Characteristics of the Complex Saddle Point of Polymer Field Theory

Abstract

For decades, polymer field theory has been proven to be a powerful tool for investigating polymeric nanostructures formed by heterogeneous polymers. By finding the saddle point of polymer fields, self-consistent field theory (SCFT) provides a mean field solution for the system. Traditionally, it has been assumed that the fields and ensemble average densities in SCFT solutions are real-valued functions. In this study, however, we unveil an intriguing possibility that the saddle point approximation leading to the SCFT solution may result in complex-valued fields. We demonstrate that for each real saddle point, there exists an infinite number of complex saddle points that share the same free energy, and these saddle points are continuously connected. Focusing on A and B homopolymer mixture and AB diblock copolymers, we explore the conditions for obtaining such saddle points and find that the fields are always Hermitian functions when there are nonvanishing imaginary parts, resembling the P T symmetric system in quantum mechanics. In the case of the homopolymer mixture, we derive an analytical expression for the complex saddle points in the high chi N limit. These findings may provide valuable insights for comprehending and analyzing the results of complex Langevin field theoretic simulations in which these complex solutions are readily accessible and can significantly impact the ensemble average of physical observables.