I am reading the famous and concise Thouless-Kohmoto-Nightingale-den Nijs (TKNN) paper Quantized Hall Conductance in a Two-Dimensional Periodic Potential, Phys. Rev. Lett. 49, 405–408 (1982), where I have several subtle questions about the details.

At the beginning, it says with a weak periodic potential and flux $\varphi = p/q, p,q\in \mathbb Z$ per unit cell, "each Landu level is split into $p$ subbands..." But on the last page, it says with strong periodic potential, $p$ and $q$ are interchanged.

  • Are there $q$ subbands in the latter case?

  • Why $\varphi \mapsto 1/\varphi$ in the strong potential case?

In the figure below,

  • what do the arrows and dots all mean?

  • How do they explain the quantized conductance?!

enter image description here

Any ideas? Many thanks! To be continued...

  • $\begingroup$ If the Fermi energy lies in a gap the Hall conductance can be written as a chern number and is therefore quantized. $\endgroup$ – jjcale Jun 7 '13 at 19:30

Answer to the question $\varphi\mapsto 1/\varphi$:

In the strong magnetic field limit, the eigenstates are Landau wavefunctions, i.e. each electron is dancing around one flux quantum. Now turn on the superlattice, we arrive at the magnetic unit cell, which encloses $p$ flux quanta (one unit cell encloses $p/q$), hence $p$ electrons.

In the strong lattice potential limit, Bloch states are eigenstates, the magnetic field enlarges the unit cell $q$ times, and there are $q$ electrons in the magnetic unit cell.

In either case, perturbations are considerd within a single band or level. If there are $p(q)$ electrons in one magnetic unit cell, there are $p(q)$ subbands.


After being frustrated by this figure also, I will explain my ideas thus far and hopefully others can help complete the picture or illuminate my misunderstandings.

The horizontal lines represent exchanges of states, band $r$ with $r\pm1$. For $p=5$, one expects exchanges at $4$ distinct energy levels which is the case in both figures. The large dots are electrons, and I believe the dashes and smaller dots are provided to distinguish the paths followed by different electrons.

In sub-figure (a) there are two closed orbits at the bottom of the first trough (one labelled by dots and the other by dashes). At the top of the first peak there are another two closed orbits. The remaining path labelled by dots is not closed and traverses the system exiting on the right. This single orbital travelling from left to right corresponds to the single Hall current expected from the central band.

In sub-figure (b) there are no closed orbits, there are three orbits that travel from right to left, corresponding to the negative currents and two travelling the opposite direction corresponding to the positive currents.

Some data

For the sake of completeness, I will provide some values from simple numerical calculations of $s_r$ for small $\frac{V}{V^{\prime}}$. The corresponding $t_r$, result from using the Diophantine equation: $r=qs_r + pt_r$, and $\sigma_H=\frac{e^2}{h}(t_r-t_{r-1})$, (using $e=h=1$).

For the case in sub-figure (a), with $\frac{p}{q} = 5$:
$s_1=1$, $t_1=0$, $(\sigma_H)_1=0$
$s_2=2$, $t_2=0$, $(\sigma_H)_2=0$
$s_3=-2$, $t_3=1$, $(\sigma_H)_3=1$
$s_4=-1$, $t_4=1$, $(\sigma_H)_4=0$
$s_5=0$, $t_5=1$, $(\sigma_H)_5=0$

And the case in sub-figure (b), with $\frac{p}{q} = \frac{5}{3}$:
$s_1=2$, $t_1=-1$, $(\sigma_H)_1=-1$
$s_2=-1$, $t_2=1$, $(\sigma_H)_2=2$
$s_3=1$, $t_3=0$, $(\sigma_H)_3=-1$
$s_4=-2$, $t_4=2$, $(\sigma_H)_4=2$
$s_5=0$, $t_5=1$, $(\sigma_H)_5=-1$


Such speculations are due to the absence in physics of ideas about the structure of electron. The latter is a vortex node in the form of trefoil. The model proposed by Lord Kelvin. This structure explains Lorentz force and all Hall effects as well. And phase transitions and topological two-dimensional fantasies should be left to mathematician. Physics are not there.


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