Counting statistics for the Anderson impurity model: Bethe ansatz and Fermi liquid study

TL;DR

This work investigates counting statistics and current noise in the Anderson impurity model using two complementary frameworks. The first part employs a Fermi-liquid/Keldysh approach to derive an exact relation between the FCS generating function $\chi(\lambda)$ and the impurity self-energy, yielding universal low-voltage results and perturbative strong-coupling expressions that connect to Nozières’ FL theory. The second part uses Bethe Ansatz to extract qualitative current-noise features, including a robust double-peak structure in the noise at finite magnetic field, by mapping to an effective one-channel problem and computing equilibrium scattering phases. While the FL approach provides a controlled low-energy expansion, the BA analysis offers insight into Kondo-regime behavior across voltages and fields, though with quantitative limitations due to equilibrium/non-equilibrium distinctions. Together, the methods illuminate the universal aspects of FCS in the AIM and reveal characteristic signatures of Kondo physics in current fluctuations.

Abstract

We study the counting statistics of charge transport in the Anderson impurity model (AIM) employing both Keldysh perturbation theory in a Fermi liquid picture and the Bethe ansatz. In the Fermi liquid approach, the object of our principal interest is the generating function for the cumulants of the charge current distribution. We derive an exact analytic formula relating the full counting statistic (FCS) generating function to the self-energy of the system in the presence of a measuring field. We first check that our approach reproduces correctly known results in simple limits, like the FCS of the resonant level system (AIM without Coulomb interaction). We then proceed to study the FCS for the AIM perturbatively in the Coulomb interaction. By comparing this perturbative analysis with a strong coupling expansion, we arrive at a conjecture for an expression for the FCS generating function at O(V^3) (V is the voltage across the impurity) valid at all orders in the interaction. In the second part of the article, we examine a Bethe ansatz analysis of the current noise for the AIM. Unlike the Fermi liquid approach, here the goal is to obtain qualitative, not quantitative, results for a wider range of voltages both in and out of a magnetic field. Particularly notable are finite field results showing a double peaked structure in the current noise for voltages satisfying eV ~ mu H$. This double peaked structure is the ``smoking gun'' of Kondo physics in the current noise and is directly analogous to the single peak structure predicted for the differential conductance of the AIM.

Figures (5)

• Figure 1: A sketch of the distribution of particles in the leads when $\mu_L > \mu_R$, where $\mu_L$ and $\mu_R$ are the chemical potentials in the two leads.
• Figure 2: Plot describing the evolution of both the noise, $S/V$, and the current, $I/V$, as a function of the applied voltage. The scale on the l.h.s. governs the noise while the scale on the r.h.s. governs the current.
• Figure 3: Plots describing the behaviour of the noise, $S/V$, as a function of the applied voltage, $V$, for a variety of magnetic field values.
• Figure 4: Plots describing the behaviour of the differential noise, -$\partial_V S$, as a function of the applied voltage, $V$, for a variety of magnetic field values.
• Figure 5: Plots describing the evolution of the differential noise peaks with increasing magnetic field. In the top panel are plots of the widths of the two peaks, in the middle panel plots of the two peaks' locations, and in the bottom panel, a single plot of both peaks' (identical) height.