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arXiv:cond-mat/9508033 · Physical Review Letters

Integrability and Coherence of Hopping between 1D Correlated Electron Systems

Frédéric Mila, Didier Poilblanc

Published Jan 8, 1996• 12 citations
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Abstract

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.

DOI
10.1103/physrevlett.76.287

Related papers

Integrability and coherence of hopping between 1D correlated electron systems (Reply Comment)

Frédéric Mila, Didier Poilblanc

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

Clarke, David, Steven P. Strong

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.

arXiv:cond-mat/9612087•Dec 10, 1996•0 citations

Confinement and transverse conductivity in coupled Luttinger liquids

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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.

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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.

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One dimensional exactly solvable models of strongly correlated electrons of $1/r^2$ hopping and exchange

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Nonintegrability and quantum spin chains

Gerhard H. Muller, J. C. Bonner, John B. Parkinson

This study concerns the concept of nonintegrability in quantum many-body systems, which is related to the important and unresolved problem of quantum chaos. Our findings strongly indicate that nonintegrability affects the reliability of many approximation techniques which have proved to be successful in the study of integrable models. This report is based on finite-size studies of the low-lying spectral excitations of both integrable and nonintegrable 1D quantum spin models. In integrable cases, the characteristic excitation pattern of the infinite system is apparent even in relatively short chains. This is generally not the case in nonintegrable systems where we observe several classes of excitations with qualitatively different character. In some situations, the nature of the lowest-lying excitations actually changes with increasing system size, which makes finite-size studies very vulnerable to misleading conclusions if care is not taken.

Journal of Applied Physics•Apr 15, 1987•6 citations
We present numerical evidence that the hopping of electrons between chains described by the t\ensuremath−Jt\ensuremath{-}Jt\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 \ensuremathα\ensuremath{\alpha}\ensuremathα which is restricted to the interval [0,1/8][0,1/8][0,1/8] when 0\ensuremath≤J\ensuremath≤20\ensuremath{\le}J\ensuremath{\le}20\ensuremath≤J\ensuremath≤2. We propose that enhanced coherence is characteristic of integrable systems and argue that it might be related to the level statistics.
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 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.
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.
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.
One dimensional exactly solvable models of strongly correlated electrons of 1/r21/r^21/r2 hopping and exchange
This study concerns the concept of nonintegrability in quantum many-body systems, which is related to the important and unresolved problem of quantum chaos. Our findings strongly indicate that nonintegrability affects the reliability of many approximation techniques which have proved to be successful in the study of integrable models. This report is based on finite-size studies of the low-lying spectral excitations of both integrable and nonintegrable 1D quantum spin models. In integrable cases, the characteristic excitation pattern of the infinite system is apparent even in relatively short chains. This is generally not the case in nonintegrable systems where we observe several classes of excitations with qualitatively different character. In some situations, the nature of the lowest-lying excitations actually changes with increasing system size, which makes finite-size studies very vulnerable to misleading conclusions if care is not taken.