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

TL;DR

This work tackles the challenge of locating minimum-energy transition paths in phase-transition problems with translational invariance, where Hessians are degenerate due to symmetry. It introduces the nullspace-preserving saddle search (NPSS) method, which separates the ascent into segments orthogonal to evolving nullspaces and then switches to a minimax search restricted to the ascent subspace to find an index-1 generalized saddle point. The method is demonstrated on Landau-Brazovskii and Lifshitz-Petrich models, showing efficient basin escape and reliable identification of transition states, often outperforming the HiSD approach. A key finding is a nullspace-preserving property before a symmetry-breaking inflection point (IP) along the minimum-energy path, which motivates potential further efficiency gains and segment-wise updates. Overall, NPSS provides a principled, segment-aware framework for phase-transition analysis under translational invariance with practical computational benefits.

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.

Figures (11)

• Figure 1: Stable ordered states and the phase diagram of the LB model. Stable ordered structures: (a). Hexagonal phase (HEX) in 2D and 3D; (b). Lamella phase (LAM); (c). Body-centered cubic spherical phase (BCC). DIS represents the disordered phase. Red dots represent the parameters $(\gamma, \tau)$ adopted in \ref{['sec:result']}.
• Figure 1: Illustration of an energy surface. The evolution path is shown as a green curve. The GLM and index-1 GSP are represented by red points on the evolution path. The boundary of GQR is marked by the blue curve. $\{\bar{U}_i\}_{i=1}^n$ are the initial states of each segment in the evolution path. The shallow circles at $\bar{U}_i$ represent a neighborhood satisfying $\|\sin \Theta(\mathcal{W}^k(U), \mathcal{W}^k( \bar{U}_i))\|_2 < 1$ for any state $U$ within the GQR.
• Figure 1: (a). Transition path from DIS to HEX computed by the NPSS method in LB model with $\tau = 0.001$, $\gamma = 0.5$, where $L = 60$, $N = 900$. (b). Transition path from DIS to DDQC computed by the NPSS method in LP model with $\varepsilon = -0.01$, $\alpha = 0.95$, where $L = 112$, $N = 1024$. The $x$-axis represents the evolution direction of phase transitions.
• Figure 1: (a). Illustration of the MEP from HEX to LQ. (b)-(g). Several states on MEP marked in (a), where A and B are two intermediate states before and after reaching the IP, respectively.
• Figure 2: Stable ordered states and the phase diagram of the LP model with $q_1 = 1$, $q_2 = 2\cos(\pi/12)$. Stable ordered structures: (a). HEX; (b). LAM; (c). Lamellar quasicrystal (LQ); (d). Dodecagonal quasicrystal (DDQC). Red dots represent the parameters $(\varepsilon, \alpha)$ adopted in \ref{['sec:result']}.

Theorems & Definitions (11)

• Definition 3.1
• Definition 3.2
• Definition 3.3
• Proposition 3.4
• Proof 1
• Remark 3.1
• Proposition 3.5
• Proof 2
• Lemma 4.1
• Proof 3