Draft:TKNN formula

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

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

$\sigma _{H}^{ab}={\frac {e^{2}}{\hbar }}\sum _{n}\int _{BZ}{\frac {d^{2}\mathbf {k} }{(2\pi )^{2}}}{\mathcal {F}}_{n}^{ab}(\mathbf {k} )$

Where the sum is made over fully occupied bands below the fermi Energy $E_{F}$

and where ${\mathcal {F}}_{n}^{ab}$ is the Berry cuvature:

${\mathcal {F}}_{n}^{ab}(\mathbf {k} )={\frac {1}{2i}}\int d^{d}\mathbf {r} {\Big (}{\frac {\partial u_{n}^{*}}{\partial k_{a}}}{\frac {\partial u_{n}}{\partial k_{b}}}-{\frac {\partial u_{n}^{*}}{\partial k_{b}}}{\frac {\partial u_{n}}{\partial k_{a}}}{\Big )}$

And where the Berry phase is quantized

$\int _{BZ}d^{2}\mathbf {k} {\mathcal {F}}_{n}^{ab}(\mathbf {k} )=2\pi \ C_{n}$

and $C_{n}$ is a Chern number which is a characteristic number for each band.

Therefore the total Hall conductivity is Quantized:

$\sigma _{H}={\frac {e^{2}}{h}}\sum _{n}C_{n}$

2nd form:

$\sigma _{H}={\frac {ie^{2}}{4\pi \hbar }}\sum _{n}\oint _{BZB}dk_{j}\int d^{2}r{\Big (}u_{n}^{*}{\frac {\partial u_{n}}{\partial k_{j}}}-{\frac {\partial u_{n}^{*}}{\partial k_{j}}}u_{n}^{*}{\Big )}$

Hofstadter butterfly

Hofstadter butterfly is the graphical solution to Harper's equation, where the energy ratio $\epsilon$ is plotted as a function of the flux ratio $2\pi \alpha$.

In this case the sum is over the full set of bands below the Fermi energy $E_{F}$ in the spectrum

Hofstadter's butterfly with the distribution of Chern numbers one per color

The electrons are modeled as a fluid of independent particles with an infinite set of phases, one per .. band or ...

Quantum Hall Effect

The TKNN formula can explain the levels of the Integer quantum Hall Effect as a set of independent electrons.

Longitudinal and transverse (Hall) resistivity, $\rho _{xx}$ and $\rho _{xy}$, of a two-dimensional electron gas as a function of magnetic field. Both vertical axes were divided by the quantum unit of conductance $e^{2}/h$ (units are misleading). The filling factor $\nu$ is displayed for the last 4 plateau.

The Fractional quantum Hall effect is considered an open research problem where the interaction between electrons becomes a major relevant factor.

See also

• Quantum Hall Effect
• Berry phase
• Adiabatic theorem
• Berry connection and curvature
• Chern number
• Integer quantum hall and Chern Simons Theories