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


 * $$\sigma^{ab}_{H} = \frac{e^2}{\hbar}\sum_n\int_{BZ}\frac{d^2\mathbf{k}}{(2\pi)^2}\mathcal{F}^{ab}_n(\mathbf{k})$$

Where the sum is made over fully occupied bands below the fermi Energy $$E_F$$

and where $$\mathcal{F}^{ab}_n$$ is the Berry cuvature:
 * $$\mathcal{F}^{ab}_n(\mathbf{k})=\frac{1}{2i}\int d^d\mathbf{r} \Big(\frac{\partial u^{*}_n}{\partial k_a}\frac{\partial u_n}{\partial k_b} - \frac{\partial u^{*}_n}{\partial k_b}\frac{\partial u_n}{\partial k_a} \Big)$$

And where the Berry phase is quantized


 * $$\int_{BZ}d^2\mathbf{k}\mathcal{F}^{ab}_n(\mathbf{k}) = 2\pi\ C_n$$

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:


 * $$\sigma_H = \frac{i e^2}{4\pi \hbar}\sum_{n} \oint_{BZB} d k_j \int d^2 r \Big( u^{*}_n\frac{\partial u_n}{\partial k_j} - \frac{\partial u^{*}_n}{\partial k_j}u^{*}_n \Big)$$

Hofstadter butterfly
In this case the sum is over the full set of bands below the Fermi energy $$E_F$$ 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
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.