Local spectral properties of Luttinger liquids: scaling versus nonuniversal energy scales

TL;DR

This paper investigates the local spectral properties of Luttinger liquids, comparing field theoretical predictions with microscopic lattice models, highlighting universal scaling behavior and nonuniversal energy scales.

Contribution

It provides an exact analysis of spectral functions using bosonization and compares these with numerical results from the extended Hubbard model, revealing phase shifts and nonuniversal energy scales.

Findings

Scaling functions are identical for translational invariant and open boundary Luttinger liquids.

Spectral functions follow field theory predictions at low energies.

Nonuniversal energy scales significantly influence spectral properties.

Abstract

Motivated by recent scanning tunneling and photoemission spectroscopy measurements on self-organized gold chains on a germanium surface we reinvestigate the local single-particle spectral properties of Luttinger liquids. In the first part we use the bosonization approach to exactly compute the local spectral function of a simplified field theoretical low-energy model and take a closer look at scaling properties as a function of the ratio of energy and temperature. Translational invariant Luttinger liquids as well as those with an open boundary (cut chain geometry) are considered. We explicitly show that the scaling functions of both setups have the same analytic form. The scaling behavior suggests a variety of consistency checks which can be performed on measured data to experimentally verify Luttinger liquid behavior. In a second part we approximately compute the local spectral function of a microscopic lattice model---the extended Hubbard model---close to an open boundary using the functional renormalization group. We show that as a function of energy and temperature it follows the field theoretical prediction in the low-energy regime and point out the importance of nonuniversal energy scales inherent to any microscopic model. The spatial dependence of this spectral function is characterized by oscillatory behavior and an envelope function which follows a power law both in accordance with the field theoretical continuum model. Interestingly, for the lattice model we find a phase shift which is proportional to the two-particle interaction and not accounted for in the standard bosonization approach to Luttinger liquids with an open boundary. We briefly comment on the effects of several one-dimensional branches cutting the Fermi energy and Rashba spin-orbit interaction.