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arXiv:0912.3530 · Physical Review B

Nonequilibrium electron spectroscopy of Luttinger liquids

So Takei, Mirco Milletarì, Bernd Rosenow

Published Jul 22, 2010• 29 citations
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Abstract

We theoretically study a Luttinger liquid (LL) driven out of equilibrium by injection of high-energy electrons. The electrons are injected into the LL locally at a fixed energy and their spectral properties are detected at another spatial point some distance away by evaluating the average tunneling current from the LL into a resonant level with tunable energy. For energies slightly below the injection energy, the dependence of the detected current on the difference between injection, and detection energies is described by a power law whose exponent depends continuously on the Luttinger parameter. In contrast, for tunneling into a chiral LL edge of a fractional quantum-Hall state from the Laughlin sequence, we find that the detected current grows linearly with energy difference, despite a decreasing density of states.

DOI
10.1103/physrevb.82.041306

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We theoretically study a Luttinger liquid (LL) driven out of equilibrium by injection of high-energy electrons. The electrons are injected into the LL locally at a fixed energy and their spectral properties are detected at another spatial point some distance away by evaluating the average tunneling current from the LL into a resonant level with tunable energy. For energies slightly below the injection energy, the dependence of the detected current on the difference between injection, and detection energies is described by a power law whose exponent depends continuously on the Luttinger parameter. In contrast, for tunneling into a chiral LL edge of a fractional quantum-Hall state from the Laughlin sequence, we find that the detected current grows linearly with energy difference, despite a decreasing density of states.
We develop a theory of tunneling spectroscopy of interacting electrons in a nonequilibrium quantum wire coupled to reservoirs. The problem is modeled as an out-of-equilibrium Luttinger liquid with spatially dependent interaction. The interaction leads to the renormalization of the tunneling density of states, as well as to the redistribution of electrons over energies. Energy relaxation is controlled by plasmon scattering at the boundaries between regions with different interaction strengths and affects the distribution function of electrons in the wire as well as that of electrons emitted from the interacting regions into noninteracting electrodes.
We study non-equilibrium noise in Chiral Luttinger Liquids using the Landauer-Buttiker Scattering approach, obtaining the current/voltage noise spectrum for a four-terminal measurement scheme. Experimental consequences of the tunneling of charges are present in the four-terminal measurement of both the low-frequency shot noise (ωωω near 0), and the high-frequency Josephson noise (ωωω near ωJ=e∗V/ℏω_J=e^*V/\hbarωJ​=e∗V/ℏ). Within perturbation theory, an algebraic singularity is present (to all orders) at the Josephson frequency ωJ=e∗V/ℏω_J=e^*V/\hbarωJ​=e∗V/ℏ, whose position depends on the charge e∗e^*e∗ of the tunneling particles, either electrons or fractionally charged quasiparticles. We show in a non-perturbative calculation for an exactly solvable point that the singularity at the quasiparticle frequency exists only in the limit of vanishing coupling, whereas the singularity at the electron frequency is present for all coupling strengths. The vanishing coupling limit corresponds to perfectly quantized Hall conductance in the case of quasiparticle tunneling between edge states in the fractional quantum Hall regime, and thus tunneling destroys the singularity at the quasiparticle frequency concomitantly with the quantized current.
We study nonequilibrium noise in chiral Luttinger liquids using the Landauer-Buttiker scattering approach, obtaining the current and voltage noise spectrum for a four-terminal measurement scheme. Experimental consequences of the tunneling of charges are present in the four-terminal measurement of both the low-frequency shot noise (\ensuremath{\omega} near 0), and the high-frequency Josephson noise (\ensuremath{\omega} near ${\mathrm{\ensuremath{\omega}}}_{\mathit{J}}$=e∗{\mathit{e}}^{\mathrm{*}}e∗V/\ensuremath{\Elzxh}). Within perturbation theory, an algebraic singularity is present at the Josephson frequency ${\mathrm{\ensuremath{\omega}}}_{\mathit{J}}$=e∗{\mathit{e}}^{\mathrm{*}}e∗V/\ensuremath{\Elzxh}, whose position depends on the charge e∗{\mathit{e}}^{\mathrm{*}}e∗ of the tunneling particles, either electrons or fractionally charged quasiparticles. These two types of tunneling are related by a strong-weak--coupling duality transformation. We show in a nonperturbative calculation for an exactly solvable point that the singularity at the quasiparticle frequency exists only in the limit of vanishing coupling, whereas the singularity at the electron frequency is present for all coupling strengths. The vanishing coupling limit corresponds to perfectly quantized Hall conductance in the case of quasiparticle tunneling between edge states in the fractional quantum Hall regime, and thus tunneling destroys the singularity at the quasiparticle frequency concomitantly with the quantized current. \textcopyright{} 1996 The American Physical Society.
We evaluate tunneling rates into/from a voltage (U)(U)(U) biased quantum wire containing a weak backscattering defect. Interacting electrons in such a wire form a true nonequilibrium state of the Luttinger liquid (LL). This state is created due to inelastic electron backscattering leading to the emission of nonequilibrium plasmons with typical frequency $\ensuremath{\hbar}\ensuremath{\omega}\ensuremath{\le}U$. The tunneling rates are split in two edges. The tunneling exponent at the Fermi edge is positive and equals that of the equilibrium LL, while the exponent at the side edge ${E}_{F}\ensuremath{-}U$ is negative if Coulomb interaction is not too strong.
The theoretical model of the short-range interacting Luttinger liquid predicts a power-law scaling of the density of states and the momentum distribution function around the Fermi surface, which can be readily tested through tunneling experiments. However, some physical systems have long-range interaction, most notably the Coulomb interaction, leading to significantly different behaviors from the short-range interacting system. In this paper, we revisit the tunneling theory for the one-dimensional electrons interacting via the long-range Coulomb force. We show that even though in a small dynamic range of temperature and bias voltage, the tunneling conductance may appear to have a power-law decay similar to short-range interacting systems, the effective exponent is scale-dependent and slowly increases with decreasing energy. This factor may lead to the sample-to-sample variation in the measured tunneling exponents. We also discuss the crossover to a free Fermi gas at high energy and the effect of the finite size. Our work demonstrates that experimental tunneling measurements in one-dimensional electron systems should be interpreted with great caution when the system is a Coulomb Luttinger liquid.
Power-law suppression of local electronic tunneling density of states (TDOS) in the zero-energy limit is a hallmark of the Luttinger liquid (LL) phase of the interacting one-dimensional electron system. We present a theoretical model which hosts the LL state with the surprising feature of enhancement rather than suppression in local TDOS originating from nonlocal and repulsive density-density interactions. Importantly, we find enhancement of TDOS in the manifold of parameter space where the system is stable in the renormalization group (RG) sense. We argue that enhancement of TDOS along with RG stability is possible only when the system has broken parity symmetry about the position of local TDOS enhancement. Such a model could be realized on the edge states of a bilayer quantum Hall system where both intralayer and interlayer density-density interactions are present mimicking the role of local and nonlocal interactions, respectively.