Transition behavior in extended two-level systems with a band-touching point

It has been recently demonstrated that the often-overlooked quadratic term in the energy dispersion of graphene-like models can influence transition dynamics. These dynamics can be analyzed using fidelity susceptibility, whose peak at the band-touching point indicates dominant contributions to nonadiabatic transitions. In this work, we extend this analysis to general gapless systems, systematically identifying the key Hamiltonian terms that drive transitions near band-touching points. Contrary to the common belief that gapless systems are inherently nonadiabatic, we derive upper bounds for adiabaticity in certain cases and establish conditions for adiabatic evolution in effective two-level systems. These bounds conveniently reflect the exponential form of the transition probabilities, with the exponent matching that of the upper-bound limit. Finally, we discuss relevant model realizations and propose a graphene-inspired tight-binding framework to realize gapless Hamiltonians.