One-Particle an Two-Particle Instability of Coupled Luttinger Liquids

TL;DR

This paper demonstrates that coupled Luttinger liquids are inherently unstable to small transverse hopping, leading to either Fermi liquid behavior or long-range order, with explicit scaling relations for crossover temperatures.

Contribution

It provides explicit scaling relations for the crossover temperatures in coupled Luttinger liquids and analyzes the effects of transverse hopping on their stability.

Findings

Luttinger liquids become unstable with small transverse hopping.

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

Infinite-range transverse hopping yields a Fermi liquid unless the anomalous exponent exceeds one.

Abstract

It is shown that the Luttinger liquid is unstable to arbitrarily small transverse hopping. It becomes either a Fermi liquid or exhibits long-range order at zero temperature. 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 the one-particle and two-particle crossover temperatures are derived in terms of transverse hopping amplitude, spin and charge velocities as well as anomalous exponents. The special case of infinite-range transverse hopping can be treated exactly and yields a Fermi liquid down to zero temperature, unless the anomalous exponent $\theta $ is larger than unity.