I'm new to Hubbard model and I have a few questions about it. From the sources I can find on the internet, Fermionic-Hubbard model is often written as (please correct me if I'm wrong!) \begin{align} H &= H_{hop} + H_{int}+ H_{chem} \\ &=\sum_{i.j\sigma} t_{ij}\, (c_{i,\sigma}^{\dagger}c_{j,\sigma}+H.c.) \,+\, \sum_i U_i \,n_{i,\uparrow}n_{i,\downarrow} \,-\, \sum_i\mu\,(n_{i,\uparrow}+n_{i,\downarrow}). \end{align}
My first question is , why is the hermitian conjugate term in $$H_{hop}$$ sometimes omitted?
The second question is, why do we need the chemical potential term in this model but not in others let's say Hartree-Fock Hamiltonian?