Diophantine equation in the integer quantum hall effect

In summary, the conversation is about a person studying the IQHE and trying to understand the diophantine equation. They have read articles by Thouless and Kohmoto, but are still unclear about the link between the diophantine equation and the darboux theorem. They have also looked into the magnetic translation group but are struggling to make the connection. They are seeking help and have been provided with a readable derivation of the diophantine equation and a reference for using the magnetic translation group argument.
  • #1
alex022
1
0
Hi everybody!

I'm studying the IQHE and I want to understand the rise of the diophantine equation. I read the thouless article but it was no so conclusive.
I've also read the kohmoto article (phys rev B 1989) and he says that that property comes from the darboux theorem but i don't understand why… I've investigated the magnetic translation group but now I cannot do the link with the diophantine equation.

Can you help me?

Thanks a lot!

alex
 
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  • #2
Here is a readable derivation of the diophantine equation in IQHE:
http://www.jstor.org/stable/51904?seq=8
See section 5 in particular
 
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  • #3
See also the reference they cite for the derivation using magnetic translation group arguement.
 

Related to Diophantine equation in the integer quantum hall effect

1. What is a Diophantine equation in the integer quantum hall effect?

A Diophantine equation in the integer quantum hall effect is a type of mathematical equation that is used to describe the behavior of electrons in a two-dimensional electron gas under the influence of a magnetic field. It is named after the Greek mathematician Diophantus, who first studied equations with integer coefficients.

2. How is a Diophantine equation used in the study of the integer quantum hall effect?

In the study of the integer quantum hall effect, a Diophantine equation is used to calculate the Hall conductance, which is a measure of the resistance of a material to an electric current. This equation takes into account the number of electrons, the strength of the magnetic field, and the energy levels of the electrons.

3. What are the applications of the Diophantine equation in the integer quantum hall effect?

The Diophantine equation in the integer quantum hall effect has many applications in physics and materials science. It is used to understand the properties of materials and to design new materials with specific conductance values. It also has applications in the development of quantum computers and other electronic devices.

4. How does the Diophantine equation relate to the quantum Hall effect?

The Diophantine equation is closely related to the quantum Hall effect, which is a phenomenon where the Hall conductance of a material is quantized in discrete steps. This effect can be explained by the behavior of electrons in a two-dimensional electron gas under the influence of a magnetic field, which is described by the Diophantine equation.

5. Are there any challenges in using the Diophantine equation in the integer quantum hall effect?

One of the main challenges in using the Diophantine equation in the integer quantum hall effect is accurately accounting for the effects of disorder and interactions between electrons. These factors can significantly affect the behavior of the system and can make it difficult to obtain precise results from the equation. Additionally, the equation may not be applicable to systems with very high magnetic fields or temperatures.

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