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arXiv:cond-mat/9612087

Comment on ``Integrability and Coherence of Hopping between 1D Correlated Electron Systems''

Clarke, David, Steven P. Strong

Published Dec 10, 1996• 0 citations
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Abstract

We comment on recent numerical studies concerning coupled 1D electron liquids (F. Mila and D. Poilblanc, Phys. Rev. Lett. 76, 287 (1996)). In particular, we point out that the importance of integrability observed in the results of these authors for the quantity $P(t)$ is really an indication of the importance of integrability for questions of ergodicity, rather than questions of coherence. We discuss why this is so, and why the quantity $< δN (t) >$ is a more appropriate function for determining whether or not interliquid hopping is coherent.

DOI
10.48550/arxiv.cond-mat/9612087

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The main point of Clarke and Strong's comment, namely that our results concerning the effect of integrability on the hopping of electrons between chains are probably not related to the notion of coherence proposed in \cite{clarke}, is important and well taken. In fact, we have already stressed in our work that integrability affects the intermediate-time behaviour but not the short-time one, while in \cite{clarke} the authors make predictions about the long-time dynamics on the basis of a calculation at short times. The logical conclusion is then that these two effects are in fact different phenomena. In more recent papers we have numerically investigated other physical quantities (spectral function, transverse conductivity in relation to the notion of coherence developed in \cite{clarke}.

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Integrability and Coherence of Hopping between 1D Correlated Electron Systems

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We present numerical evidence that the hopping of electrons between chains described by the $t\ensuremath{-}J$ model is much more coherent in the integrable cases ( $J\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0$ and $J\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2$) than in the nonintegrable ones. This effect is not related to the value of the exponent $\ensuremath{\alpha}$ which is restricted to the interval $[0,1/8]$ when $0\ensuremath{\le}J\ensuremath{\le}2$. We propose that enhanced coherence is characteristic of integrable systems and argue that it might be related to the level statistics.

arXiv:cond-mat/9508033•Physical Review Letters•Jan 8, 1996•12 citations
We comment on recent numerical studies concerning coupled 1D electron liquids (F. Mila and D. Poilblanc, Phys. Rev. Lett. 76, 287 (1996)). In particular, we point out that the importance of integrability observed in the results of these authors for the quantity P(t)P(t)P(t) is really an indication of the importance of integrability for questions of ergodicity, rather than questions of coherence. We discuss why this is so, and why the quantity <δN(t)>< δN (t) ><δN(t)> is a more appropriate function for determining whether or not interliquid hopping is coherent.
The main point of Clarke and Strong's comment, namely that our results concerning the effect of integrability on the hopping of electrons between chains are probably not related to the notion of coherence proposed in \cite{clarke}, is important and well taken. In fact, we have already stressed in our work that integrability affects the intermediate-time behaviour but not the short-time one, while in \cite{clarke} the authors make predictions about the long-time dynamics on the basis of a calculation at short times. The logical conclusion is then that these two effects are in fact different phenomena. In more recent papers we have numerically investigated other physical quantities (spectral function, transverse conductivity in relation to the notion of coherence developed in \cite{clarke}.
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.
We present numerical evidence that the hopping of electrons between chains described by the $t\ensuremath{-}J$ model is much more coherent in the integrable cases ( J=0J\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0J=0 and J=2J\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2J=2) than in the nonintegrable ones. This effect is not related to the value of the exponent $\ensuremath{\alpha}$ which is restricted to the interval [0,1/8][0,1/8][0,1/8] when $0\ensuremath{\le}J\ensuremath{\le}2$. We propose that enhanced coherence is characteristic of integrable systems and argue that it might be related to the level statistics.