Complex Solutions of Self-Consistent Field Theory

Abstract

For the past few decades, self-consistent field theory (SCFT) has proven to be a powerful tool for the exploration of polymeric nanostructures. It provides a mean field solution to the physical system, and thus one naturally assumes that the fields and ensemble average densities of SCFT solution must be real-valued functions. However, we unveil an intriguing possibility that the saddle-point approximation in SCFT may result in complex solutions. In this study, we demonstrate that for each real saddle-point solution, there may exist an infinite number of complex solutions sharing the same free energy, and thus the concept of "saddle-point" in the functional space can be generalized as "saddle-line" in the complex world. Focusing on AB homopolymer mixtures and AB diblock copolymer systems, we explore the conditions for obtaining such complex solutions and their unique characteristics. In the case of the AB homopolymer mixture, we derive an analytic expression for the complex solution in the high χN limit. These findings offer valuable insights for comprehending and analyzing results from complex Langevin field theoretic simulations, where these complex solutions are readily accessible.