Past exam of the quantum Condensed Matter Field Theory course of the University of Cambridge / 2016 / Paper 1 / 1 / d / Solution

For occupations small compared with $S$, use $S^z=S-n$, $S^{-}\simeq \sqrt{2S}{a}^{†}$ and $S^{+}\simeq \sqrt{2S}a$. The longitudinal bond contributes $-J\Delta S^2+JS\Delta (n_m+n_{m+1})+O(S^0)$; the transverse pair terms contribute $JS(a_m{a}_{m+1}+a_m^{†}{a}_{m+1}^{†})+O(S^0)$. Thus the direct real-space expansion isTake $a_m=N^{-1/2}{\sum }_k{e}^{ikm}{a}_k$, with $k=2\pi j/N$ over one full Brillouin zone and the lattice spacing set to one in $k$. Symmetrizing $k$ and $-k$ changes the phase in the pair sum to $\cos k$, givingnormal ordering the second Nambu spinor component then yieldsThe printed matrix has a factor-of-two error on its diagonal. With a full-zone sum its printed $2\Delta$ would produce $4\Delta a_k^{†}{a}_k$, and even its vacuum expectation would differ from the direct expansion by $J\Delta NS$. A half-zone convention cannot repair both coefficients: one must then group each $k,-k$ pair and double its off-diagonal pair coefficient as well. The formula above follows from the specified Hamiltonian and has the expected gapless isotropic limit; subsequent physical results use this consistent expansion. Terms quartic in bosons first enter at order $S^0$.