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arXiv:cond-mat/9210016 · Physical Review Letters

Spin-charge separation in a model of two coupled chains

Michele Fabrizio, Alberto Parola

Published Jan 11, 1993• 42 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.70.226

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A model of interacting electrons living on two chains coupled by a transverse hopping t\ensuremath⊥{\mathit{t}}_{\mathrm{\ensuremath{\perp}}}t\ensuremath⊥​ is solved exactly by a bosonization technique. It is shown that t\ensuremath⊥{\mathit{t}}_{\mathrm{\ensuremath{\perp}}}t\ensuremath⊥​ does modify the shape of the Fermi surface also in the presence of interaction, although charge and spin excitations keep different velocities u\ensuremathρ{\mathit{u}}_{\mathrm{\ensuremath{\rho}}}u\ensuremathρ​, u\ensuremathσ{\mathit{u}}_{\mathrm{\ensuremath{\sigma}}}u\ensuremathσ​. Two different regimes occur: At short distances, x\ensuremath{\ll}\ensuremath{\xi}=(u\ensuremathρ{\mathit{u}}_{\mathrm{\ensuremath{\rho}}}u\ensuremathρ​-u\ensuremathσ{\mathit{u}}_{\mathrm{\ensuremath{\sigma}}}u\ensuremathσ​)/4t\ensuremath⊥{\mathit{t}}_{\mathrm{\ensuremath{\perp}}}t\ensuremath⊥​, the two chain model is not sensitive to t\ensuremath⊥{\mathit{t}}_{\mathrm{\ensuremath{\perp}}}t\ensuremath⊥​, 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.
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.
Using the fermion-spin transformation to implement spin-charge separation of constrained electrons, a model of two t-J chains with interchain single-electron hopping is studied by Abelian bosonization. After spin-charge decoupling the charge dynamics can be trivially solved, while the spin dynamics is determined by a strong-coupling fixed point where the correlation functions can be calculated explicitly. This is a generalization of the Luther-Emery line for two coupled t-J chains. The interchain single-electron hopping changes the asymptotic behavior of the interchain spin-spin correlation functions and the electron Green function, but their exponents are independent of the coupling strength.
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.
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 g(4) 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 g(2) and g(4) interactions, coherent interchain hopping coexists with anomalous scaling and non-Fermi liquid behavior in the chain direction.
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.
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.