  Numerical simulations of the full quantum properties of interacting many-body
systems by means of field-theoretic Monte-Carlo techniques are often limited
due to a sign problem. Here we simulate properties of a dilute two-dimensional
Bose gas in the vicinity of the Berezinskii-Kosterlitz-Thouless (BKT)
transition by means of the Complex Langevin (CL) algorithm, thereby extending
our previous CL study of the three-dimensional Bose gas to the
lower-dimensional case. The purpose of the paper is twofold. On the one hand,
it adds to benchmarking of the CL method and thus contributes to further
exploring the range of applicability of the method. With the respective
results, the universality of the equation of state is recovered, as well as the
long-wave-length power-law dependence of the single-particle momentum spectrum
below the BKT transition. Analysis of the rotational part of the current
density corroborates vortex unbinding in crossing the transition. Beyond these
measures of consistency we compute quantum corrections to the critical density
and chemical potential in the weakly coupled regime. Our results show a shift
of these quantities to lower values as compared to those obtained from
classical field theory. It points in the opposite direction as compared to the
shift of the critical density found by means of the path-integral Monte-Carlo
method at larger values of the coupling. Our simulations widen the perspective
for precision comparisons with experiment.
