The Utility of Hidden Symmetry in Polymer Field Theory for Self-Consistent Field Theory and Field-Theoretic Simulation

Polymer field theory has been pivotal in investigating the nanostructures of heterogeneous polymers, providing robust analytical and computational methodologies. The two main approaches are Field-Theoretic Simulation (FTS), which samples fluctuating fields, and Self-Consistent Field Theory (SCFT), which identifies saddle points of polymer fields. Traditionally, SCFT assumes real-valued fields and isolated saddle points in the field configuration space. We challenge this assumption, demonstrating that saddle points form a continuously connected low-dimensional family sharing the same Hamiltonian value. We show that this behavior is a natural consequence of the analyticity and translational invariance of the Hamiltonian, which together demand its invariance under generalized translations by displacements with complex components. This hidden symmetry offers key insights into the behavior of Complex Langevin FTS (CL-FTS), where fields are sampled around these connected saddle points. We also propose a translation scheme for CL-FTS to mitigate its instability, aiming to improve polymer field simulations and better align them with experimental nanostructures.