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

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 $t_{\perp }.$ We evaluate the exact asymptotic low-energy behavior of the self-energy in the anisotropic infinite-dimension limit. This limit extends the dynamical 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 α 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 carrying out a resummation of all diagrams of the expansion in $t_{\perp }$ contributing to the anisotropic $\overrightarrow{D}\infty$ 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 $t_{\perp }$ 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. 83, 128 (1999)], and includes all mathematical details.