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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Berry curvature plays a key role in determining the Hall effect.
“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
The integral of Berry curvature over the 2D Brillouin zone determines the Chern number, a global topological invariant underlying quantization of Hall transport for a filled 2D band.
“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]
When a lattice is put in a commensurate magnetic field, each occupied band j contributes an integer to the total Hall conductance.
“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]
The sign of the Hall conductance depends on the sign of the effective Dirac mass, which here is set by the direction of the Zeeman field at the surface.
“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]
The total conductance of a slab is always an integer.
“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]
These transitions can be regarded as a realization of the two-dimensional quantum Hall effect in zero net field discussed by Haldane.
“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 rhombus geometry, the spin-orbit coupling from X to Y vanishes because the cross products d1×d2 from the two paths cancel.
“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]
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Colloquium : Zoo of quantum-topological phases of matter
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Quantum Hall effects in a Weyl semimetal: Possible application in pyrochlore iridates
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From DFT to machine learning: recent approaches to materials science–a review
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Higher-order topology in bismuth
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Topological bands for ultracold atoms
Cooper, N. R., Dalibard, J., & Spielman, I. B. (2019) · Reviews of Modern Physics · doi:10.1103/revmodphys.91.015005 · Cited by 940 · see citing works
“As first shown by Thouless et al. (1982), a band insulator exhibits the integer quantum Hall effect if the total Chern number of the filled bands is nonzero, with a Hall conductivity quantized at C times the fundamental unit of conductance e2/h.”
BackgroundTopological phases in acoustic and mechanical systems
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“Corner R3 Corner R1 Corner R4 Corner R2 Corner R1 Corner R3 d e (1) Lower band (2) In-gap modes (3) Upper band”
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The Quantum Spin Hall Effect: Theory and Experiment
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- 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 › Atomic and Molecular Physics, and Optics › 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 › Atomic and Molecular Physics, and Optics › 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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