Hidden Symmetry in Polymer Field Theory and Its Implications for Self-Consistent Field Theory and Field-Theoretic Simulation

Abstract

Polymer field theory has long been a cornerstone for investigating the nanostructures of heterogeneous polymers, offering robust methodologies for analysis. There are two primary approaches within polymer field theory: Field-Theoretic Simulation (FTS), which samples an ensemble of fluctuating fields, and Self-Consistent Field Theory (SCFT), which identifies the saddle points of polymer fields. Traditionally, SCFT has been used to determine the mean field solution of polymer systems, under the assumption that the fields and ensemble average densities in SCFT solutions are real-valued functions, and that the real-valued saddle points are isolated points in the field configuration space. Contrary to this assumption, we demonstrate that the saddle points form a continuously connected family sharing the same Hamiltonian value. This study reveals that the Hamiltonian of polymer field theory inherently possesses symmetry under both real and imaginary spatial translations. This hidden symmetry not only generates a family of interesting complex SCFT solutions but also provides insight into the behavior of Complex Langevin FTS (CL-FTS), where field configurations are sampled around these connected saddle points. Additionally, we propose a translation scheme for CL-FTS to manage its intrinsic instability. Our findings aim to facilitate more realistic polymer field simulations, better aligning them with experimentally observed polymeric nanostructures.