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Physical review. B, Solid state

Single-particle states in weakly coupled linear chains

H. G. Schuster

Published Jan 15, 1976• 2 citations
Publisher

Abstract

The validity of the quasiparticle picture for electrons in weakly coupled linear chains is investigated using for a single chain the Tomonaga-Luttinger model of interacting one-dimensional electrons. For interchain coupling without jumps between the chains it is proved exactly that no stable quasiparticles exist. The excitation spectrum is exhausted by density oscillations and the single-particle occupation number shows a power-law behavior. For finite-interchain hopping, lowest-order-perturbation theory yields a quasiparticle peak in the spectral density. Its weight is $z={[1\ensuremath{-}2\ensuremath{\alpha}\mathrm{ln}(\frac{{W}_{\ensuremath{\perp}}}{{W}_{\ensuremath{\parallel}}})]}^{\ensuremath{-}1}$, where $\ensuremath{\alpha}$ is the interaction parameter and ${W}_{\ensuremath{\parallel}}$ and ${W}_{\ensuremath{\perp}}$ are the widths of the conduction band in momentum direction parallel and perpendicular to the chains. The relevance of our results for real quasi-one-dimensional conductors will be discussed.

DOI
10.1103/physrevb.13.628

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The validity of the quasiparticle picture for electrons in weakly coupled linear chains is investigated using for a single chain the Tomonaga-Luttinger model of interacting one-dimensional electrons. For interchain coupling without jumps between the chains it is proved exactly that no stable quasiparticles exist. The excitation spectrum is exhausted by density oscillations and the single-particle occupation number shows a power-law behavior. For finite-interchain hopping, lowest-order-perturbation theory yields a quasiparticle peak in the spectral density. Its weight is z=[1\ensuremath−2\ensuremathαln(W\ensuremath⊥W\ensuremath∥)]\ensuremath−1z={[1\ensuremath{-}2\ensuremath{\alpha}\mathrm{ln}(\frac{{W}_{\ensuremath{\perp}}}{{W}_{\ensuremath{\parallel}}})]}^{\ensuremath{-}1}z=[1\ensuremath−2\ensuremathαln(W\ensuremath∥​W\ensuremath⊥​​)]\ensuremath−1, where \ensuremathα\ensuremath{\alpha}\ensuremathα is the interaction parameter and W\ensuremath∥{W}_{\ensuremath{\parallel}}W\ensuremath∥​ and W\ensuremath⊥{W}_{\ensuremath{\perp}}W\ensuremath⊥​ are the widths of the conduction band in momentum direction parallel and perpendicular to the chains. The relevance of our results for real quasi-one-dimensional conductors will be discussed.
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.
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)].
Variational wave function is proposed to describe electronic properties of an array of one-dimensional conductors coupled by transverse hopping and interaction. For weak or intermediate in-chain interaction the wave function has the following structure: Tomonaga-Luttinger bosons with momentum higher than some variational quantity $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\Lambda}}$ are in their ground state while other bosons (with $|k|<\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\Lambda}})$ form kinks --- fermionlike excitations of the Tomonaga-Luttinger boson field. The nature of the ground state for these quasiparticles can be determined by solving a three-dimensional effective Hamiltonian. Since the anisotropy of the effective Hamiltonian is small the use of mean-field theory is justified. For repulsive interaction possible phases are density wave and p-wave superconductivity. Our method allows us to calculate the low-energy part of different electronic Green's functions. In order to do this it is sufficient to apply standard perturbation theory technique to the effective Hamiltonian. When the in-chain interaction is strong $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\Lambda}}$ vanishes and no fermionic excitation is present in the system. In this regime the dynamics is described by transversally coupled Tomonaga-Luttinger bosons.
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 single-particle hopping between two chains is investigated by exact-diagonalization techniques supplemented by finite-size scaling analysis. In the case of two coupled strongly correlated chains of spinless fermions, the Taylor expansion of the expectation value of the single-particle interchain hopping operator of an electron at momentum kF{k}_{F}kF​ in powers of the interchain hopping ${t}_{\ensuremath{\perp}}$ is shown to become unstable in the thermodynamic limit. The single-chain anomalous exponent $\ensuremath{\alpha}$ [characterizing the low-energy density of state $N(\ensuremath{\omega})\ensuremath{\sim}{\ensuremath{\omega}}^{\ensuremath{\alpha}}]$ is shown to be the key parameter that governs the finite-size scaling behavior. In the regime $\ensuremath{\alpha}<{\ensuremath{\alpha}}_{2p}$ $({\ensuremath{\alpha}}_{2p}\ensuremath{\simeq}0.41)$ where transverse two-particle hopping is less relevant than single-particle hopping, the finite-size effects can be described in terms of a universal scaling function. From this analysis it is found that the single-particle transverse hopping behaves as ${t}_{\ensuremath{\perp}}^{\ensuremath{\alpha}/(1\ensuremath{-}\ensuremath{\alpha})},$ in agreement with a random-phase-approximation--like treatment of the interchain coupling. For $\ensuremath{\alpha}>{\ensuremath{\alpha}}_{2p},$ the scaling law is proved to change its functional form, thus signaling the onset of coherent transverse two-particle hopping.
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.