Past exam of the quantum Condensed Matter Field Theory course of the University of Cambridge / 2012 / Paper 1 / 1 / d / Solution

Put $d_k=\sqrt{(g-\cos k{)}^2+{\sin }^{2}k}$. The matrix in the Ising-chain Nambu spinor has eigenvalues $\pm d_k$. A canonical Bogoliubov transformation, with paired angles satisfying $\cos {\vartheta }_k=(g-\cos k)/d_k$ and $\sin {\vartheta }_k=-\sin k/d_k$, diagonalizes it. Accounting for full-zone fermionic Nambu energy counting givesThus the bulk Ising-chain ground-state energy isExact finite-chain states additionally obey their fermion parity sector.

The Ising-chain quasiparticle dispersion has bulk gap $2J|1-g|$, closing at the Ising-chain quantum critical point. Finite-system parity splittings and the bulk gap are distinct scales. The three requested limits are illustrated below.