Occupancy-preserving second-order bosonic hopping
= Occupancy-preserving second-order bosonic hopping
{title2=$H_{\rm occ}^{(2)}=t[f(\Delta)XY-f(\Delta+2)YX]$}
With $X=\sum_s a_s^\dagger b_s$, $Y=X^\dagger$ and a real coefficient $f$, the fixed-occupancy second-order operator is $t[f(\Delta)XY-f(\Delta+2)YX]$. <Normal ordering> gives $XY=N_a+P$ and $YX=N_b+P$, where $P=\sum_{ss^\prime}a_s^\dagger a_{s^\prime}b_{s^\prime}^\dagger b_s$ exchanges internal states without changing site totals.