At zero field, $r_k=1$ and \b[$\epsilon_k=2J$ for every $k$]. The <Ising-chain quasiparticle dispersion> is flat: the bulk quadratic excitation costs no momentum-dependent energy. In spin language these are domain-wall excitations. A periodic finite spin configuration has an <even number> of domain walls, so the single bulk <quasiparticle> energy $2J$ does not imply an allowed isolated one-wall spin state; the first such closed-chain spin excitation costs $4J$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 75 / 1 / c / i / Solution

 Solution (source code)