Usually, we define the momentum $k$-space Brillouin zone (by Fourier transformed from the real space $x$ with a wavefunction $\psi(x)$ to the momentum $k$-space) for:

(1) quadratic non-interacting (free) systems (such as those can be written in terms of BdG equation.)

and

(2) translational invariant systems (so one can define the conjugate momentum $k$ as a good quantum number).

Question: Could we define the momentum $k$-space Brillouin zone for

non-quadratic interacting systems

but translational invariant systems? (Namely can we modify (1) to interacting, but keep (2)?)

Usually, we define the momentum $k$-space Brillouin zone (by Fourier transformed from the real space $x$ with a wavefunction $\psi(x)$ to the momentum $k$-space) for:

(1) quadratic non-interacting (free) systems (such as those can be written in terms of BdG equation.)

and

(2) translational invariant systems (so one can define the conjugate momentum $k$ as a good quantum number).

Question: Could we define the momentum $k$-space Brillouin zone for

non-quadratic and interacting systems

but translational invariant systems? (Namely can we modify (1) to interacting, but keep (2)?)