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arXiv:cond-mat/0202175 · Journal of Physics Condensed Matter

Dynamic scaling in the vicinity of the Luttinger liquid fixed point

Tom Busche, Lorenz Bartosch, Peter Kopietz

Published Aug 28, 2002• 16 citations
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Abstract

We calculate the single-particle spectral function A ( k , ω) of a one-dimensional Luttinger liquid by means of a functional renormalization group (RG) approach. Given an infrared energy cut-off Λ = Λ 0 e − l , our approach yields the spectral function in the scaling form, A Λ ( k F + p , ω) = τ Z l à l ( p ξ, ωτ), where k F is the Fermi momentum, Z l is the wavefunction renormalization factor, τ = 1/Λ is the timescale and ξ = v F /Λ is the length scale associated with Λ. At the Luttinger liquid fixed point ( l → ∞ ) our RG result for A ( k , ω) exhibits the correct anomalous scaling properties, and for k = ± k F agrees exactly with the well known bosonization result at weak coupling. Our calculation demonstrates that the field rescaling is essential for obtaining the crossover from Fermi liquid behaviour to Luttinger liquid behaviour from a truncation of the hierarchy of exact RG flow equations as the infrared cut-off Λ is reduced.

DOI
10.1088/0953-8984/14/36/309

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We calculate the single-particle spectral function A ( k , ω) of a one-dimensional Luttinger liquid by means of a functional renormalization group (RG) approach. Given an infrared energy cut-off Λ = Λ 0 e − l , our approach yields the spectral function in the scaling form, A Λ ( k F + p , ω) = τ Z l à l ( p ξ, ωτ), where k F is the Fermi momentum, Z l is the wavefunction renormalization factor, τ = 1/Λ is the timescale and ξ = v F /Λ is the length scale associated with Λ. At the Luttinger liquid fixed point ( l → ∞ ) our RG result for A ( k , ω) exhibits the correct anomalous scaling properties, and for k = ± k F agrees exactly with the well known bosonization result at weak coupling. Our calculation demonstrates that the field rescaling is essential for obtaining the crossover from Fermi liquid behaviour to Luttinger liquid behaviour from a truncation of the hierarchy of exact RG flow equations as the infrared cut-off Λ is reduced.
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 ${t}_{\ensuremath{\perp}}.$ 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 \ensuremath{\alpha} 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 ${t}_{\ensuremath{\perp}}$ contributing to the anisotropic $\stackrel{\ensuremath{\rightarrow}}{D}\ensuremath{\infty}$ 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 ${t}_{\ensuremath{\perp}}$ 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.
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