Today I was puzzling over quantisation conditions for electrons in magnetic fields and learnt a lot from a paper, Topological Berry phase and semiclassical quantization of cyclotron orbits for two dimensional electrons in coupled band models

A few things I learnt:

The phase mismatch gamma which occurs in the semi-classical quantisation condition (for the wavefunction) is related to the

Aside: I am still confused as to exactly what a caustic is and how to visualise it.

This phase can be observed in the quantum Hall effect and deHaas van Alphen effect. In graphene it is found to have a different value (gamma=0) from conventional metals (gamma=1/2). This is usually stated as being due to Berry phase effects. But there is more to the story...

The phase parameter gamma_L which occurs in the energy quantisation condition is NOT necessarily the same as gamma. This is only the topological part of the Berry phase.

For a system with a gap the total Berry phase depends on the magnitude of the gap, whereas gamma_L does not.

A few things I learnt:

The phase mismatch gamma which occurs in the semi-classical quantisation condition (for the wavefunction) is related to the

**Maslov index**(number of**caustics)**in the classical periodic orbit.Aside: I am still confused as to exactly what a caustic is and how to visualise it.

This phase can be observed in the quantum Hall effect and deHaas van Alphen effect. In graphene it is found to have a different value (gamma=0) from conventional metals (gamma=1/2). This is usually stated as being due to Berry phase effects. But there is more to the story...

The phase parameter gamma_L which occurs in the energy quantisation condition is NOT necessarily the same as gamma. This is only the topological part of the Berry phase.

For a system with a gap the total Berry phase depends on the magnitude of the gap, whereas gamma_L does not.

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