  In this work, we propose an efficient nullspace-preserving saddle search
(NPSS) method for a class of phase transitions involving translational
invariance, where the critical states are often degenerate. The NPSS method
includes two stages, escaping from the basin and searching for the index-1
generalized saddle point. The NPSS method climbs upward from the generalized
local minimum in segments to overcome the challenges of degeneracy. In each
segment, an effective ascent direction is ensured by keeping this direction
orthogonal to the nullspace of the initial state in this segment. This method
can escape the basin quickly and converge to the transition states. We apply
the NPSS method to the phase transitions between crystals, and between crystal
and quasicrystal, based on the Landau-Brazovskii and Lifshitz-Petrich free
energy functionals. Numerical results show a good performance of the NPSS
method.
