Specific heat of a one-dimensional interacting Fermi system: Role of anomalies

We revisit the issue of the temperature dependence of the specific heat C (T) for interacting fermions in one dimension. The charge component Cc (T) scales linearly with T, but the spin component Cs (T) displays a more complex behavior with T as it depends on the backscattering amplitude, g1, which scales down under renormalization group transformation and eventually behaves as g1 (T) ∼1 log T. We show, however, by direct perturbative calculations that Cs (T) is strictly linear in T to order g12 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 2 kF. The running amplitude g1 (T) appears only at third order and gives rise to an extra T log3 T term in Cs (T). This agrees with the results obtained by a variety of bosonization techniques. We also show how to obtain the same expansion in g1 within the sine-Gordon model.