Nonadiabatic transitions during a passage near a critical point

The passage through a critical point of a many-body quantum system leads to abundant nonadiabatic excitations. Here, we explore a regime, in which the critical point is not crossed although the system is passing slowly very close to it. We show that the leading exponent for the excitation probability can then be obtained by standard arguments of the Dykhne formula, but the exponential prefactor is no longer simple and behaves as a power law on the characteristic transition rate. We derive this prefactor for the nonlinear Landau-Zener model by adjusting Dykhne's approach. Then, we introduce an exactly solvable model of the transition near a critical point in the Stark ladder. We derive the number of excitations for it without approximations and find qualitatively similar results for the excitation scaling.