= Solution
\b[A linear number-difference energy gives only a one-body level renormalization.] Let $\mathcal E(\Delta)=E_c+\gamma\Delta$, with $\gamma\ne0$. Then $f=t/(2\gamma)$ is constant, so the exchange coefficient cancels. Since $[X,Y]=\Delta$,
$$
\boxed{H^{(2)}_{\rm occ}=\frac{t^2}{2\gamma}\Delta.}
$$
The same-direction <commutators> vanish as well, so this is the whole second-order term. This agrees with expanding the two-site single-particle splitting $\sqrt{\gamma^2+t^2}$ with the appropriate assignment of levels. A collection of noninteracting <bosons> remains noninteracting after a change of single-particle basis. The case $\gamma=0$ is resonant and cannot be treated by this elimination.
\b[A nonlinear energy generally makes the virtual denominators occupancy-dependent.] Then $f(\Delta)$ and $f(\Delta+2)$ need not agree, and the occupancy-preserving effective <Hamiltonian> contains density-dependent shifts and a spin-exchange coefficient
$$
J(\Delta)=2t\,[f(\Delta)-f(\Delta+2)]
$$
multiplying $\mathbf s_a\cdot\mathbf s_b$. Its sign depends on the function and sector; nonlinearity alone does not fix a universal exchange sign. The full second-order result can additionally contain pair hopping between different occupation sectors.
For a concrete repulsive example, take $\mathcal E=U\Delta^2/4$, $U>0$, and restrict to two <bosons> with one on each site. Here $f(0)=-t/U$, $f(2)=t/U$, $N_a=N_b=1$, giving
$$
\boxed{H^{(2)}_{\rm occ}
=-\frac{t^2}{U}\left(3+4\mathbf s_a\cdot\mathbf s_b\right).}
$$
The triplet energy shift is $-4t^2/U$ and the singlet shift is zero. This is ferromagnetic <bosonic superexchange> in this example: the symmetric <spin> sector can lower its energy through virtual double occupancy. The sign must not be imported from the fermionic Hubbard problem. The example specifies an additional energy function solely to illustrate the general expression; it is not an asserted numerical input.
\Image[/past-exam-of-the-theoretical-physics-ii-course-of-the-university-of-cambridge/2016/paper-1-bosonic-exchange.png]
{title=Original virtual-hopping construction and exact versus perturbative spin-sector energies for the illustrative repulsive two-boson model}
Back to article page
