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arXiv:cond-mat/9703205 · Physical review. B, Condensed matter

Exactly solvable model of N coupled Luttinger chains

Lorenz Bartosch, Peter Kopietz

Published Jun 15, 1997• 6 citations
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Abstract

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.

DOI
10.1103/physrevb.55.15360

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We calculate the exact Green function of a special model of N coupled Luttinger chains with arbitrary interchain hopping t\ensuremath⊥{\mathrm{t}}_{\mathrm{\ensuremath{\perp}}}t\ensuremath⊥​. 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 \ensuremathγ(N){\ensuremath{\gamma}}^{(\mathrm{N})}\ensuremathγ(N). However, the Luttinger liquid state does not preclude coherent interchain hopping. We also show that \ensuremathγ(N){\ensuremath{\gamma}}^{(\mathrm{N})}\ensuremathγ(N)\ensuremath{\rightarrow}0 for N\ensuremath{\rightarrow}\ensuremath{\infty}, so that in the limit of infinitely many chains we obtain a Fermi liquid.
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)].
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.
The coherence of transport between two one-dimensional interacting Fermi liquids, coupled by single-particle hopping and interchain interaction, is examined in the context of two exactly soluble models. It is found that the coherence of the interchain hopping depends on the interplay between interchain hopping and interchain interaction terms, and not simply on the ground state spectral properties of an isolated chain. Specifically, the splitting of levels associated with interchain hopping in a g4{\mathrm{g}}_{4}g4​ soluble model is found to be enhanced by the introduction of interchain interaction. It is also shown that, for an exactly solvable model with both g2{\mathrm{g}}_{2}g2​ and g4{\mathrm{g}}_{4}g4​ interactions, coherent interchain hopping coexists with anomalous scaling and non-Fermi liquid behavior in the chain direction.
We consider the problem of tunneling between spinless one-dimensional (1D) Luttinger-liquid chains. We show how to map the problem onto the four-state chiral clock model together with a free boson for the total charge mode. We use a variety of results, some of them exact, from the integrable chiral Potts model and the study of commensurate/incommensurate transitions to deduce the physics of coupled Luttinger liquids. For those intrachain interaction strengths for which interchain tunneling is relevant, we find that it can lead to the formation of symmetric and antisymmetric bands with split Fermi surfaces, depending on the relative strengths of interchain tunneling and interchain interactions. With split Fermi surfaces, interchain transport is coherent. It is not possible to have two gapless Fermi surfaces with the same Fermi momentum when interchain tunneling is relevant. However, interchain interactions can drive the formation of a gap, in which case interchain transport will be incoherent. We comment on the possible relevance of our results to c-axis transport in high-Tc{T}_{c}Tc​ superconductors.
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 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.