An efficient saddle search method for ordered phase transitions involving translational invariance

TL;DR

This paper introduces an efficient nullspace-preserving saddle search method for phase transitions with translational invariance, effectively handling degeneracy and identifying transition states in complex systems.

Contribution

The paper presents a novel NPSS method that improves saddle point search efficiency for degenerate critical states involving translational invariance.

Findings

Successfully applied to crystal and quasicrystal phase transitions

Demonstrates quick escape from basins and accurate transition state convergence

Shows good performance on Landau-Brazovskii and Lifshitz-Petrich models

Abstract

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.