Thermal bosonisation in the sine-Gordon and massive Thirring models

We study bosonisation in the massive Thirring and sine-Gordon models at finite temperature T and nonzero fermion chemical potential $\mu$. For that purpose we use both canonical operator and path integral approaches, paying particular attention to the issues of thermal normal ordering and renormalisation. At T>0 and $\mu=0$, the massive Thirring model bosonises to the sine-Gordon model with the same T=0 identification between coupling constants. We prove that not only the partition functions of the two models coincide, as was recently shown, but also that thermal averages of zero-charge operators can be identified. In particular, analysis of the point-split regularised fermion current then leads to the thermal equivalence between sine-Gordon kinks and Thirring fermions. At $\mu \neq 0$, T>0 and working in perturbation theory about the massless Thirring model, we show that the bosonised theory is the sine-Gordon model plus an additional topological term which accounts for the existence of net fermion charge excitations (the fermions or the kinks) in the thermal bath. This result generalises one recently obtained for the massless case, and it is the two-dimensional version of the low-energy QCD chiral Lagrangian at finite baryon density.