Draft:TKNN formula
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Last edited by UtherSRG (talk | contribs) 5 months ago. (Update) |
The TKNN formula is a formula for topological band theory due to D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs. It was invented to explain various theoretical results such as the Hofstadter butterfly and experimental results such as the Integer Quantum Hall Effect.
It's one of the leading results in the motivation for the Nobel prize of 2016 in physics and is foundational result in regards to topological insulators.
The formula
[edit]Quantized Conductance in a Two-Dimensional Periodic Potential
Physical Review Letters 49, 405 (1982) D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs
Where the sum is made over fully occupied bands below the fermi Energy
and where is the Berry cuvature:
And where the Berry phase is quantized
and is a Chern number which is a characteristic number for each band.
Therefore the total Hall conductivity is Quantized:
2nd form:
Hofstadter butterfly
[edit]In this case the sum is over the full set of bands below the Fermi energy in the spectrum
The electrons are modeled as a fluid of independent particles with an infinite set of phases, one per .. band or ...
Quantum Hall Effect
[edit]The TKNN formula can explain the levels of the Integer quantum Hall Effect as a set of independent electrons.
The Fractional quantum Hall effect is considered an open research problem where the interaction between electrons becomes a major relevant factor.
See also
[edit]- Quantum Hall Effect
- Berry phase
- Adiabatic theorem
- Berry connection and curvature
- Chern number
- Integer quantum hall and Chern Simons Theories [3]