  The single impurity problem in a spinful Tomonaga-Luttinger liquid is studied
numerically using path-integral Monte Carlo methods. The advantage of our
approach is that the system allows for extensive analyses of charge and spin
conductance in the non-perturbative regime. By closely examining the behavior
of conductances at low temperatures, in the presence of a finite backward
scattering barrier due to the impurity, we identified four distinct phases
characterized by either perfect transmission or reflection of charge and spin
channels. Our phase diagram for an intermediate scattering strength is
consistent with the standard perturbative renormalization group (RG) analysis
in the limit of weak and strong backward scattering, in the sense that all our
phase boundaries interpolate the two limiting cases. Further investigations
show, however, that precise location and form of our phase boundaries are not
trivially explained by the standard RG analysis, e.g., some part of the phase
diagram looks much similar to the weak backscattering limit, whereas some other
part is clearly derived from the opposite limit. In order to give a more
intuitive interpretation of such behaviors, we also reconsidered our impurity
problem from the viewpoint of a quantum Brownian motion picture.
