  We show for the one-dimensional quantum $XY$ model with $s=1/2$ that the
energy gap between the ground and first excited states behaves anomalously as a
function of the system size at the first-order quantum phase transition.
Although it is generally the case that the energy gap closes exponentially at a
quantum first-order transition, the gap in the present model behaves
non-monotonically as a function of the system size, apparently very irregularly
in some cases. This property of the gap is similar to that of the
infinite-range quantum $XY$ model, in which the gap closes polynomially,
exponentially, or even factorially fast depending on the choice of the series
of system sizes toward the thermodynamic limit. This observation is surprising
in consideration of the apparent maturity of our understanding of the
one-dimensional quantum $XY$ model. Our result is also important from the
viewpoint of quantum annealing, where the rate of gap closing determines the
efficiency of computation.
