Energy gap at first-order quantum phase transitions: An anomalous case

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 infiniterange 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 for relating the rate of gap closing to the order of quantum phase transitions as is often implied in many studies, particularly in relation to the computational complexity of quantum annealing in its implementation as quantum adiabatic computation.