Question

Most explanations of the integer quantum Hall effect start out in the grand canonical ensemble, where the plateaus arise when the chemical potential (or equivalently the Fermi energy) is in the gaps between the Landau levels. However, they then usually continue with a statement that working at fixed chemical potential is actually imprecise as the system is in a canonical ensemble (i.e. fixed particle number,) and so we need to include disorder to get the plateaus. While I understand why disorder is needed if we work at fixed particle number, I don't understand why the experimental systems are assumed to be in the canonical ensemble to begin with. Don't electrons flow in/out of the material when you measure resistivity? If this is the case, why is working in the canonical ensemble justified?

Answer 1

The assumption that experimental systems in the integer quantum Hall effect (IQHE) are in the canonical ensemble is justified because the number of electrons in the system remains fixed. Disorder is introduced in the system to explain the plateaus observed in the IQHE.

Step-by-step explanation:

The assumption that experimental systems in the integer quantum Hall effect (IQHE) are in the canonical ensemble is justified because the number of electrons in the system remains fixed. Although electrons can flow in and out of the material when measuring resistivity, the overall number of electrons does not change significantly. The canonical ensemble assumes a fixed particle number, which is a reasonable approximation for IQHE systems.

Disorder is introduced in the system to explain the plateaus observed in the IQHE. Disorder helps in creating localized states that act as impurities, which trap electrons and lead to the formation of plateaus in the Hall resistance. The disorder breaks the translational symmetry and allows for the quantization of the Hall resistance in the presence of a magnetic field.