Topological Physics – Phenomena in physics controlled by the topology (often: the homotopy theory) of the physical system.
General theory:
In electromagnetism:
A physical system in solid state physics consisting of electrons confined to an essentially 2-dimensional surface and subject to a perpendicular magnetic field.
In an appropriate limit of low temperature aspects of this system are described by the topological quantum field theory called Chern-Simons theory. In this limit the system has been proposed as constituting a possible implementation of topological quantum computation.
The bulk/edge behaviour in a quantum Hall effect is is that of a topological insulator. (While topological insulator materials typically show this behaviour without the need of a strong magnetic field.)
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Review:
Klaus von Klitzing, The quantized Hall effect, Rev. Mod. Phys. 58, 519 (1986) (doi:10.1103/RevModPhys.58.519)
The quantum Hall effect (pdf)
See also:
Wikipedia, Quantum Hall effect,
Wikipedia Fractional quantum Hall effect
Original experimental detection:
Klaus von Klitzing, G. Dorda, and M. Pepper, New Method for High-Accuracy Determination of the Fine-Structure Constant Based on Quantized Hall Resistance, Phys. Rev. Lett. 45, 494 (1980) (doi:10.1103/PhysRevLett.45.494)
M. A. Paalanen, D. C. Tsui, and A. C. Gossard, Quantized Hall effect at low temperatures, Phys. Rev. B 25, 5566(R) April 1982 (doi:10.1103/PhysRevB.25.5566)
While an intuitive understanding for the quantization of the Hall conductance has been given in
a theoretical derivation of the effect was obtained only much later in
with closely related results in
Review of this theory behind the quantum Hall effect:
Yosi Afron, Why is the Hall conductance quantized?, 2017 (pdf, pdf)
Spyridon Michalakis, Why is the Hall conductance quantized?, Nature Reviews Physics 2, 392–393 (2020) (doi:10.1038/s42254-020-0212-6)
S. Klevtsov, X. Ma, G. Marinescu, P. Wiegmann, Quantum Hall effect and Quillen metric Commun. Math. Phys. 349, 819–855 (2017) doi
A quick review of the description via Chern-Simons theory with further pointers is in the introduction of
J. Bellissard introduced an approach via noncommutative geometry and Connes-Chern character:
In terms of Berry phase and Chern numbers in
Realization via AdS/CFT in condensed matter physics:
References on anyon-excitations (satisfying braid group statistics) in the quantum Hall effect (for more on the application to topological quantum computation see the references there):
The prediction of abelian anyon-excitations in the quantum Hall effect (i.e. satisfying braid group statistics in 1-dimensional linear representations of the braid group):
B. I. Halperin, Statistics of Quasiparticles and the Hierarchy of Fractional Quantized Hall States, Phys. Rev. Lett. 52, 1583 (1984) (doi:10.1103/PhysRevLett.52.1583)
Erratum Phys. Rev. Lett. 52, 2390 (1984) (doi:10.1103/PhysRevLett.52.2390.4)
Daniel Arovas, J. R. Schrieffer, Frank Wilczek, Fractional Statistics and the Quantum Hall Effect, Phys. Rev. Lett. 53, 722 (1984) (doi:10.1103/PhysRevLett.53.722)
The original discussion of non-abelian anyon-excitations in the quantum Hall effect (i.e. satisfying braid group statistics in higher dimensional linear representations of the braid group, related to modular tensor categories):
Review:
Last revised on August 19, 2021 at 08:27:19. See the history of this page for a list of all contributions to it.