Fermi oscillator

A Fermi oscillator has nilpotent lowering operator $B$, anticommutator $B{B}^{†}+B^{†}B=1$, and occupation Hamiltonian $B^{†}B$ with levels zero and one.

The fermionic occupation Hamiltonian satisfies $H^2=H$, so its spectrum lies in $\{0,1\}$.

The lowering operator sends the occupied state to the vacuum and kills the vacuum; its adjoint performs the reverse and kills the occupied state.