apparent the quantization when the Fermi energy lies in a gap. Explicit expressions have been obtained for the Hall conductance for both large and small $\frac{U}{\ensuremath{\hbar}{\ensuremath{\omega}}_{c}}$.
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“The corresponding Lorentz-like force in reciprocal space provides an anomalous velocity term which is responsible, e.g., for the anomalous and integer (Thouless et al., 1982) quantum Hall effects.”
From Topological photonics (2019) · cites this paper as Thouless et al., 1982
Experimental — accuracy not yet measuredThe cited source says what the claim attributes to it.
In the citing paper
“Topological consequences of Weyl nodes began to be explored with the realization that Berry curvature plays a key role in determining the Hall effect (Karplus and Luttinger, 1954; Thouless et al. , 1982), and the Weyl nodes are related to 'diabolic points' discussed by Berry as sources of Berry flux (Berry, 1985).”
From Weyl and Dirac semimetals in three-dimensional solids (2018) · cites this paper as Thouless et al., 1982
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In the citing paper
“Soon after, the extremely robust quantization of the Hall conductance was related to the topology of bands in momentum space by Thouless, Kohmoto, Nightingale, and den Nijs (TKNN) (Thouless et al., 1982).”
From Topological photonics (2019) · cites this paper as Thouless et al., 1982
In the citing paper
“In a twodimensional (2D) quantum Hall system, for example, the integral of the Berry curvature over the 2D Brillouin zone determines the Chern number: a global topological invariant that underlies the quantization of Hall transport for a 2D filled energy band 2 .”
From Experimental measurement of the Berry curvature from anomalous transport (2017) · cites this paper as [2]
In the citing paper
“When a lattice is put in a commensurate magnetic field (one with rational flux per unit cell, in units of the flux quantum so that Bloch's theorem applies), each occupied band j contributes an integer (48)”
From Three-Dimensional Topological Insulators (2011) · cites this paper as [48]
In the citing paper
“The sign of the Hall conductance depends on the sign of the effective Dirac mass, 10,31,32 which here is set by the direction of the Zeeman field at the surface.”
From Antiferromagnetic topological insulators (2010) · cites this paper as [10]
In the citing paper
“Note that the total conductance of a slab is always an integer, as required. 10,11,33”
From Antiferromagnetic topological insulators (2010) · cites this paper as [10]
In the citing paper
“These transitions can be regarded as a realization of the two-dimensional quantum Hall effect in zero net field discussed by Haldane. 10”
From Antiferromagnetic topological insulators (2010) · cites this paper as [10]
In the citing paper
“In this geometry this coupling vanishes as the cross products d 1 × d 2 from the two paths cancel.”
From Antiferromagnetic topological insulators (2010) · cites this paper as [10]
In the citing paper
“Within linear response theory and ignoring interparticle interactions, one can show that the Hall conductance σ xy of a two-dimensional insulator is (Thouless et al., 1982) σ xy ¼ - e 2 h X n C n”
From Topological photonics (2019) · cites this paper as Thouless et al., 1982
In the citing paper
“These works related the integer appearing in the Hall conductance to a topological invariant of the system, the Chern number, that is an integer-valued quantity which describes the global structure of the wave function in momentum space over the Brillouin zone.”
From Topological photonics (2019) · cites this paper as Thouless et al. (1982)
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“…Thouless et al proposed that the Hall conductivity exhibits the extremely robust quantization characteristics when the time-reversal ( T ) symmetry is broken, which is related to a nonzero topological invariant of the system [the Chern number or Thou-less–Kohomoto–Nightingale–den Nijs number [21]].”
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“They are quite different from the standard QPTs, since they do not involve symmetry breaking and are characterised by global order parameters [8].”
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Metadata
- Authors
- Thouless, D. J., Kohmoto, M., Nightingale, M. P., & den Nijs, M.
- Affiliations
- University of Washington (US)
- Venue
- Physical Review Letters
- Publisher
- American Physical Society (APS)
- Published
- 1982-08-09
- Volume
- 49
- Issue
- 6
- Pages
- 405-408
- ISSN
- 0031-9007 (ISSN-L)
- OpenAlex
- W2037807652
- Cited by
- 6,973
- References
- 10
- Citations by year
- 2026: 367 · 2025: 539 · 2024: 532 · 2023: 534 · 2022: 464 · 2021: 498 · 2020: 563 · 2019: 412 · 2018: 395 · 2017: 383 · 2016: 290 · 2015: 270 · 2014: 221 · 2013: 228 · 2012: 165
- FWCI
- 14.72
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Physical Review Letters (publishedVersion)
- Journal of Media Literacy Education (submittedVersion, cc-by)
- Funders
- National Science Foundation (7920785)
- Division of Materials Research
- Topics
- Physical Sciences › Physics and Astronomy › Condensed Matter Physics › Quantum and electron transport phenomena
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Semiconductor Quantum Structures and Devices
- Physical Sciences › Physics and Astronomy › Condensed Matter Physics › Surface and Thin Film Phenomena
- Keywords
- Hall conductance quantization, two-dimensional electron gas, uniform magnetic field, Kubo formula, Landau levels, quantum Hall effect
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