  While the Mott transition from a Fermi liquid is correctly believed to obtain
without the breaking of any continuous symmetry, we show that in fact a
discrete emergent $\mathbb Z_2$ symmetry of the Fermi surface is broken. The
extra $\mathbb Z_2$ symmetry of the Fermi liquid appears to be little known
although it was pointed out by Anderson and Haldane and we use it here to
classify all possible Fermi liquids topologically by invoking K-homology. It is
this $\mathbb Z_2$ symmetry breaking that signals the onset of particle-hole
asymmetry, a widely observed phenomenon in strongly correlated systems. In
addition from this principle, we are able to classify which interactions
suffice to generate the $\mathbb Z_2$-symmetry-broken phase. As this is a
symmetry breaking in momentum space, the local-in-momentum space interaction of
the Hatsugai-Kohmoto (HK) model suffices as well as the Hubbard interaction as
it contains the HK interaction. Both lie in the same universality class as can
be seen from exact diagonalization. We then use the Bott topological invariant
to establish the stability of a Luttinger surface. Our proof demonstrates that
the strongly coupled fixed point only corresponds to those Luttinger surfaces
with co-dimension $p+1$ with $p$ odd. Because they both lie in the same
universality class, we conclude that the Hubard and HK models are controlled by
this fixed point.
