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arXiv:cond-mat/0412620 · Physical Review B

Self-consistent Fermi surface renormalization of two coupled Luttinger liquids

Sascha Ledowski, Peter Kopietz, Álvaro Antônio Bandeira Ferraz

Published Jun 1, 2005• 16 citations
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Abstract

Using functional renormalization group methods, we present a self-consistent calculation of the true Fermi momenta ${k}_{F}^{a}$ (antibonding band) and ${k}_{F}^{b}$ (bonding band) of two spinless interacting metallic chains coupled by small interchain hopping. In the regime where the system is a Luttinger liquid, we find that $\ensuremath{\Delta}={k}_{F}^{b}\ensuremath{-}{k}_{F}^{a}$ is self-consistently determined by $\ensuremath{\Delta}={\ensuremath{\Delta}}_{1}{[1+{g}_{0}^{2}\mathrm{ln}{({\ensuremath{\Lambda}}_{0}∕\ensuremath{\Delta})}^{2}]}^{\ensuremath{-}1}$, where ${g}_{0}$ is the dimensionless interchain backscattering interaction, ${\ensuremath{\Delta}}_{1}$ is the Hartree-Fock result for ${k}_{F}^{b}\ensuremath{-}{k}_{F}^{a}$, and ${\ensuremath{\Lambda}}_{0}⪢\ensuremath{\Delta}$ is an ultraviolet cutoff. For ${g}_{0}^{2}\mathrm{ln}{({\ensuremath{\Lambda}}_{0}∕{\ensuremath{\Delta}}_{1})}^{2}⪢1$ even weak interachain backscattering leads to a strong reduction of the distance between the Fermi momenta.

DOI
10.1103/physrevb.71.235106

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Using functional renormalization group methods, we present a self-consistent calculation of the true Fermi momenta kFa{k}_{F}^{a}kFa​ (antibonding band) and kFb{k}_{F}^{b}kFb​ (bonding band) of two spinless interacting metallic chains coupled by small interchain hopping. In the regime where the system is a Luttinger liquid, we find that \ensuremathΔ=kFb\ensuremath−kFa\ensuremath{\Delta}={k}_{F}^{b}\ensuremath{-}{k}_{F}^{a}\ensuremathΔ=kFb​\ensuremath−kFa​ is self-consistently determined by \ensuremathΔ=\ensuremathΔ1[1+g02ln(\ensuremathΛ0∕\ensuremathΔ)2]\ensuremath−1\ensuremath{\Delta}={\ensuremath{\Delta}}_{1}{[1+{g}_{0}^{2}\mathrm{ln}{({\ensuremath{\Lambda}}_{0}∕\ensuremath{\Delta})}^{2}]}^{\ensuremath{-}1}\ensuremathΔ=\ensuremathΔ1​[1+g02​ln(\ensuremathΛ0​∕\ensuremathΔ)2]\ensuremath−1, where g0{g}_{0}g0​ is the dimensionless interchain backscattering interaction, \ensuremathΔ1{\ensuremath{\Delta}}_{1}\ensuremathΔ1​ is the Hartree-Fock result for kFb\ensuremath−kFa{k}_{F}^{b}\ensuremath{-}{k}_{F}^{a}kFb​\ensuremath−kFa​, and \ensuremathΛ0⪢\ensuremathΔ{\ensuremath{\Lambda}}_{0}⪢\ensuremath{\Delta}\ensuremathΛ0​⪢\ensuremathΔ is an ultraviolet cutoff. For g02ln(\ensuremathΛ0∕\ensuremathΔ1)2⪢1{g}_{0}^{2}\mathrm{ln}{({\ensuremath{\Lambda}}_{0}∕{\ensuremath{\Delta}}_{1})}^{2}⪢1g02​ln(\ensuremathΛ0​∕\ensuremathΔ1​)2⪢1 even weak interachain backscattering leads to a strong reduction of the distance between the Fermi momenta.
Using a nonperturbative functional renormalization group approach involving both fermionic and bosonic fields we calculate the interaction-induced change of the Fermi surface of spinless fermions moving on two chains connected by weak interchain hopping ${t}_{\ensuremath{\perp}}$. For a model containing interband backward scattering only we show that the distance $\ensuremath{\Delta}$ between the Fermi momenta associated with the bonding and the antibonding band can be strongly reduced, corresponding to a large reduction of the effective interchain hopping ${t}_{\ensuremath{\perp}}^{*}\ensuremath{\propto}\ensuremath{\Delta}$. A self-consistent one-loop approximation neglecting marginal vertex corrections and wave-function renormalizations predicts a confinement transition for sufficiently large interchain backscattering, where the renormalized ${t}_{\ensuremath{\perp}}^{*}$ vanishes. However, a more accurate calculation taking vertex corrections and wave-function renormalizations into account predicts only weak confinement in the sense that $0<\ensuremath{\mid}{t}_{\ensuremath{\perp}}^{*}\ensuremath{\mid}⪡\ensuremath{\mid}{t}_{\ensuremath{\perp}}\ensuremath{\mid}$. Our method can be applied to other strong-coupling problems where the dominant scattering channel is known.
The T=0 phase diagram for two Hubbard chains with interchain hopping ${\mathit{t}}_{\mathrm{\ensuremath{\perp}}}$ and on-site repulsion U is derived within a renormalization-group (RG) approach. We explicitly allow for renormalization of the Fermi momenta. A nonzero ${\mathit{t}}_{\mathrm{\ensuremath{\perp}}}$ drives the chains towards strong coupling, and only for large ${\mathit{t}}_{\mathrm{\ensuremath{\perp}}}$ is a Luttinger liquid recovered. By use of weak-coupling RG, a qualitative picture of the phase diagram can be obtained. Our calculations show two different strong-coupling regimes: one with nonvanishing effective interchain hopping (〈${\mathit{t}}_{\mathrm{\ensuremath{\perp}}}$〉\ensuremath{\ne}0), not confining, for small U, and another with 〈${\mathit{t}}_{\mathrm{\ensuremath{\perp}}}$〉=0, confining, for large U. Numerical diagonalizations for a small cluster support the RG results.
Interchain hopping in systems of coupled chains of correlated electrons is investigated by exact diagonalization and quantum Monte Carlo methods. For two weakly coupled Hubbard chains at commensurate densities (e.g., n=1/3), the splitting at the Fermi level between bonding and antibonding bands is strongly reduced (but not suppressed) by repulsive interactions extending to a few lattice spacings. The magnitude of this reduction is directly connected to the exponent \ensuremath{\alpha} of the one-dimensional Luttinger liquid. However, we show that the incoherent part of the single-particle spectral function is much less affected by the interchain coupling. This suggests that incoherent interchain hopping could occur for intermediate \ensuremath{\alpha} values. \textcopyright{} 1996 The American Physical Society.
One-particle interchain hopping in a system of coupled Luttinger liquids is investigated using exact diagonalizations techniques. First, the two-chain problem of spinless fermions is studied in order to see the behavior of the band splitting as a function of the exponent $\ensuremath{\alpha}$ which characterizes the one-dimensional (1D) Luttinger liquid. Moderate intrachain interactions can lead to a strong reduction of this splitting. The onset of the confinement within the individual chains (defined by a vanishing splitting) seems to be governed by $\ensuremath{\alpha}$. We give numerical evidence that interchain coherent hopping can be totally suppressed for $\ensuremath{\alpha}\ensuremath{\sim}0.4$ or even smaller $\ensuremath{\alpha}$ values. The transverse conductivity is shown to exhibit a strong incoherent part. Even when coherent interchain hopping is believed to occur (at small $\ensuremath{\alpha}$ values), it is shown that the coherent Drude weight is always significantly smaller than the incoherent weight. Implications for the optical experiments in quasi-1D organic or high-Tc{T}_{c}Tc​ superconductors are outlined.
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 calculate the exact Green function of a special model of N coupled Luttinger chains with arbitrary interchain hopping ${\mathrm{t}}_{\mathrm{\ensuremath{\perp}}}$. The model is exactly solvable via bosonization if the interchain interaction does not fall off in the direction perpendicular to the chains. For any finite N we find Luttinger liquid behavior and explicitly calculate the anomalous dimension ${\ensuremath{\gamma}}^{(\mathrm{N})}$. However, the Luttinger liquid state does not preclude coherent interchain hopping. We also show that ${\ensuremath{\gamma}}^{(\mathrm{N})}$\ensuremath{\rightarrow}0 for N\ensuremath{\rightarrow}\ensuremath{\infty}, so that in the limit of infinitely many chains we obtain a Fermi liquid.