Spin-charge separation, anomalous scaling, and the coherence of hopping in exactly solved two-chain models

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 ${\mathrm{g}}_{4}$ 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 ${\mathrm{g}}_{2}$ and ${\mathrm{g}}_{4}$ 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 ${\mathrm{g}}_{4}$ 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 ${\mathrm{g}}_{2}$ and ${\mathrm{g}}_{4}$ interactions, coherent interchain hopping coexists with anomalous scaling and non-Fermi liquid behavior in the chain direction.

arXiv:cond-mat/9611071Physical review. B, Condensed matterMay 15, 19973 citations

One-particle interchain hopping in coupled Hubbard chains

Didier Poilblanc, H. Endres, Frédéric Mila +3 more

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.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.

arXiv:cond-mat/9605106Physical review. B, Condensed matterOct 15, 19968 citations

Exactly solvable model of N coupled Luttinger chains

Lorenz Bartosch, Peter Kopietz

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.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.

arXiv:cond-mat/9703205Physical review. B, Condensed matterJun 15, 19976 citations

Confinement and transverse conductivity in coupled Luttinger liquids

Sylvain Capponi, Didier Poilblanc, Frédéric Mila

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- ${T}_{c}$ superconductors are outlined.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-${T}_{c}$ superconductors are outlined.

arXiv:cond-mat/9608004Physical review. B, Condensed matterDec 15, 199611 citations

Universal scaling behavior of coupled chains of interacting fermions

Sylvain Capponi, Didier Poilblanc, Enrico Arrigoni

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 ${k}_{F}$ 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.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 ${k}_{F}$ 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.

arXiv:cond-mat/9709169Physical review. B, Condensed matterMar 15, 199810 citations

Tunneling between Luttinger liquids

Paul Fendley, Chetan Nayak

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- ${T}_{c}$ superconductors.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-${T}_{c}$ superconductors.

arXiv:cond-mat/0008157Physical review. B, Condensed matterFeb 20, 20011 citation

{\mathrm{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

{\mathrm{g}}_{2}

and

{\mathrm{g}}_{4}

interactions, coherent interchain hopping coexists with anomalous scaling and non-Fermi liquid behavior in the chain direction.

{T}_{c}

superconductors are outlined.

{k}_{F}

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.

{T}_{c}

superconductors.