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arXiv:1105.1695 · Journal of Mathematical Physics

Full counting statistics in the resonant-level model

Denis Bernard, Benjamin Doyon

Published Nov 6, 2012• 30 citations
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Abstract

We derive the large deviation function, which provides the large-time full counting statistics for the charge transfer, in the non-equilibrium steady state of the resonant-level model. The general form of this function in free fermion models, in terms of transmission coefficients, was proposed by Levitov and Lesovik in 1993 using a particular measurement set-up involving an interacting spin. It was later suggested to hold as well for a proper quantum mechanical measurement of the transferred charge. We give a precise proof of both statements in the resonant-level model. We first give a full description of the model and its steady state. That is, we explain how the decoupled system prepared with a charge differential evolves, with the impurity coupling, towards the Hershfield non-equilibrium density matrix, in the sense of averages of finitely supported operators. We describe how this holds both for the usual resonant-level model with a point-like impurity, and for a regularized model with an impurity spread on a finite region, shedding light on subtleties associated to the point-like impurity. We then prove Levitov-Lesovik formula by recasting the problem into calculating averages of finitely supported operators.

DOI
10.1063/1.4763471

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We derive the large deviation function, which provides the large-time full counting statistics for the charge transfer, in the non-equilibrium steady state of the resonant-level model. The general form of this function in free fermion models, in terms of transmission coefficients, was proposed by Levitov and Lesovik in 1993 using a particular measurement set-up involving an interacting spin. It was later suggested to hold as well for a proper quantum mechanical measurement of the transferred charge. We give a precise proof of both statements in the resonant-level model. We first give a full description of the model and its steady state. That is, we explain how the decoupled system prepared with a charge differential evolves, with the impurity coupling, towards the Hershfield non-equilibrium density matrix, in the sense of averages of finitely supported operators. We describe how this holds both for the usual resonant-level model with a point-like impurity, and for a regularized model with an impurity spread on a finite region, shedding light on subtleties associated to the point-like impurity. We then prove Levitov-Lesovik formula by recasting the problem into calculating averages of finitely supported operators.
Exact numerical results for the full counting statistics (FCS) for a one-dimensional tight-binding model of noninteracting electrons are presented without using an idealized measuring device. The two initially separate subsystems are connected at t=0t=0t=0 and the exact time evolution for the large but finite combined system is obtained numerically via a finite dimensional determinant. Even for surprisingly short times the approximate description of the time evolution with the help of scattering states agrees well with the exact result for the local current matrix elements and the FCS. An additional approximation has to be made to recover the Levitov-Lesovik formula in the limit where the system size becomes infinite and afterwards the long time limit is addressed. The derivation of the Levitov-Lesovik formula is generalized to more general geometries such as a Y junction enclosing a magnetic flux.
Abstract We study charge transport and fluctuations of the (3+1)-dimensional massive free Dirac theory. In particular, we focus on the steady state that emerges following a local quench whereby two independently thermalized halves of the system are connected and let to evolve unitarily for a long time. Based on the two-time von Neumann measurement statistics and exact computations, the scaled cumulant generating function associated with the charge transport is derived. We find that it can be written as a generalization of Levitov–Lesovik formula to the case in three spatial dimensions. In the massless case, we note that only the first four scaled cumulants are nonzero. Our results provide also a direct confirmation for the validity of the extended fluctuation relation in higher dimensions. An application of our approach to Lifshitz fermions is also briefly discussed.
We analyze the full counting statistics (FCS) of the charge transport through the Anderson impurity model (AIM) and similar systems with a single conducting channel. The object of principal interest is the generating function for the cumulants of charge current distribution. We derive an exact analytic formula relating the FCS generating function to the self energy of the system in the presence of the measuring field. We first check that our approach reproduces correctly known results in simple limits, such as the FCS of the resonant level system (AIM without Coulomb interaction). We then proceed to study the FCS for the AIM both perturbatively in the Coulomb interaction and in the Kondo regime at the Toulouse point (we also study a related model of a spinless single-site quantum dot coupled to two half-infinite metallic leads in the Luttinger liquid phase at a special interaction strength). At zero temperature the FCS turns out to be binomial for small voltages. For the generic case of arbitrary energy scales the FCS is shown to be captured very well by generalizations of the Levitov-Lesovik type formula. Surprisingly, the FCS for the AIM indicates a presence of coherent electron pair tunnelling in addition to conventional single-particle processes. By means of perturbative expansions around the Toulouse point we succeeded in showing the universality of the binomial FCS at zero temperature in linear response. Based on our general formula for the FCS we then argue for a more general binomial theorem stating that the linear response zero-temperature FCS for any interacting single-channel setup is always binomial.
This thesis investigates the full-counting statistics (FCS) of charge and spin transport using nonequilibrium Green's function (NEGF) in non-interacting systems. \nGenerating function (GF) encodes all the distribution information and is the key of FCS. For the charge statistics, GF is expressed with respect to the modified evolution operator on the complex contour using the two time measurement scheme, the modified evolution operator is expressed in terms of the modified Hamiltonian. Using Keldysh NEGF technique, GF has the form of Fredholm determinant in terms of Greens functions and self-energies in the time domain. In the long time limit, the celebrated Levitov-Lesovik's formula and the fluctuation relation are derived from this formalism. GF of charge statistics for the quantum point contact system in the transient regime is obtained as well. The transient dynamics in lead-dot-lead transport system, including the cumulants of transferred charges and waiting time distribution defined under the transient regime for different temperatures is investigated numerically. A detailed description on how to calculate the GF in the time domain is presented. Generalizations of the formalism to the ferromagnet-normal-ferromagnet system to investigate the FCS of charge current, spin current and spin transfer torque (STT) are presented. GFs in time domain are Fredholm determinants expressed by NEGF as well. As an application of FCS, a formalism using FCS of STT in the long time limit to calculate the switching probability of a nanomagnet system is proposed. The formalism enables us to calculate the switching probability for nonequilibrium systems which are non-Gaussian. From the stochastic Landau-Lifshitz-Gilbert (LLG) equation, the contributions to the change of the anisotropic energy, one is from the power gain due to Gilbert damping and the other is from power dissipation due to the spin transfer torque, were derived. Optimal path approximation which requires the nanomagnet a modestly large volume is presented and the approximation greatly reduces the numerical complexities. This formalism could be used to predict the switching probability numerically for real systems in combination with the first principle calculation.
We present a general technique to obtain the zero temperature cumulant generating function of the full counting statistics of charge transfer in interacting impurity models out of equilibrium from time-dependent simulations on a lattice. We demonstrate the technique with application to the self-dual interacting resonant level model, where very good agreement between numerical simulations using the density matrix renormalization group and those obtained analytically from the thermodynamic Bethe ansatz is found. We show from the exact form of counting statistics that the quasiparticles involved in transport carry charge 2e in the low bias regime and e/2 in the high bias regime.
We develop a new perturbative method for studying any steady states of quantum impurities, in or out of equilibrium. We show that steady-state averages are completely fixed by basic properties of the steady-state (Hershfield's) density matrix along with dynamical "impurity conditions." This gives the full perturbative expansion without Feynman diagrams (matrix products instead are used), and "resums" into an equilibrium average that may lend itself to numerical procedures. We calculate the universal current in the interacting resonant level model (IRLM) at finite bias V to first order in Coulomb repulsion U for all V and temperatures. We find that the bias, like the temperature, cuts off low-energy processes. In the IRLM, this implies a power-law decay of the current at large V (also recently observed by Boulat and Saleur at some finite value of U).