Complex Saddle Structures in Self-Consistent Field Theory and Their Role in Field-Theoretic Simulations

Abstract

Polymer field theory has become a fundamental tool for exploring nanoscale structures in heterogeneous polymer systems, and various analytical and computational tools have been developed for the last few decades. One of its central methodologies is Field-Theoretic Simulations (FTS), which explore fluctuations in field configurations, and we recently developed a Langevin FTS method boosting the simulation using deep neural network. Another long-standing theoretical tool is Self-Consistent Field Theory (SCFT), which seeks saddle points within the field landscape. Conventional SCFT approaches typically assume that these saddle points are isolated and that the underlying fields are real-valued. In this work, we revisit and revise this premise, showing that saddle points actually form a connected, low-dimensional manifold characterized by a constant Hamiltonian. This phenomenon arises naturally from the Hamiltonian’s analyticity and invariance under spatial translations, which together imply a broader symmetry: invariance under generalized translations with complex-valued displacements. Recognizing this hidden symmetry sheds light on the workings of Complex Langevin FTS (CL-FTS), where sampling in its natural implementation occurs near these extended saddle structures. Building on this understanding, we introduce a translation method designed to stabilize CL-FTS dynamics, with the goal of enhancing simulation accuracy and bringing theoretical predictions closer to experimentally observed polymer morphologies.