  We show that the rate of closing of the energy gap between the ground state
and the first excited state, as a function of system size, behaves in many
qualitatively different ways at first-order quantum phase transitions of the
infinite-range quantum XY model. Examples include polynomial, exponential and
even factorially fast closing of the energy gap, all of which coexist along a
single axis of the phase diagram representing the transverse field. This
variety emerges depending on whether or not the transverse field assumes a
rational number as well as on how the series of system size is chosen toward
the thermodynamic limit. We conclude that there is no generically applicable
rule to relate the rate of gap closing and the order of quantum phase
transitions as is often implied in many studies, in particular in relation to
the computational complexity of quantum annealing in its implementation as
quantum adiabatic computation.
