Core Concepts
The authors present an efficient nullspace-preserving saddle search (NPSS) method to locate the index-1 generalized saddle point for phase transitions involving translational invariance, where the critical states are often degenerate.
Abstract
The authors propose an efficient nullspace-preserving saddle search (NPSS) method to study phase transitions involving translational invariance, where the critical states are often degenerate.
The NPSS method consists of two stages:
Stage I - Escaping from the basin:
- The NPSS method climbs upward from the generalized local minimum (GLM) in segments based on the changes of principal angles measuring the difference of nullspace.
- Within each segment, the ascent direction is chosen to be orthogonal to the nullspace of the initial state in the current segment.
- When the principal angle between nullspaces of the current and initial states exceeds a threshold, a new segment is initiated.
- These operations ensure the effectiveness of the ascent direction and avoid the costs of updating the nullspace at each step, enabling a quick escape from the basin.
Stage II - Searching for the index-1 generalized saddle point:
- After escaping from the generalized quadratic region of the GLM, the minimum eigenvalue of its Hessian becomes negative.
- The NPSS method ascends along the ascent subspace V spanned by the eigenvector corresponding to the negative eigenvalue and descends along the orthogonal complement V⊥.
- This allows the system to effectively converge to the transition state.
The authors demonstrate the power of the NPSS method for phase transitions with translational invariance using the Landau-Brazovskii and Lifshitz-Petrich models. The NPSS method can be applied to a range of models involving phase transitions with translational symmetry.
Stats
The Landau-Brazovskii free energy functional is given by:
ELB(u) = 1/|Ω| ∫Ω [1/2 ((1 + Δ)u)2 + τ/2 u2 - γ/3! u3 + 1/4! u4] dr
The Lifshitz-Petrich free energy functional is given by:
ELP(u) = 1/|Ω| ∫Ω [1/2 ((q2_1 + Δ)(q2_2 + Δ)u)2 - ε/2 u2 - α/3 u3 + 1/4 u4] dr
Quotes
"The NPSS method climbs upward from the generalized local minimum (GLM) in segments based on the changes of principal angles measuring the difference of nullspace."
"Within each segment, the ascent direction is chosen to be orthogonal to the nullspace of the initial state in the current segment."
"After escaping from the generalized quadratic region of the GLM, the minimum eigenvalue of its Hessian becomes negative."