We reexamine the well-studied one-dimensional spin-1/2 XY model to reveal its nontrivial energy spectrum,
in particular the energy gap between the ground state and the first excited state. In the case of the isotropic
XY model, the XX model, the gap behaves very irregularly as a function of the system size at a second order
transition point. This is in stark contrast to the usual power-law decay of the gap and is reminiscent of the similar
behavior at the first order phase transition in the infinite-range quantum XY model. The gap also shows nontrivial
oscillatory behavior for the phase transitions in the anisotropic model in the incommensurate phase. We observe
a close relation between this anomalous behavior of the gap and the correlation functions. These results, those
for the isotropic case in particular, are important from the viewpoint of quantum annealing where the efficiency
of computation is strongly affected by the size dependence of the energy gap
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