Regular networks of Luttinger liquids

TL;DR

This paper studies arrays of Luttinger liquids arranged in a lattice, deriving renormalization group equations for their scattering matrices, predicting strongly renormalized behavior and power-law suppression of conductivity at low temperatures.

Contribution

It extends single-node Luttinger liquid analysis to regular lattices, revealing how band structure influences renormalization and low-energy properties.

Findings

Renormalization group equations match single-node results.

Power-law suppression of conductivity at low temperatures.

Insulating states occur at specific integer fillings.

Abstract

We consider arrays of Luttinger liquids, where each node is described by a unitary scattering matrix. In the limit of small electron-electron interaction, we study the evolution of these scattering matrices as the high-energy single particle states are gradually integrated out. Interestingly, we obtain the same renormalization group equations as those derived by Lal, Rao, and Sen, for a system composed of a single node coupled to several semi-infinite 1D wires. The main difference between the single node geometry and a regular lattice is that in the latter case, the single particle spectrum is organized into periodic energy bands, so that the renormalization procedure has to stop when the last totally occupied band has been eliminated. We therefore predict a strongly renormalized Luttinger liquid behavior for generic filling factors, which should exhibit power-law suppression of the conductivity at low temperatures E_{F}/(k_{F}a) << k_{B}T << E_{F}, where a is the lattice spacing and k_{F}a >> 1. Some fully insulating ground-states are expected only for a discrete set of integer filling factors for the electronic system. A detailed discussion of the scattering matrix flow and its implication for the low energy band structure is given on the example of a square lattice.