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arXiv:cond-mat/0008157 · Physical review. B, Condensed matter

Tunneling between Luttinger liquids

Paul Fendley, Chetan Nayak

Published Feb 20, 2001• 1 citation
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Abstract

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.

DOI
10.1103/physrevb.63.115102

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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.
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.
According to Anderson ([1], p. 54) "the equivalent of BCS theory for a fully spin-charge separated Luttinger liquid has not been formally worked out". We consider a model for two one-dimensional spinning Luttinger liquids coupled via a Cooper tunnelling Hamiltonian. We show that the partition function is the four-dimensional integral of an exponential whose exponent has an extremal point obtained solving an anomalous non-BCS self-consistence equation. If the extremal point is a global minimum the model is completely solved by the saddle point theorem and the anomalous gap generation is proved. We find that the Luttinger interaction enhances strongly T c if the intrachain Luttinger interaction is much bigger than the interchain interaction.
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 demonstrate that for general spin-charge separated Luttinger liquids there exists a critical value of the inter-Luttinger-liquid single particle hopping, ${\mathit{t}}_{\mathrm{\ensuremath{\perp}}}^{\mathit{c}}$, below which there is no coherent single particle hopping between the liquids. In the absence of coherent single particle hopping, two Luttinger liquids coupled by ${\mathit{t}}_{\mathrm{\ensuremath{\perp}}}$ will not exhibit split Fermi surfaces. For many Luttinger liquids, no band dipsersing between the liquids will form, and thus the system will retain a one-dimensional Fermi surface. This will have dramatic implications for the physical properties of such a system.
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.