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

TL;DR

This paper investigates how a quadratic correction to the energy dispersion affects the Green's function in the Tomonaga-Luttinger model, revealing a renormalization of the anomalous dimension at weak coupling.

Contribution

It provides an explicit calculation of the Green's function with quadratic dispersion and shows how it modifies the anomalous dimension in the TLM.

Findings

Quadratic dispersion leads to a renormalization of the anomalous dimension.

At weak coupling, the anomalous dimension is reduced by a factor involving lambda and gamma.

The approach includes treatment of chemical potential changes via functional bosonization.

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.