  We study the effect of antiferromagnetic longitudinal coupling on the
one-dimensional transverse field Ising model with nearest-neighbour couplings.
In the topological phase where, in the thermodynamic limit, the ground state is
twofold degenerate, we show that, for a finite system of N sites, the
longitudinal coupling induces N level crossings between the two lowest lying
states as a function of the field. We also provide strong arguments suggesting
that these N level crossings all appear simultaneously as soon as the
longitudinal coupling is switched on. This conclusion is based on perturbation
theory, and a mapping of the problem onto the open Kitaev chain, for which we
write down the complete solution in terms of Majorana fermions.
