= Solution
Write $a_k=g-\cos k$, $s_k=\sin k$ and $r_k=\sqrt{a_k^2+s_k^2}$. The <Ising-chain Bogoliubov diagonalization> uses the <eigenvalues> $\pm r_k$ of the two-by-two Nambu matrix. For distinct paired modes $k\ne-k$, choose $\cos\theta_k=a_k/r_k$, $\sin\theta_k=-s_k/r_k$ and $\theta_{-k}=-\theta_k$. The rotation $U_k=e^{-i\theta_k\sigma_y/2}$ satisfies $U_k^\dagger M_kU_k=r_k\sigma_z$, and its first transformed component is
$$
\gamma_k=\cos(\theta_k/2)c_k-i\sin(\theta_k/2)c_{-k}^\dagger.
$$
The opposite signs of the paired angles preserve the <canonical anticommutation relations>. <Self-paired spinless fermion modes> have zero pairing and are treated directly as occupied or empty number levels; they do not require this paired-angle formula.
Both $k$ and $-k$ occur in the sum, so the physical <quasiparticle> coefficient is $2Jr_k$, not $Jr_k$. Consequently
$$
\boxed{H=\sum_k\epsilon_k(\gamma_k^\dagger\gamma_k-1/2),\qquad
\epsilon_k=2J\sqrt{1+g^2-2g\cos k},\qquad E_0=-\frac12\sum_k\epsilon_k.}
$$
This is the bulk quadratic result under the stipulated boundary simplification. Restoring the finite-chain parity sectors adjusts allowed modes and global excitation constraints. In the <thermodynamic limit>, the <ground-state energy> per site is
$$
\frac{E_0}{N}=-\frac J{2\pi}\int_{-\pi}^{\pi}\sqrt{1+g^2-2g\cos k}\,dk.
$$
For $g\ge0$, the bulk <quasiparticle> gap is $2J|g-1|$. The requested three regimes are shown in the original sketch and discussed separately below.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2013/iii/paper-75-ising-dispersion.png]
{title=Ising-chain quasiparticle dispersion is flat at zero field and gapless at the critical field}
{height=360}
Back to article page
