Finite-ring parity qualification of an Ising quasiparticle gap

A bulk quasiparticle energy is not automatically the first excitation gap of a finite periodic spin chain. The Jordan–Wigner transformation fixes a boundary condition through fermion parity and restricts occupations to that same parity. At zero transverse field an Ising domain wall costs 2J, but periodic spin configurations contain an even number of walls; the first wall-pair excitation costs 4J. An ordered finite chain also has a distinct small symmetry-sector splitting at nonzero field. The thermodynamic dispersion and these finite-system scales answer different questions.