Path-integral method for soliton-bearing systems. Higher-order corrections in the sine-Gordon model in the classical limit

We investigate systematically the higher-order corrections in $t=\frac{T}{E_s}$ in the one-dimensional sine-Gordon system in the classical limit, where $T$ is the temperature and $E_s$ is the soliton energy. Making use of the collective-coordinate technique, we obtain the expression of the momentum-dependent soliton density $n_s\mathrm{}\left(P\right)\mathrm{}$, which includes the higher-order corrections. This $n_s\mathrm{}\left(P\right)\mathrm{}$ gives rise to the $t$-dependent corrections in the soliton free energy, which coincide exactly with the corrections found recently by the use of the transfer-integral technique.