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arXiv:1401.4581

Non-equilibrium noise in the (non-)Abelian fractional quantum Hall effect

Olaf Smits, Johannes K. Slingerland, Steven H. Simon

Published Jan 18, 2014• 2 citations
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Abstract

We analyse the noise of the edge current of a generic fractional quantum Hall state in a tunnelling point contact system. We show that the non-symmetrized noise in the edge current for the system out-of-equilibrium is completely determined by the noise in the tunnelling current and the Nyquist-Johnson (equilibrium) noise of the edge current. Simply put, the noise in the tunnelling current does not simply add up the equilibrium noise of the edge current. A correction term arises associated with the correlation between the tunnelling current and the edge current. We show, using a non-equilibrium Ward identity, that this correction term is determined by the anti-symmetric part of the noise in the tunnelling current. This leads to a non-equilibrium fluctuation-dissipation theorem and related expressions for the excess and shot noise of the noise in the edge current. Our approach makes use of simple properties of the edge, such as charge conservation and chirality, and applies to generic constructions of the edge theory which includes edges of non-Abelian states and edges with multiple charged channels. Two important tools we make use of are the non-equilibrium Kubo formula and the non-equilibrium Ward identity. We discuss these identities in the appendix.

DOI
10.48550/arxiv.1401.4581

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We analyse the noise of the edge current of a generic fractional quantum Hall state in a tunnelling point contact system. We show that the non-symmetrized noise in the edge current for the system out-of-equilibrium is completely determined by the noise in the tunnelling current and the Nyquist-Johnson (equilibrium) noise of the edge current. Simply put, the noise in the tunnelling current does not simply add up the equilibrium noise of the edge current. A correction term arises associated with the correlation between the tunnelling current and the edge current. We show, using a non-equilibrium Ward identity, that this correction term is determined by the anti-symmetric part of the noise in the tunnelling current. This leads to a non-equilibrium fluctuation-dissipation theorem and related expressions for the excess and shot noise of the noise in the edge current. Our approach makes use of simple properties of the edge, such as charge conservation and chirality, and applies to generic constructions of the edge theory which includes edges of non-Abelian states and edges with multiple charged channels. Two important tools we make use of are the non-equilibrium Kubo formula and the non-equilibrium Ward identity. We discuss these identities in the appendix.
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.
One of the manifestations of the integrability of electron dynamics in one dimension is the existence of non-equilibrium stationary states. Such states emerge near a quantum point contact coupling two quantum Hall edges with different chemical potentials. We use a non-equilibrium bosonization technique to show that the effective temperature in such states at the fractional quantum Hall edge has a universal linear dependence on the current through the contact. In contrast, the temperature of one-dimensional equilibrium systems scales as the square root of the corresponding dissipating power. This striking difference from the equilibrium behavior can be used to experimentally detect these unique non-equilibrium states.
We study nonequilibrium noise in the tunneling current between the edges of a quantum Hall liquid in the Pfaffian state, which is a strong candidate for the plateau at $\ensuremath{\nu}=5∕2$. To first nonvanishing order in perturbation theory (in the tunneling amplitude) we find that one can extract the value of the fractional charge of the tunneling quasiparticles. We note, however, that no direct information about non-Abelian statistics can be retrieved at this level. If we go to higher order in the perturbative calculation of the nonequilibrium shot noise, we find effects due to non-Abelian statistics. They are subtle, but eventually may have an experimental signature on the frequency-dependent shot noise. We suggest how multiterminal noise measurements might yield a more dramatic signature of non-Abelian statistics and develop some of the relevant formalism.
The Kubo fluctuation-dissipation theorem relates the current fluctuations of a system in an equilibrium state with the linear AC-conductance. This theorem holds also out of equilibrium provided that the system is in a stationary state and that the linear conductance is replaced by the (dynamic) conductance with respect to the nonequilibrium state. We provide a simple proof for that statement and then apply it in two cases. We first show that in an excess noise measurement at zero temperature, in which the impedance matching is maintained while driving a mesoscopic sample out of equilibrium, it is the nonsymmetrized noise power spectrum which is measured, even if the bare measurement, i.e. without extracting the excess part of the noise, obtains the symmetrized noise. As a second application we derive a commutation relation for the two components of fermionic or bosonic currents which holds in every stationary state and which is a generalization of the one valid only for bosonic currents. As is usually the case, such a commutation relation can be used e.g. to derive Heisenberg uncertainty relationships among these current components.
This thesis deals with the study of transport properties of integer and fractional QH edge states and it is based on the work I performed during my Ph.D. studies. The focus of this thesis is on Luttinger liquids far from equilibrium and their relaxation dynamics. Since Boltzmann, a fundamental aspect of statistical mechanics has been the understanding of the emergence of an equilibrium state. Interactions play a crucial role in the thermalization process that drives a system through states described by the Gibbs equilibrium ensemble. Therefore, it seems counterintuitive that a strongly interacting system, such as the Luttinger liquid, should not present any relaxation dynamics. This peculiar fact is due to the integrability of the Luttinger model, i.e. the existence of an infinite number of conserved quantities that precludes the equilibration process. However, in the past few years it has become clear that integrable systems can present some kind of relaxation, even though not towards the Gibbs equilibrium ensemble. Remarkably, the necessity of correctly taking into account some particular non-equilibrium configurations, also revealed the necessity of modifying bosonization, a technique widely used to study strongly interacting systems in one dimension. In this work we focus on three different cases: • Relaxation of high energy electrons injected in a ν = 1/3 chiral Luttinger liquid and in a standard Luttinger liquid. • Heating and the emergence of effective temperatures in a Quantum Hall system at fractional filling fraction ν = 2/3 partitioned by a Quantum Point Contact. • Effect of relaxation on shot-noise measurement of the quasi-particle charge in a ν = 2 QH state.
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.