Crossover to Fermi-liquid behavior for weakly-coupled Luttinger liquids in the anisotropic large-dimension limit

TL;DR

This paper investigates the transition from one-dimensional Luttinger liquids to higher-dimensional Fermi liquids using an anisotropic large-dimension limit, revealing the emergence of quasiparticles and Fermi-liquid behavior at low energies.

Contribution

It extends the dynamical mean field approach to coupled Luttinger liquids, providing an almost analytic solution for the crossover to Fermi-liquid behavior in the anisotropic infinite-dimension limit.

Findings

Fermi-liquid fixed point is reached below the crossover temperature.

Spectral function shows sharp quasiparticle peaks across the Fermi surface.

Lowest-order perturbation theory is unreliable below the crossover temperature.

Abstract

We study the problem of the crossover from one- to higher-dimensional metals by considering an array of Luttinger liquids (one-dimensional chains) coupled by a weak interchain hopping {\tp.} We evaluate the exact asymptotic low-energy behavior of the self-energy in the anisotropic infinite-dimension limit. This limit extends the dinamical mean field concept to the case of a chain embedded in a self-consistent medium. The system flows to a Fermi-liquid fixed point for energies below the dimensional crossover temperature, and the anomalous exponent $\al$ renormalizes to zero, in the case of equal spin and charge velocities. In particular, the single-particle spectral function shows sharp quasiparticle peaks with nonvanishing weight along the whole Fermi surface, in contrast to the lowest-order result. Our result is obtained by carring out a resummation of all diagrams of the expansion in \tp contributing to the anisotropic $D\to\infty$ limit. This is done by solving, in an almost completely analytic way, an asymptotically exact recursive equation for the renormalized vertices, within a skeleton expansion. Our outcome shows that perturbation expansions in \tp restricted to lowest orders are unreliable below the crossover temperature. The extension to finite dimensions is discussed. This work extends our recent Letter [Phys. Rev. Lett. {\bf 83}, 128 (1999)], and includes all mathematical details.