Discovering new phases via computing second-order stationary states of Landau-Brazovskii model

TL;DR

This paper introduces a novel trust region method for the Landau-Brazovskii model, enabling the discovery of a new stable cubic phase and improving the identification of stable structures in complex energy landscapes.

Contribution

The paper presents an implicit-explicit trust region algorithm that guarantees convergence to second-order stationary points, facilitating the discovery of new phases in the LB model.

Findings

Discovered the cubic FDDD phase in the LB model.

Constructed an updated phase diagram including the new phase.

Demonstrated robustness of the algorithm in locating stable phases.

Abstract

In this work, we report a stable ordered structure -- the cubic FDDD phase -- that has not previously been identified in the Landau-Brazovskii (LB) model, a fundamental and important model for studying crystals and their phase transitions. The key to this discovery is the proposed implicit-explicit trust region method for computing second-order stationary points in the high-dimensional nonconvex energy landscape of the LB model. Different from existing first-order gradient-based algorithms, which only guarantee convergence to first-order stationary points and may therefore stagnate at saddle points, the proposed method is theoretically guaranteed to converge to second-order stationary points corresponding to local minima. Numerical experiments verify the theoretical properties of the algorithm and demonstrate its robustness in locating stable phases from different initial conditions. Based on the discovered FDDD phase, we further construct an updated phase diagram of the LB model and identify its stable region. These results show that targeting second-order stationary points provides an effective computational paradigm for exploring complex free-energy landscapes and uncovering new stable phases.