Past exam of the theoretical physics II course of the University of Cambridge / 2016 / Paper 1 / 3 / d

The second-order terms aresoHere $S$ is the first-order generator. The full second-order commutator includes terms changing the number difference by four. To retain only fixed site occupancies, keep the products $XY$ and $YX$.

With $f=f(\Delta )$ and $f_{+}=f(\Delta +2)$, the mixed commutators giveThus occupancy-preserving second-order bosonic hopping givesFor bosons, normal ordering givesConsequently an explicit form in terms of the supplied function isThe operator $P$ exchanges spin labels between sites while preserving their total occupancies. Defining dimensionless Schwinger-boson spins ${\mathbf{s}}_a=\frac{1}{2}{a}_s^{†}{\mathbf{\sigma }}_{s{s}^{\prime }}{a}_{s^{\prime }}$ and similarly for $B$,No factor of ${\hslash }^2$ is needed for these dimensionless spins; for physical spin operators ${\mathbf{S}}_a=\hslash {\mathbf{s}}_a$ it becomes $2{\mathbf{S}}_a\cdot {\mathbf{S}}_b/{\hslash }^2$. The spin exchange belongs in the occupancy-preserving part, whereas two hops in the same direction do not.