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.

翻译：本文提出了一种高效的零空间保持鞍点搜索（NPSS）方法，用于处理涉及平移不变性的一类相变问题，其中临界态通常具有简并性。NPSS方法包含两个阶段：逃离势阱和搜索指数为1的广义鞍点。该方法从广义局部极小点分段向上攀爬，以克服简并带来的挑战。在每个分段中，通过使上升方向与该分段初始状态的零空间保持正交，确保有效的上升方向。该方法能够快速逃离势阱并收敛至过渡态。基于Landau-Brazovskii和Lifshitz-Petrich自由能泛函，我们将NPSS方法应用于晶体之间以及晶体与准晶体之间的相变问题。数值结果表明NPSS方法具有良好的性能。