= Solution
For nonzero <Magnetic field>[field], the <Spin wave>[spin-wave] <ground state> is the <Fock vacuum>[vacuum] of all $b_k$. The <Bogoliubov transformation>[inverse Bogoliubov transformation] gives $\langle a_k^\dagger a_q\rangle=\delta_{kq}\sinh^2\theta_k$ and vanishing one-point oscillator averages. <Translation invariance>[Translational invariance] therefore gives the <One-dimensional planar spin-wave depletion>[depletion]
$$
\boxed{\delta S=\langle n_j\rangle=\frac1L\sum_k\sinh^2\theta_k
=\frac1{2L}\sum_k\left[\frac{H_f+2J-J\cos k}{\sqrt{(H_f+2J)[H_f+2J(1-\cos k)]}}-1\right].}
$$
Rotating back to the physical axes yields, to the retained <Spin wave>[spin-wave] order,
$$
\boxed{\langle S_j^x\rangle=s(S-\delta S),\qquad \langle S_j^y\rangle=0,\qquad\langle S_j^z\rangle=0.}
$$
The zero transverse components also follow from <rotation> by $\pi$ about the physical $x$ axis in a symmetry-preserving <ground state>. The nonzero <One-dimensional planar spin-wave depletion>[depletion] describes <Quantum fluctuation>[zero-point quantum fluctuations]: the original <boson> <Fock vacuum>[vacuum] is not the <ground state> because the <Bosonic Bogoliubov diagonalization>[anomalous terms] create pairs. It is nonnegative and must satisfy $\delta S\ll S$ for the expansion to be reliable. At strong <Magnetic field>[field] it tends to zero, with $\delta S\sim J^2/(8H_f^2)$ in the infinite-chain limit.
For fixed $H_f>0$, the <Sum>[momentum sum] has a finite <thermodynamic limit> obtained by $L^{-1}\sum_k\to(2\pi)^{-1}\int_{-\pi}^{\pi}dk$. At zero <Magnetic field>[field] the uniform mode already has no oscillator <Fock vacuum>[vacuum], as explained in part d. Even after removing that mode, the nonzero modes behave as
$$
\sinh^2\theta_k=\frac{1}{2\sqrt2|k|}-\frac12+O(|k|),\qquad
\delta S\sim\frac{\log L}{2\pi\sqrt2}+O(1).
$$
Thus assigning a finite ordered <moment> by simply setting $h=0$ in the infinite <Sum>[momentum sum] is invalid. The divergence indicates failure of this expansion about a fixed planar orientation; it is not a negative physical <magnetization>. At finite size an exact zero-field state chosen to have definite total $S^z$ has zero planar one-point <magnetization> by the <Global spin-rotation symmetry about one axis>[continuous rotation symmetry]. Establishing its full <correlations> requires a treatment beyond the divergent ordered-state approximation.
\Image[/past-exam-of-the-theoretical-physics-major-option-course-of-the-university-of-oxford/2017-spin-wave-gap-and-depletion.png]
{title=<Spin wave>[Spin-wave] <dispersion relation>[dispersion] and <one-dimensional planar spin-wave depletion>[zero-point depletion] for the <XY model>[planar ferromagnetic chain] in a <magnetic field>[longitudinal field]}
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