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arXiv:cond-mat/0510394 · Physical Review B

Effects of non-Abelian statistics on two-terminal shot noise in a quantum Hall liquid in the Pfaffian state

Cristina Bena B, Chetan Nayak

Published Apr 27, 2006• 35 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.73.155335

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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 \ensuremathν=5∕2\ensuremath{\nu}=5∕2\ensuremathν=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.
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 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.
The tunneling current and shot noise of the current between two fractional quantum hall (FQH) edges in the $\ensuremath{\nu}=12/5$ FQH state in an electronic Mach--Zehnder interferometer are studied. It is shown that the tunneling current and shot noise can be used to probe the existence of k=3k=3k=3 parafermion statistics in the $\ensuremath{\nu}=12/5$ FQH state. More specifically, the dependence of the current on the Aharonov--Bohm flux in the Read--Rezayi state is asymmetric under the change in the sign of the applied voltage. This property is absent in the Abelian--Laughlin states. Moreover, the Fano factor can exceed 12.7 electron charges in the $\ensuremath{\nu}=12/5$ FQH state. This number well exceeds the maximum possible Fano factor in all Laughlin states and the $\ensuremath{\nu}=5/2$ Moore--Read state, which was previously shown to be eee and 3.2e3.2e3.2e, respectively.
Fractional quantum Hall quasiparticles are generally characterized by two quantum numbers: electric charge QQQ and scaling dimension $\mathrm{\ensuremath{\Delta}}$. For the simplest states (such as the Laughlin series), the scaling dimension determines the anyonic statistics of the quasiparticle (the statistical phase $\ensuremath{\theta}=2\ensuremath{\pi}\mathrm{\ensuremath{\Delta}}$). For more complicated states (featuring counterpropagating modes or non-Abelian statistics), knowing the scaling dimension is not enough to extract the quasiparticle statistics. Nevertheless, even in those cases, knowing the scaling dimension facilitates distinguishing different candidate theories for describing the quantum Hall state at a particular filling (such as PH-Pfaffian and anti-Pfaffian at $\ensuremath{\nu}=\frac{5}{2}$). Here, we propose a scheme for extracting the scaling dimension of quantum Hall quasiparticles from thermal tunneling noise produced at a quantum point contact. Our scheme makes only minimal assumptions about the edge structure and features the level of robustness, simplicity, and model independence comparable with extracting the quasiparticle charge from tunneling shot noise.
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 the scaling behavior in the tunneling amplitude when quasiparticles tunnel along a straight path between the two edges of a fractional quantum Hall annulus. Such scaling behavior originates from the propagation and tunneling of charged quasielectrons and quasiholes in an effective field analysis. In the limit when the annulus deforms continuously into a quasi-one-dimensional (1D) ring, we conjecture the exact functional form of the tunneling amplitude for several cases, which reproduces the numerical results in finite systems exactly. The results for Abelian quasiparticle tunneling is consistent with the scaling analysis; this allows for the extraction of the conformal dimensions of the quasiparticles. We analyze the scaling behavior of both Abelian and non-Abelian quasiparticles in the Read–Rezayi -parafermion states. Interestingly, the non-Abelian quasiparticle tunneling amplitudes exhibit non-trivial k -dependent corrections to the scaling exponent.