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Physical review. B, Condensed matter

Role of transverse hopping in a two-coupled-chains model

Michele Fabrizio

Published Dec 1, 1993• 198 citations
Publisher

Abstract

We study the effect of a transverse hopping ${\mathit{t}}_{\mathrm{\ensuremath{\perp}}}$ in two chains of both spinless and spinning repulsively interacting fermions, by means of renormalization group and bosonization techniques. We show that, independent of the presence of spin, ${\mathit{t}}_{\mathrm{\ensuremath{\perp}}}$ strongly modifies the asymptotic long-wavelength behavior of the two chains, opening gaps in the excitation spectra. The origin of the instability of the gapless Luttinger-liquid behavior is identified in the flavor (==chain index) anisotropy induced by ${\mathit{t}}_{\mathrm{\ensuremath{\perp}}}$. In the case of spinning fermions, it leads to dominant pair fluctuations, in spite of the repulsive interaction. The role of spin is further analyzed in a model of two coupled chains showing, in the absence of ${\mathit{t}}_{\mathrm{\ensuremath{\perp}}}$, spin-charge separation without anomalous exponents. We solve this model exactly by the bosonization technique, and we find that the interesting analytical properties induced by spin-charge separation persist in the presence of transverse hopping, although ${\mathit{t}}_{\mathrm{\ensuremath{\perp}}}$ does modify the shape of the Fermi surface. The asymptotic expression of the single-particle Green function is also obtained.

DOI
10.1103/physrevb.48.15838

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We study the effect of a transverse hopping t\ensuremath⊥{\mathit{t}}_{\mathrm{\ensuremath{\perp}}}t\ensuremath⊥​ in two chains of both spinless and spinning repulsively interacting fermions, by means of renormalization group and bosonization techniques. We show that, independent of the presence of spin, t\ensuremath⊥{\mathit{t}}_{\mathrm{\ensuremath{\perp}}}t\ensuremath⊥​ strongly modifies the asymptotic long-wavelength behavior of the two chains, opening gaps in the excitation spectra. The origin of the instability of the gapless Luttinger-liquid behavior is identified in the flavor (==chain index) anisotropy induced by t\ensuremath⊥{\mathit{t}}_{\mathrm{\ensuremath{\perp}}}t\ensuremath⊥​. In the case of spinning fermions, it leads to dominant pair fluctuations, in spite of the repulsive interaction. The role of spin is further analyzed in a model of two coupled chains showing, in the absence of t\ensuremath⊥{\mathit{t}}_{\mathrm{\ensuremath{\perp}}}t\ensuremath⊥​, spin-charge separation without anomalous exponents. We solve this model exactly by the bosonization technique, and we find that the interesting analytical properties induced by spin-charge separation persist in the presence of transverse hopping, although t\ensuremath⊥{\mathit{t}}_{\mathrm{\ensuremath{\perp}}}t\ensuremath⊥​ does modify the shape of the Fermi surface. The asymptotic expression of the single-particle Green function is also obtained.
A model of interacting electrons living on two chains coupled by a transverse hopping ${\mathit{t}}_{\mathrm{\ensuremath{\perp}}}$ is solved exactly by a bosonization technique. It is shown that ${\mathit{t}}_{\mathrm{\ensuremath{\perp}}}$ does modify the shape of the Fermi surface also in the presence of interaction, although charge and spin excitations keep different velocities ${\mathit{u}}_{\mathrm{\ensuremath{\rho}}}$, ${\mathit{u}}_{\mathrm{\ensuremath{\sigma}}}$. Two different regimes occur: At short distances, x\ensuremath{\ll}\ensuremath{\xi}=(${\mathit{u}}_{\mathrm{\ensuremath{\rho}}}$-${\mathit{u}}_{\mathrm{\ensuremath{\sigma}}}$)/4${\mathit{t}}_{\mathrm{\ensuremath{\perp}}}$, the two chain model is not sensitive to ${\mathit{t}}_{\mathrm{\ensuremath{\perp}}}$, while for larger separation x\ensuremath{\gg}\ensuremath{\xi} interchain hopping is relevant and generates further singularities in the electron Green function besides those due to spin-charge decoupling.
A model of two interacting one-dimensional fermion systems (``Luttinger liquids'') coupled by single-particle hopping is investigated. Bosonization allows a number of exact statements to be made. In particular, for forward scattering only, the model contains two massless boson sectors and an Ising-type critical sector. For general repulsive interactions, there is a spin excitation gap and either d-type pairing or orbital antiferromagnetic fluctuations dominate. It is argued that the same behavior is also found for strong interactions. A possible scenario for the crossover to a Fermi liquid in a many-chain system is discussed. \textcopyright{} 1996 The American Physical Society.
Using bosonization techniques, we show that strong forward scattering interactions between one-dimensional spinless Luttinger liquids (LL) can stabilize a phase where charge-density wave, superconducting, and transverse single particle hopping perturbations are irrelevant. This new phase retains its LL-like properties in the directions of the chains, but with relations between exponents modified by the transverse interactions, whereas it is a perfect insulator in the transverse direction. The mechanism that stabilizes this phase is strong transverse charge-density wave fluctuations at incommensurate wave vector, which frustrates crystal formation by preventing lock-in of the in-chain density waves.
Two chains of spinless Tomonaga model coupled by a transverse hopping are investigated in terms of phase Hamiltonian where separation of Fermi momentum in the presence of the hopping is taken into account explicitly. It is shown that there appears a gap in one of the two excitations by using a renormalization group method. As a result, correlation of charge density wave with interchain ordering and out of phase between the chains (superconductivity with intrachain ordering and in phase between the chains) becomes the most dominant one in the case of the repulsive (attractive) intrachain interaction.
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 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)].