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we investigate an important property of the inflection point, where symmetry-breaking begins to occur and nullspace is no longer maintained.

翻译：研究相变的瓶颈在于跨越势垒的过程，即从局部极小值盆地逃逸并找到鞍点。突破这一瓶颈需要针对具体相变特性设计高效算法。本文针对一类涉及平移不变性的相变提出了一种高效的零空间保持鞍点搜索（NPSS）方法。这类相变中的临界态通常具有简并性。NPSS通过确保上升方向与初始极小值的核空间正交，克服了简并带来的困难，从而高效地逃离盆地并定位鞍点。基于Landau-Brazvoskii和Lifshitz-Petrich自由能泛函，我们将NPSS方法应用于晶体间以及晶体与准晶间的相变。数值结果表明该方法具有良好的性能。最后，我们研究了拐点的一个重要特性——在该点对称性破缺开始发生，且零空间不再保持。