  In the study of open quantum systems, the polaron transformation has recently
attracted a renewed interest as it offers the possibility to explore the strong
system-bath coupling regime. Despite this interest, a clear and unambiguous
analysis of the regimes of validity of the polaron transformation is still
lacking. Here we provide such a benchmark, comparing second order perturbation
theory results in the original untransformed frame, the polaron frame and the
variational extension with numerically exact path integral calculations of the
equilibrium reduced density matrix. Equilibrium quantities allow a direct
comparison of the three methods without invoking any further approximations as
is usually required in deriving master equations. It is found that the second
order results in the original frame are accurate for weak system-bath coupling,
the full polaron results are accurate in the opposite regime of strong
coupling, and the variational method is capable of interpolating between these
two extremes. As the bath becomes more non-Markovian (slow bath), all three
approaches become less accurate.
