Projected complex Langevin sampling method for bosons in the canonical and microcanonical ensembles

We introduce a projected complex Langevin (CL) numerical sampling method-a fictitious Langevin dynamics scheme that uses numerical projection to sample a constrained stationary distribution with highly oscillatory character. Despite the complex-valued degrees of freedom and associated sign problem, the projected CL method succeeds as a natural extension of real-valued projected Langevin processes. In the newly proposed method, complex-valued Lagrange multipliers are determined to enforce constraints to machine precision at each iteration. To illustrate the efficacy of this approach, we adapt the projected CL method to sample coherent state quantum field theories describing interacting Bose gases, which are realized in modern cold-atom experiments. We apply projected CL to two scenarios with holomorphic constraints, namely the canonical and microcanonical ensembles, and we show that projected CL reproduces the correct thermodynamic observables. We further observe improved numerical stability and accuracy at larger time steps when compared to the previous state-of-the-art method for performing constrained CL sampling.