  We show that the general low-energy Bogoliubov-de Genness Hamiltonian in a
multiband superconductor with broken time reversal and preserved inversion
symmetry is a generator of real four-dimensional representation of $SO(4)$. In
the particular representation such an effective Hamiltonian is a purely
imaginary matrix, and it is proportional to the antisymmetric tensor of a
fictitious electromagnetic field which one can define in the momentum space.
The quantum time evolution of the low-energy quasiparticle state becomes this
way closely related to the classical relativistic motion of a charged particle
in the presence of the Lorentz force that would be derived from such an
electromagnetic field configuration. The condition for the emergence of a
Bogoliubov-Fermi surface can then be understood as orthogonality of the
fictitious electric and magnetic fields, which would allow zero Lorentz force.
The corresponding zero-energy eigenstates are identified as the physical
timelike and the unphysical spacelike solutions of the Lorentz force equation.
We study the looming instability of the inversion-symmetric Bogoliubov-Fermi
surface in presence of electron-electron interaction by formulating a concrete
interacting model on the Lieb lattice that features the requisite $SO(4)$
kinetic energy term together with nearest-neighbor two-body repulsion. The
latter is shown to favor dynamical breaking of the inversion symmetry. The
inversion symmetry in our lattice model indeed becomes spontaneously broken at
zero temperature at infinitesimal repulsion, with the original Bogoliubov-Fermi
surface deformed and reduced in size. General features of this symmetry
breaking phenomenon are discussed and a comparison with other works in
literature is presented.
