with the spontaneous breakdown of translation invariance? The usual breaking of a continuous global symmetry is associated with the occurrence of massless Goldstone bosons. For example, when translation invariance is spontaneously broken by the formation of a crystal lattice, phonons arise as massless excitations. In the quantum mechanical system studied here, there is no room for phonons because it has only a finite number of degrees of freedom. Indeed, the infinitely degenerate ground states are separated from the rest of the spectrum by a gap $\omega$. Still, just like a system with spontaneous symmetry breaking, the charged particle in a magnetic field may chose spontaneously from a continuous variety of degenerate ground states.

Coherent States
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Coherent states are well-known from the harmonic oscillator, and have also been constructed for the Landau level problem [@Fel70]. As usual, the coherent states are constructed as eigenstates of the annihilation operators, i.e. $$a |\lambda \lambda'\rangle = \lambda |\lambda \lambda'\rangle, \
b |\lambda \lambda'\rangle = \lambda' |\lambda \lambda'\rangle, \ 
\lambda, \lambda' \in {{\mathbb{C}}}.$$ In coordinate space, the coherent states can be expressed as $$\langle \vec x|\lambda \lambda' \rangle = A
\exp\left[- \frac{M \omega}{4} (x^2 + 2 i x y + y^2) + 
\sqrt{\frac{M \omega}{2}}
