  The switching polarization of a ferroelectric is a characterization of the
current that flows due to changes in polarization when the system is switched
between two states. Computation of this change in polarization in crystal
systems has been enabled by the modern theory of polarization, where it is
expressed in terms of a change in Berry phase as the material switches. It is
straightforward to compute this change of phase, but only modulo $2\pi$,
requiring a branch choice from among a lattice of values separated by $2\pi$.
The measured switching polarization depends on the actual path along which the
material switches, which in general involves nucleation and growth of domains
and is therefore quite complex. In this work, we present a physically motivated
approach for predicting the experimentally measured switching polarization that
involves separating the change in phase between two states into as many
gauge-invariant smaller phase changes as possible. As long as the magnitudes of
these smaller phase changes remain smaller than $\pi$, their sum forms a phase
change which corresponds to the change one would find along any path involving
minimal evolution of the atomic and electronic structure. We show that for
typical ferroelectrics, including those that would have otherwise required a
densely sampled path, this technique allows the switching polarization to be
computed without any need for intermediate sampling between oppositely
polarized states.
