Exact calculation of current correlations and admittance in the fractional quantum Hall regime

The gist The study focuses on calculating finite frequency current-current correlations and quantum admittance in the fractional quantum Hall regime using a refermionization method, which provides relations between these quantities.

Model Framework

The two-dimensional electron gas in the fractional quantum Hall regime is modeled using the Tomonaga-Luttinger theory Hamiltonian (1). This Hamiltonian describes incompressible chiral edge modes with a backscattering amplitude at the constriction denoted by ΓB (2). The refermionization method is applied for ν = 1/2 to calculate transport properties, although it yields interesting features on the relation between admittance and current correlations in such a system (3).

Quantum Admittance Calculation

The quantum admittance Y(ω) is obtained by calculating the photo-assisted current, which is the response to an ac voltage superimposed to a dc voltage Vdc (2). The expression for Y(ω) involves integrals over time and frequency, incorporating the transmission amplitude t(omega) through the constriction (2).

Current-Current Correlations

The Fourier transform of nonsymmetrized correlations between currents in branches i and j is given by Sij (3). Auto-correlators S1 and S2 are related to charges free to propagate along the edge states, while auto-correlators S5 and S6 are those of charges that have been reflected or transmitted through the constriction (4). Cross-correlations exhibit various relationships, such as S3(ω) = 0 and S7(ω) = S8(ω) (9).

Discussion of Excess Correlators

The excess correlators, defined as the difference between values at finite dc voltage Vdc and zero voltage, are shown to be equal four to four (12). The excess correlators associated with diagrams 5, 6, 7 et 8 are related to S0 and the real part of the excess admittance (13). Furthermore, the excess correlators associated with diagrams 9, 10, 11 and 12 are related to the excess admittance as ∆Sn in [9,12] = hωN(hω)∆Y (14), and diagrams 13, 14, 15 and 16 are related to ∆Sn in [9,8] = hωN(hω)∆Y∗ (15).

Frequency Limits

In the low temperature limit, the excess correlators are entirely determined by the excess admittance (20). The singularities observed at ±eVdc/h in weak backscattering regimes are directly related to the singularities of the quantum admittance (21). In both low and high temperature limits, all current correlators are fully determined by the admittance and explicit relations between them have been established (23).

Conclusion

The study demonstrates that all current correlators are closely related because they can be expressed in terms of the transmission amplitude through the constriction which couples the edge states (5). The singularities observed in the correlators in weak backscattering and low temperature regimes at frequencies equal to ±eVdc/h are directly related to the singularities of the quantum admittance (21). Moreover, all current correlators are fully determined by the admittance and explicit relations between them have been established (23).

ACKNOWLEDGMENT The authors would like to thank T. Martin and I. Safi (5).

REFERENCES

arXiv:1305.4829v2 3 Oct 2013

Exact calculation of current correlations and admittance in the fractional quantum Hall regime Redouane Zamoum Aix-Marseille Universit´e, Universit´e de Toulon CNRS, CPT UMR 7332, 13288, Marseille, France Email: zamoum.redouane@gmail.com Adeline Cr´epieux Aix-Marseille Universit´e, Universit´e de Toulon CNRS, CPT UMR 7332, 13288, Marseille, France Email: adeline.crepieux@cpt.univ-mrs.fr

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arXiv:1305.4829v2 [cond-mat.mes-hall] 3 Oct 2013 Exact calculation of current correlations and admittance in the fractional quantum Hall regime Redouane Zamoum Aix-Marseille Universit´e, Universit´e de Toulon CNRS, CPT UMR 7332, 13288, Marseille, France Email: zamoum.redouane@gmail.com Adeline Cr´epieux Aix-Marseille Universit´e, Universit´e de Toulon CNRS, CPT UMR 7332, 13288, Marseille, France Email: adeline.crepieux@cpt.univ-mrs.fr

Abstract—In this work we focus on the finite frequency current-current correlations between edge states in a fractional quantum Hall two dimensional gas and on their relations to the quantum admittance Using a refermionization method we calculate these quantities within the same framework Our results apply whatever the values of backscattering amplitude frequency voltage and temperature allowing us to reach different regimes Auto-correlations and cross-correlations exhibit distinct frequency dependencies that we discuss in detail I. INTRODUCTION Both the current noise and the admittance provide information about the dynamics of mesoscopic conductors However whereas the current fluctuations have been widely studied in interacting systems it is not the case of the quantum admittance for which only few theoretical works are available [1], [2], [3], [4], [5] Moreover, the inter-relations between these quantities have not been deeply explored despite the fact that they verify the simple relation: 2¯hωRe[Y (ω)] = S(−ω) − S(ω) which means that a non-vanishing real part of the admittance Y is responsible of the asymmetry of the noise S obtained in interacting systems [6], [7], [8] The admittance is a quantity which deserves to be known especially as its experimental determination expends rapidly Indeed following the pioneer works by Gabelli and co-workers [9], [10], [11] measurements of the quantum admittance in various mesoscopic systems have been achieved recently In particular high-frequency admittance has been measured for a quantum point contact in the quantum Hall regime [12] determination of the admittance has been performed for a carbon nanotube double quantum dot [13] both highfrequency admittance and noise have been measured in superconductor/insulator/superconductor junction [14] and dynamic admittance has been studied in a quantum dot coupled to a two dimensional electron gas [15] It is thus needed to develop in parallel theoretical studies of quantum admittance Here we consider a two dimensional electron gas in the fractional quantum Hall regime and we carefully look at the inter-relations between admittance which was calculated in Ref. [4] and current-current correlations between edge states along which the fractional charges are propagated The paper is organized as follows: In Sec. II, we present the model used to describe the chiral edge states in the fractional quantum Hall regime and the method of calculation of the current-current correlations generated by a constriction (see figure 1) In Sec. III, we give the expressions of the auto- and cross-correlators in terms of current admittance and integral which involve all the transmission amplitude through the constriction In Sec. IV, we discuss the results in various limits We conclude in Sec. V

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Fig. 1 Schematic representation of all possible current-current correlators between two chiral edge states coupled by a constriction The diagrams 1, 2, 5 and 6 gives the auto-correlators the remaining diagrams correspond to the cross correlators The values 0 and τ in the black boxes indicate the associated time arguments of the current operators in Eq. (3) where f is the Fermi-Dirac distribution function and t(omega) = (iΓB/2)/(¯homega +iΓB/2) is the transmission amplitude through the constriction

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Next we calculate the Fourier transform of the nonsymmetrized correlations between currents in branches i and j associated to Eq.

1. Direct Noise Characterization from Admittance: The improved system can directly determine the value of the finite-frequency noise from the measurement of the admittance which is easier to perform, as stated in Section V, by using explicit relations derived from the paper's findings to find noise characteristics.

2. Finite Frequency Noise Calculation via Admittance: The system can calculate excess noise components using equations like ∆ST(ω) = ∆S0(ω) − ¯hωRe[∆Y (ω)] and ∆SB(ω) = ∆ST (ω) − 4¯hωN(¯hω)Re[∆Y (ω)], allowing for noise characterization based on measurable admittance data.

3. Frequency-Dependent Noise Profile Prediction: The AI can predict the frequency profile of excess correlators, specifically showing how they are entirely determined by the admittance in both low and high temperature limits, enabling prediction of singularities at frequencies equal to ±eV¯ dc/¯h based on the real part of the admittance.

4. Backscattering Sensitivity Analysis: The system can analyze how noise behavior changes with backscattering strength, noting that the singularities observed at ¯hω = ±eV¯ dc in figures 2(a), 2(b) and 2(c) for weak backscattering amplitudes are those of the admittance, and these singularities disappear when the backscattering amplitude increases.

5. Temperature Dependence Modeling: The system can model the noise response across temperature regimes, determining that in the low-temperature limit, the excess correlators are entirely determined by the excess admittance, and in the high-temperature limit, they are related to the fluctuation-dissipation theorem since the real part of the admittance correspond to the ac conductance.