Landau level

A charged particle in a uniform magnetic field has equally spaced orbital energy levels with guiding-centre degeneracy.

The $n=0$ level of the transverse harmonic oscillator in a uniform magnetic field, with energy $\hslash |qB|/(2m)$ before longitudinal kinetic energy or spin contributions.

For a particle of charge $q$ in a vector potential $\mathbf{A}$, the canonical momentum in the free Hamiltonian is replaced by $\mathbf{p}-q\mathbf{A}$, giving

A particle of charge $q$ and mass $m$ in a uniform magnetic field of magnitude $B$ has cyclotron angular frequency ${\omega }_c=|qB|/m$.

For $\mathbf{B}=B{\mathbf{e}}_z$, the vector potential $\mathbf{A}=(0,Bx,0)$ is a Landau gauge. Translation invariance in $y$ turns the transverse Hamiltonian into a family of harmonic oscillators in $x$ whose centres are labelled by the conserved $y$-momentum.

In a planar region of area $A$, each spinless Landau level has guiding-centre degeneracyMagnetic periodic boundary conditions require this flux count to be an integer.

For a spin-one-half particle with the Pauli magnetic term and gyromagnetic factor two, the orbital energies $\hslash {\omega }_c(n+1/2)$ split into $\hslash {\omega }_{c}n$ and $\hslash {\omega }_c(n+1)$. The lowest level has one spin branch; every positive level has two.

A uniform electric field perpendicular to a uniform magnetic field shifts each cyclotron oscillator's guiding centre and makes its energy depend linearly on the conserved momentum. The formerly degenerate Landau levels therefore become tilted bands.

In uniform perpendicular electric and magnetic fields, a charged particle's guiding centre drifts with velocity ${\mathbf{v}}_D=\mathbf{E}\times \mathbf{B}/B^2$, independently of its charge and mass. For $\mathbf{E}=E{\mathbf{e}}_x$ and $\mathbf{B}=B{\mathbf{e}}_z$, this gives $v_y=-E/B$.

In Landau gauge with periodic boundary conditions of length $L_y$, adjacent guiding centres differ by $\Delta k_y=2\pi /L_y$. A perpendicular electric field tilts the energy by $-(E/B)\hslash k_y$, so adjacent states formerly in one degenerate Landau level are separated by $2\pi \hslash |E/B|/L_y$.

For a constant magnetic field,is the symmetric gauge.

For charge $e$, defineIn the symmetric gauge,

The kinetic momentum $\mathbf{\Pi }=\mathbf{p}-q\mathbf{A}$ equals $m\mathbf{v}$ for a nonrelativistic charged particle and is invariant under an electromagnetic gauge transformation.

For a lattice vector $r$,acts in the symmetric gauge asIt commutes with $(p-eA{)}^2/(2m)+V(x)$ when $V(x+r)=V(x)$.

For constant $B$,

Magnetic translations along two lattice vectors commute when their cell flux obeys