 Occupancy-preserving second-order bosonic hopping

With $X={\sum }_s{a}_s^{†}{b}_s$, $Y=X^{†}$ 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 $P={\sum }_{s{s}^{\prime }}{a}_s^{†}{a}_{s^{\prime }}{b}_{s^{\prime }}^{†}{b}_s$ exchanges internal states without changing site totals.