Coherent states formulation of polymer field theory

Abstract

We introduce a stable and efficient complex Langevin (CL) scheme to enable the first direct numerical simulations of the coherent-states (CS) formulation of polymer field theory. In contrast with Edwards' well-known auxiliary-field (AF) framework, the CS formulation does not contain an embedded nonlinear, non-local, implicit functional of the auxiliary fields, and the action of the field theory has a fully explicit, semi-local and finite-order polynomial character. In the context of a polymer solution model, we demonstrate that the new CS-CL dynamical scheme for sampling fluctuations in the space of coherent states yields results in good agreement with now-standard AF-CL simulations. The formalism is potentially applicable to a broad range of polymer architectures and may facilitate systematic generation of trial actions for use in coarse-graining and numerical renormalization-group studies.

Numerical Simulation of Finite-Temperature Field Theory for Interacting Bosons

“…Within this approach, both the real and imaginary parts of the field are treated as independent and are individually analytically continued to the full complex plane; a stochastic process then samples field configurations, evolving to near-stationary-phase trajectories, and operator averages are computed over the fictitious time. Following our algorithmic development for a similar form of classical polymer field theories [26], we have found that an off-diagonal relaxation scheme that decouples ϕ and ϕ to linear order allows for stable time stepping of the complex Langevin equations. In this scheme, the CL equations of motion become…”

Linear Scaling Self-Consistent Field Theory with Spectral Contour Accuracy

“…The details of how to convert an AF theory to a CS theory are given explicitly by Man et al, 12 and following their procedure yields the partition function…”

“…Previous authors reported a method with spectral accuracy in space and first-order accuracy in contour by using a combination of Fourier transforms and finite differences (FDs). 12 We propose an alternative method that replaces the FD approximation with a Chebyshev−Tau approximation in the contour dimension to obtain a scheme that has spectral accuracy in both space and contour. 16 Volume fraction operators can also be computed with spectral accuracy using Clenshaw−Curtis quadrature.…”

“…The details of how to convert an AF theory to a CS theory are given explicitly by Man et al, and following their procedure yields the partition function where β = 1/ k B T is the inverse thermal energy. In eq , the contour dimension, s , has been scaled by the polymer length, N , and all positions, lengths, and volumes have been scaled by the unperturbed radius of gyration.…”

“…To perform an SCFT calculation, we seek a saddle point with respect to the six fields that are arguments of H CS . As is frequently done in the AF case, we perform a relaxation in fictitious time to reach the steady state saddle point. , This time relaxation can in principle be augmented with appropriately distributed noise to perform a complex Langevin field theoretic simulation of the exact field theory, , but here we restrict consideration to the mean-field approximation of SCFT. The relaxation equations are with analogous equations for ϕ B and ϕ B * .…”