The bottleneck of studying phase transitions is the barrier-crossing process composed of escaping from the basin of the local minimum and finding the saddle point. Breaking the bottleneck requires designing efficient algorithms relevant to the properties of concrete phase transition. In this work, we propose an efficient nullspace-preserving saddle search (NPSS) method for a class of phase transitions involving translational invariance. These critical states in these phase transitions are usually degenerate. The NPSS overcomes the difficulty of degeneration by ensuring the ascent direction orthogonal to the kernel space of the initial minimum, then efficiently escapes from the basin and finds the saddle point. We apply the NPSS method to the phase transitions between crystals, and between crystal and quasicrystal, based on the Landau-Brazvoskii and Lifshitz-Petrich free energy functionals. Numerical results show a good performance of the proposed method. Finally, we investigate an important property of the inflection point, where symmetry-breaking begins to occur and nullspace is no longer maintained.

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