The spectral function of the Tomonaga-Luttinger model revisited: power laws and universality

TL;DR

This paper examines how the momentum dependence of interactions affects the spectral function in the Tomonaga-Luttinger model, revealing that curvature in interactions destroys universal power-law behavior away from the Fermi momentum.

Contribution

It demonstrates that the momentum dependence of the two-particle interaction significantly influences spectral functions, challenging assumptions of universality in Luttinger liquids.

Findings

Curvature in the interaction destroys power-law scaling.

Spectral line shape depends on interaction details away from Fermi momentum.

Universal low-energy properties remain unaffected by interaction momentum dependence.

Abstract

We reinvestigate the momentum-resolved single-particle spectral function of the Tomonaga-Luttinger model. In particular, we focus on the role of the momentum-dependence of the two-particle interaction V(q). Usually, V(q) is assumed to be a constant and integrals are regularized in the ultraviolet `by hand' employing an ad hoc procedure. As the momentum dependence of the interaction is irrelevant in the renormalization group sense this does not affect the universal low-energy properties of the model, e.g. exponents of power laws, if all energy scales are sent to zero. If, however, the momentum k is fixed away from the Fermi momentum k_F, with |k-k_F| setting a nonvanishing energy scale, the details of V(q) start to matter. We provide strong evidence that any curvature of the two-particle interaction at small transferred momentum q destroys power-law scaling of the momentum resolved spectral function as a function of energy. Even for |k-k_F| much smaller than the momentum space range of the interaction the spectral line shape depends on the details of V(q). The significance of our results for universality in the Luttinger liquid sense, for experiments on quasi one-dimensional metals, and for recent attempts to compute the spectral function of one-dimensional correlated systems taking effects of the curvature of the single-particle dispersion into account (nonlinear Luttinger liquid phenomenology) is discussed.