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

It is shown 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 the one-particle and two-particle crossover temperatures are derived in terms of transverse hopping, spin and charge velocities, and anomalous exponents. The special case of infinite-range transverse hopping is treated exactly and yields a Fermi liquid down to $T=0$, unless the anomalous exponent θ>1.