How does a quadratic term in the energy dispersion modify the single-particle Green's function of the Tomonaga-Luttinger model?

Abstract

We calculate the effect of a quadratic term in the energy dispersion on the low-energy behavior of the Green's function of the spinless Tomonaga-Luttinger model (TLM). Assuming that for small wave-vectors q = k - k_F the fermionic excitation energy relative to the Fermi energy is v_F q + q^2 / (2m), we explicitly calculate the single-particle Green's function for finite but small values of lambda = q_c /(2k_F). Here k_F is the Fermi wave-vector, q_c is the maximal momentum transfered by the interaction, and v_F = k_F / m is the Fermi velocity. Assuming equal forward scattering couplings g_2 = g_4, we find that the dominant effect of the quadratic term in the energy dispersion is a renormalization of the anomalous dimension. In particular, at weak coupling the anomalous dimension is tilde{gamma} = gamma (1 - 2 lambda^2 gamma), where gamma is the anomalous dimension of the TLM. We also show how to treat the change of the chemical potential due to the interactions within the functional bosonization approach in arbitrary dimensions.