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What is the physical meaning of magnetic length $\ell_B=\frac{\hbar c}{e B}$ in 2D electron system under magnetic field? When $\ell_B \longrightarrow a$, where $a$ is the lattice constant, does that mean the Landau sub-band is nearly flat?

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The physical meaning is the length of electron trajectory along which this electron gains phase factor comparable with $2\pi$ from the magnetic field. Normally, it is rather large. When is is as small as lattice constant, it means that the magnetic field is comparable with electric field in the atom which is rather rare situation.

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There is a deeper meaning of the magnetic length $l_B$. It is the lattice constant of a two-dimensional (2D) artificial structure. Let's consider a particle moving in 2D plane under a perpendicular uniform magnetic field of strength $B$. In classical mechanics, the system is obviously translationally invariant under any translation. However, in quantum mechanics, it is not so, because what enters the Schrodinger equation is the corresponding vector potential, which breaks this symmetry. Nevertheless, it is still possible to define an artificial lattice with primitive vectors $\vec{a}$ and $\vec{b}$, so that the translation by any $\vec{R} = m\vec{a} + n\vec{b}$ remains a symmetry, provided the area of the unit cell is $|\vec{a}\times\vec{b}|=l^2_B$. This means that, $l_B$ measures the lattice constant of this artificial lattice.

For more details, please see the book The Geometric Phase in Quantum Systems by A. Bohm et al. Also, this paper is very useful: PRL49:405(1982).

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