Is a supernova explosion an essential condition for the formation of a neutron stars?

I am trying to understand why chemical potential $= U/2$ is considered to be at half-filling in the case of the Hubbard Model Hamiltonian. So when I substitute this in its Hamiltonian, this is the equation I am getting $$ H = K.E-\frac{U}{2}\Sigma [n_{j,up}n_{j,down} + (n_{j,up} - n_{j,down})^{...