Luttinger liquids coupled by hopping

TL;DR

This paper demonstrates that Luttinger liquids are inherently unstable to small transverse hopping, providing explicit scaling relations for crossover temperatures and addressing limitations of previous renormalization group analyses.

Contribution

It introduces an improved approach that explicitly considers spin-charge velocity differences, showing the instability of Luttinger liquids to transverse hopping and deriving relevant scaling relations.

Findings

Luttinger liquids are unstable to arbitrarily small transverse hopping.

Finite crossover temperatures can exist even with differing spin and charge velocities.

Results align with recent experimental observations.

Abstract

The stability of the Luttinger liquid to small transverse hopping has been studied from several points of view. The renormalization group approach in particular has been criticized because it does not take explicitly into account the difference between spin and charge velocities and because the interaction should be turned-on before the transverse hopping if the stability of the Luttinger liquid is a non-perturbative effect. An approach that answers both of these objections is explained here. It shows that the Luttinger liquid is unstable to arbitrarily small transverse hopping. The crossover temperatures below which either transverse coherent band motion or long-range order start to develop can be finite even when spin and charge velocities differ. Explicit scaling relations for these one-particle and two-particle crossover temperatures are derived in terms of transverse hopping, spin and charge velocities and anomalous exponents. The renormalization group results are recovered as special cases when spin and charge velocities are identical. The results compare well with recent experiments presented at this conference. Magnetic field effects are alluded to.