Complex Implications\nof Translational Invariance in\nPolymer Field Theory

The behavior of complex-Langevin field-theoretic simulations\n(CL-FTSs)\nof polymer liquids is sensitive to the nature of saddle-point field\nconfigurations, which are solutions of self-consistent field theory\n(SCFT). Recent work [Kang et al. <i>Macromolecules</i> <b>2024</b>, 57, 3850] has shown that SCFT saddle-points with real\nfields are generally not isolated solutions but rather members of\na low-dimensional family of continuously connected complex-valued\nsaddle-points sharing the same Hamiltonian value. We show that this\nbehavior is a natural consequence of the analyticity and translational\ninvariance of the Hamiltonian, which together demand its invariance\nunder generalized translations by displacements with complex components.\nWe also present a numerical algorithm that minimizes the deleterious\neffects of this generalized symmetry on the stability of CL-FTSs.