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arXiv:cond-mat/9607141 · Journal of Physics Condensed Matter

New evidence for `confined coherence' in weakly coupled Luttinger liquids

David G. Clarke, Steven P. Strong

Published May 12, 1997• 2 citations
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Abstract

On the basis of a calculation of the exact interliquid hopping rate and an approximate single particle Green's function, we present new evidence for the existence of a phase of relevant but incoherent inter-Luttinger liquid transport. This phase of ``confined coherence'' occurs when the Luttinger liquid exponent alpha satisfies alpha_c

DOI
10.1088/0953-8984/9/19/006

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On the basis of a calculation of the exact interliquid hopping rate and an approximate single particle Green's function, we present new evidence for the existence of a phase of relevant but incoherent inter-Luttinger liquid transport. This phase of ``confined coherence'' occurs when the Luttinger liquid exponent alpha satisfies alpha_c
We present a pedagogical account of our approach to the problem of Luttinger liquids coupled by interliquid single particle hopping. It is shown that the key issue is that of coherence/incoherence of interliquid hopping, and not of relevance/irrelevance in a renormalization group sense. A clear signal of coherence, present in the case of coupled Fermi liquids, is absent for Luttinger liquids, and we argue for the existence of an incoherent regime when the interliquid hopping rate is sufficiently small. The problem is relevant to any sufficiently anisotropic, strongly correlated metal, and in particular to understanding the anomalous c-axis conductivity in the cuprate superconductors, and the physics of the quasi-1D organic conductors.
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.
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 interliquid conductivity for Luttinger liquids, within the so-called ``confinement'' regime where interliquid hopping is completely incoherent. We argue that the interliquid conductivity behaves as ${\ensuremath{\sigma}}_{\ensuremath{\perp}}(\ensuremath{\omega})\ensuremath{\sim}{\ensuremath{\omega}}^{4\ensuremath{\alpha}}$, where $\ensuremath{\alpha}$ is the Luttinger liquid exponent. We discuss the effect of finite temperature, which is found to introduce a coherent weight into the conductivity. These results are in good agreement with experimental measurements of the frequency dependent normal state ccc-axis conductivity in the high-temperature superconductors YBa2{\mathrm{a}}_{2}a2​Cu3{\mathrm{u}}_{3}u3​O ${}_{7\ensuremath{-}\ensuremath{\delta}}$ and La ${}_{2\ensuremath{-}x}$Sr x{}_{x}x​CuO4{\mathrm{O}}_{4}O4​.
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.
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.