  By the adiabatic theorem, the probability of non-adiabatic transitions in a
time-dependent quantum system vanishes in the adiabatic limit. The Landau-Zener
(LZ) formula gives the leading functional behavior of the probability close to
this limit. On the other hand, in counterdiabatic dynamics, one achieves
effectively adiabatic evolution at finite driving speed by adding an extra
field which suppresses non-adiabatic transitions: the adiabatic gauge potential
(AGP). We investigate the mechanism by which the AGP suppresses the transition
probability, changing it from the LZ formula to exactly zero. Quantitatively,
we find that adding the AGP to the Hamiltonian modifies the LZ formula by a
universal prefactor, independent of the adiabatic parameter, which vanishes in
the counterdiabatic regime. Qualitatively, this prefactor can be understood as
arising from the Aharonov-Bohm phases generated by the AGP between different
paths in the complex time plane. Finally, we show that these results extend to
a class of integrable time-dependent quantum Hamiltonians by proving that the
AGP preserves their integrability condition.
