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arXiv:cond-mat/9508013 · Physical Review Letters

Finite Size Analysis of Luttinger Liquids with a Source of 2kf Scattering

G. Gómez-Santos

Published May 27, 1996• 14 citations
arXivHTMLPDF

Abstract

Numerical analysis of the spectrum of large finite size Luttinger liquids $(g<1)$ in the presence of a single source of ${2k}_{f}$ scattering has been made possible due to an effective integration of high degrees of freedom. The presence of irrelevant operators and their manifestation in transport are issues treated independently. We confirm the existence of two irrelevant operators: particle hopping and charge oscillations, with regions of dominance separated by $g\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2$. Temperature dependence of conductance is shown to be controlled by hopping alone, giving $G\ensuremath{\sim}{T}^{2/g\ensuremath{-}2}$. Frequency dependence is affected by both operators, giving $G\ensuremath{\sim}{\ensuremath{\omega}}^{2/g\ensuremath{-}2}$ for $g>1/2$, and $G\ensuremath{\sim}{\ensuremath{\omega}}^{2}$ for $g<1/2$.

DOI
10.1103/physrevlett.76.4223

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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.
Luttinger liquids occupy a notable place in physics as one of the most understood classes of quantum many-body systems. The experimental mission of measuring its main prediction, power laws in observable quantities, has already produced a body of exponents in different semiconductor and metallic structures. Here, we combine tunneling spectroscopy with density-dependent transport measurements in the same quantum wires over more than two orders of magnitude in temperature to very low electron temperatures down to ∼40 mK. This reveals that, when the second 1D subband becomes populated, the temperature dependence splits into two ranges with different exponents in the power-law dependence of the conductance, both dominated by the finite-size effect of the end-tunneling process. This result demonstrates the importance of measuring the Luttinger parameters as well as the number of modes independently through spectroscopy in addition to the transport exponent in the characterization of Luttinger liquids. This opens a pathway to unambiguous interpretation of the exponents observed in quantum wires.
The stability of the Luttinger liquid to small transverse hopping has been studied from several points of view. The renormalization group approach in particular has been criticized because it does not take explicitly into account the difference between spin and charge velocities and because the interaction should be turned-on before the transverse hopping if the stability of the Luttinger liquid is a non-perturbative effect. An approach that answers both of these objections is explained here. It shows that the Luttinger liquid is unstable to arbitrarily small transverse hopping. The crossover temperatures below which either transverse coherent band motion or long-range order start to develop can be finite even when spin and charge velocities differ. Explicit scaling relations for these one-particle and two-particle crossover temperatures are derived in terms of transverse hopping, spin and charge velocities and anomalous exponents. The renormalization group results are recovered as special cases when spin and charge velocities are identical. The results compare well with recent experiments presented at this conference. Magnetic field effects are alluded to.
We obtain a closed-form analytical expression for the zero-temperature Fourier transform of the 2kF2{k}_{F}2kF​ component of the density-density correlation function in a Luttinger liquid with different spin and charge velocities. For frequencies near the spin and charge singularities, approximate analytical forms are given and compared with the exact result. We find power-law-like singularities leading to either divergence or cusps, depending on the values of the Luttinger parameters, and compute the corresponding exponents. Exact integral expressions and numerical results are given for the finite-temperature case as well. We show, in particular, how the temperature rounds the singularities in the correlation function.
In a first part, I study the electronic transport through a one-dimensional, finite-length quantum wire of correlated electrons (Luttinger liquid) coupled at arbitrary position via tunnel barriers to two semi-infinite, one-dimensional as well as stripe-like (two-dimensional) leads, thereby bringing theory closer towards systems resembling setups realized in experiments. In particular, I compute the temperature dependence of the linear conductance of a system without bulk impurities. The appearance of new temperature scales introduced by the lengths of the overhanging parts of the leads and the wire implies a conductance function which is much more complex than the simple power-law behavior obtained in earlier approaches. My results can be used to optimize the experimental setups designed for a verification of Luttinger-liquid scaling. In a second part, I suggest a setup to study Luttinger-liquid behavior in quantum wires which allows to determine the Luttinger-liquid parameter from two independent measurements: transport through a quantum dot embedded in the wire and the charge on the dot. To this end, I identified novel Luttinger-liquid power-laws in the charging of the dot. In a more technical part, I extended an adaption of the functional renormalization group such that it can be used on correlated electrons in complex geometries like the wire-lead structures described above and arbitrary dot structures coupled to Luttinger-liquid leads.