Specific heat of a one-dimensional interacting Fermi system

We re-visit the issue of the temperature dependence of the specific heat $% C(T)$ for interacting fermions in 1D. The charge component $C_c (T)$ scales linearly with $T$, but the spin component $C_s (T)$ displays a more complex behavior with $T$ as it depends on the backscattering amplitude, $g_1$, which scales down under RG transformation and eventually behaves as $g_1 (T) \sim 1/\log T$. We show, however, by direct perturbative calculations that $C_s (T)$ is strictly linear in $T$ to order $g^2_1$ as it contains the renormalized backscattering amplitude not on the scale of $T$, but at the cutoff scale set by the momentum dependence of the interaction around $2k_F$. The running amplitude $g_1 (T)$ appears only at third order and gives rise to an extra $T/\log^3 T$ term in $C_s (T)$. This agrees with the results obtained by a variety of bosonization techniques. We also show how to obtain the same expansion in $g_1$ within the sine-Gordon model.