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arXiv:cond-mat/9910114 · Physical review. B, Condensed matter

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

Enrico Arrigoni

Published Mar 15, 2000• 32 citations
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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 ${t}_{\ensuremath{\perp}}.$ 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.

DOI
10.1103/physrevb.61.7909

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We consider the low-energy region of an array of Luttinger liquids coupled by a weak interchain hopping. The leading-logarithmic divergences can be resummed to all orders, within a self-consistent perturbative expansion in the hopping, in the large-dimension limit. The anomalous exponent scales to zero below the one-particle crossover temperature. As a consequence, coherent quasiparticles with finite weight appear along the whole Fermi surface. Extending the expansion self-consistently to all orders turns out to be crucial to restore the correct Fermi-liquid behavior.
We use higher-dimensional bosonization to study the normal state of electrons in weakly coupled metallic chains interacting with long-range Coulomb forces. Particular attention is paid to the crossover between Luttinger and Fermi-liquid behavior as the interchain hopping ${t}_{\ensuremath{\perp}}$ is varied. Although in the physically interesting case of finite but small ${t}_{\ensuremath{\perp}}$ the quasiparticle residue does not vanish, the single-particle Green's function exhibits the signature of Luttinger-liquid behavior (i.e., anomalous scaling and spin-charge separation) in a large intermediate parameter regime. Using realistic parameters, we find that the scaling behavior in this regime is characterized by an anomalous dimension of the order of unity, as suggested by recent experiments on quasi-one-dimensional conductors. Our calculation also gives insights into the approximations inherent in higher-dimensional bosonization; in particular, we show that the replacement of a curved Fermi surface by a finite number MMM of flat patches can give rise to unphysical nesting singularities in the single-particle Green's function, which disappear only in the limit $M\ensuremath{\rightarrow}\ensuremath{\infty}$ or if curvature effects are included. We also compare our approach with other methods. This work extends our recent letter [Phys. Rev. Lett. 74, 2997 (1995)].
We study the crossover from the one-dimensional to the two-dimensional Hubbard model in the photoemission spectra of weakly coupled chains. The chains with on-site repulsion are treated using the spin-charge factorized wave function, known to provide an accurate description of the dynamics of the model in the strong coupling limit, while the hoppings between the chains are considered as a perturbation. We calculate dynamical spectral functions at all energies in the random-phase approximation by resuming an infinite set of diagrams. Even though the hoppings drive the system from a fractionalized Luttinger-liquid-like system to a Fermi-liquid-like system at low energies, significant characteristics of the one-dimensional system remain in the two-dimensional system. Furthermore, we find that, of introducing (frustrated) hoppings beyond the nearest-neighbor one, the interference effects increase the energy and momentum range of the one-dimensional character.
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=0T=0T=0, unless the anomalous exponent $\ensuremath{\theta}>1$.
We analyze the problem of Luttinger liquids coupled via a single-particle hopping parameter ${t}_{\ensuremath{\perp}}$ and introduce a systematic diagrammatic expansion in powers of ${t}_{\ensuremath{\perp}}$. An analysis of the scaling of the diagrams at each order allows us to determine the power-law behavior versus ${t}_{\ensuremath{\perp}}$ of the interchain hopping and of the Fermi surface warp. In particular, for strong interactions, we find that the exponents are dominated by higher-order diagrams producing an enhanced coherence and a failure of linear-response theory. Our results are valid at any finite order in ${t}_{\ensuremath{\perp}}$ for the self-energy.
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.