Theory of non-equilibrium transport in the SU(N) Kondo regime

Theory of non-equilibrium transport in the SU(N) Kondo regime

Christophe Mora    Pavel Vitushinsky    Xavier Leyronas    Aashish A. Clerk    Karyn Le Hur Laboratoire Pierre Aigrain, ENS, Université Denis Diderot 7, CNRS; 24 rue Lhomond, 75005 Paris, France Department of Physics, McGill University, Montréal, Québec, Canada, H3A 2T8 Laboratoire de Physique Statistique de l’Ecole Normale Supérieure associé au CNRS et aux Universités Paris 6 et Paris 7, 24 rue Lhomond, F-75005 Paris, France Department of Physics, Yale University, New Haven, Connecticut, USA, 06520
Abstract

Using a Fermi liquid approach, we provide a comprehensive treatment of the current and current noise through a quantum dot whose low-energy behaviour corresponds to an SU() Kondo model, focusing on the case relevant to carbon nanotube dots. We show that for general , one needs to consider the effects of higher-order Fermi liquid corrections even to describe low-voltage current and noise. We also show that the noise exhibits complex behaviour due to the interplay between coherent shot noise, and noise arising from interaction-induced scattering events. We also treat various imperfections relevant to experiments, such as the effects of asymmetric dot-lead couplings.

pacs:
71.10.Ay, 71.27.+a, 72.15.Qm

I Introduction

The Kondo effect has long served as a paradigm in the field of strongly correlated electron physics. It is perhaps the simplest example of a system where many-body interactions can give rise to highly non-trivial behavior: its essence involves nothing more than a localized magnetic impurity which is exchange coupled to conduction electrons in a metal. Despite having been studied for over 40 years, interest in Kondo physics shows no sign of abating. A large part of this continued interest has been fueled by recent advances allowing the controllable realization of unusual Kondo effects in nanostructures. These include multi-channel Kondo effects nozieres1980 (), where there are many conserved flavours of conduction electrons: such systems can give rise to non-Fermi liquid physics, and have recently been realized using semiconductor quantum dots potok2007 (). Another class of exotic Kondo effects are so called SU() Kondo effects, where . Such systems involve only a single channel of conduction electrons, but the effective spin of the impurity and conduction electrons is greater than . While such systems are still described at low-energies by a Fermi liquid fixed point, the properties of this Fermi liquid are modified in several interesting ways compared to the spin- case mora2009b (). The case has received particular attention due to its realizability in double borda2003 (); lehur2004 (); lopez2005 () and triple quantum dots numata2009 () as well as carbon nanotube quantum dots choi2005 (); jarillo2005 (); choi2006 (); finkelstein2007 (); lehur2007 ().

Research on Kondo physics has also been spurred by the possibility of studying experimentally its behaviour when driven out-of-equilibrium, where non-equilibrium is either achieved by the application of a drain-source voltage across a quantum dot silvano2002 (); gg2008 (), or by externally radiating a quantum dot elzerman2000 (). The non-equilibrium induced by a voltage has been the subject of a number of recent theoretical works rosch2003 (); kehrein2005 (); mehta2006 (); doyon2007 (); boulat2008 (); anders2008 (); lehur2008 ().

In this paper, we will focus on a topic which combines two of the above avenues of Kondo research: we will study non-equilibrium charge transport through a voltage-biased quantum dot exhibiting an SU() Kondo effect, focusing on the low-temperature regime where the physics is described by an effective Fermi liquid theory. We present calculations for both the non-linear conductance as well as for the current noise. As has been stressed in a number of recent papers sela2006 (); mora2008 (); vitu2008 (), the fluctuations of current through a Kondo quantum dot are extremely sensitive to the two-particle interactions associated with the underlying Fermi liquid theory. This was first discussed in the case of the standard SU() Kondo effect by Sela et al. sela2006 () , and was even measured for this system in a recent experiment by Zarchin et al. heiblum2008 (). As discussed in Refs. vitu2008 (); mora2008 (), the situation becomes even more interesting for , as now one must deal with the interplay between coherent partition noise (due to the zero-energy transmission coefficient through the dot not being one) and the interaction-induced scattering events. Of particular interest is the case , which can be realized in carbon nanotube quantum dots. Very recently, current noise in such a system has been measured experimentally by Delattre et al. delattre2009 (), though not in the low-temperature Fermi liquid regime we describe here.

The results presented here both clarify and extend those presented in Refs. mora2008 (); vitu2008 (), as well as provide details underlying the calculational approach. Particular attention is given to the role of higher-order Fermi liquid corrections, something that was not correctly treated in previous works (see erratum, Ref. mora2009 ()). We show clearly how in the case, such corrections lead to an effective shift of the Kondo resonance with applied bias voltage. As a result, the non-linear conductance does not increase with voltage, as would be expected from a simple picture of the Kondo resonance as a resonant level sitting above the Fermi energy. These Fermi-liquid energy shifts are absent in the usual Kondo effect. We also describe the experimentally-relevant case where there is an asymmetry in the coupling between the quantum dot and the source and drain electrodes. Such an asymmetry has not been investigated thoroughly in previous works.

The remainder of this paper is structured as follows. In Sec. II, we outline the basic description of our model and the Fermi liquid approach. Sec. III and IV are devoted to providing a detailed discussion of our results for both the conductance and the shot noise, as well as details on their derivation. In Sec. V, we summarize our main results for the conductance and shot noise of a SU(N) Kondo quantum dot, and conclude.

Ii Model description

ii.1 Kondo Hamiltonian

We give here a compact synopsis of the quantum dot model we study, and how it gives rise to Kondo physics. The dot connected to the leads is described by the following Anderson Hamiltonian Krishna1980 ()

(1)

is the annihilation operator for an electron of spin and energy (measured from the Fermi energy ) confined on the left/right lead. is the electron operator of the dot and the corresponding density. denotes the charging energy, the single particle energy on the dot and the tunneling matrix elements from the dot to the left/right lead. The general case of asymmetric leads contacts is parametrized by , with . recovers the symmetric case. The rotation in the basis of leads electrons

(2)

decouples the operators from the dot variables. The Kondo screening then involves only the variables. In the symmetric case, , and represent respectively even and odd wavefunctions with the dot placed at .

We consider in this work the Kondo limit where the charging energy is by far the largest energy scale. Below this energy, the charge degree of freedom on the dot is quenched to an integer value and does not fluctuate. For , the number of electrons is . The virtual occupation of other charge states by exchange tunneling with the leads is accounted for by the standard Schrieffer-Wolff transformation schrieffer1966 () (or second order perturbation theory). It transforms Eq. (1) to the Kondo Hamiltonian where

(3)

and . This Hamiltonian acts in the subspace constrained by . In this paper, we concentrate on the choice for which potential scattering terms vanish after the Schrieffer-Wolff transformation. Including potential scattering in the formalism is possible, for example along the line of Ref. affleck1993 (). It however remains outside the scope of this work where we focus on the asymmetric dot-lead couplings.

The lead electrons transform under the fundamental representation of SU(N). With exactly electrons, the localized spin on the dot transforms as a representation of SU(N) corresponding to a single column Young tableau of boxes. A basis of generators for this SU(N) representation is formed by the traceless components with . This basis can be used parcollet1998 () to rewrite Eq. (3) as an antiferromagnetic coupling

(4)

between the impurity (dot) spin and the spin operator of the lead electrons taken at , . The matrices are generators of the fundamental representation of SU(N), while are matrices acting on states with electrons.

Starting from high energies, grows under renormalization. It presages the complete screening of the dot spin by the formation of a many-body SU(N) singlet in the ground state. A large body of studies has shown that the strong coupling fixed point that dominates at low energy is a Fermi liquid one. Exact results from the Bethe-Ansatz bazhanov2003 () find low energy exponents that characterize a Fermi liquid. Writing the Kondo Hamiltonian (3) in terms of current, Affleck affleck1990 () has shown by completing the square that the impurity spin can be absorbed by lead electrons. The resulting (conformal field) theory is that of free fermions and it is believed to describe the strong coupling fixed point. It shows a simple translation of energies in the spectrum corresponding to an electron phase shift imposed by the Friedel sum rule

(5)

The identification of the leading irrelevant operator at this fixed point yields Fermi liquid behavior affleck1993 (). Alternatively and following Ref. nozieres1980 () the ground state of Eq. (4) has been shown parcollet1998 () to be a singlet state. Turning the coupling to the leads does not destabilize this singlet leading again to Fermi liquid exponents. Finally, Numerical Renormalization Group (NRG) calculations have confirmed this picture for SU(2) wilson1975 () and SU(4) borda2003 (); lehur2004 (); choi2005 (); choi2006 ().

ii.2 Fermi liquid theory

We now discuss in detail the Fermi liquid theory for the Kondo effect, first introduced by Nozières nozieres1974a+nozieres1978 (). It describes the low energy regime - the vicinity of the strong coupling fixed point - and allows one to make quantitative predictions even in an out-of-equilibrium situation. In Ref. mora2009b (), the Fermi liquid theory of Nozières has been extended with the introduction of the next-to-leading order corrections to the strong coupling fixed point. These corrections are necessary in the SU(N) case for observables like the current and the noise since their energy (, or ) dependence is mostly quadratic.

The Kondo many-body singlet (also called the ‘Kondo cloud’) having been formed, we wish to describe how lead electrons scatter off it. At low energies, two channels open: an elastic and an inelastic one. Both take place at the dot position . Elastic scattering is described by an energy-dependent phase shift. At the Fermi level , it is equal to , see Eq. (5). We expand the phase shift around the Fermi energy,

(6)

where the energy is measured from . and are dimensionless coefficients of order one.

It is instructive to think of the elastic scattering off the Kondo singlet in terms of an effective non-interacting resonant level model (RLM), where this effective resonance represents the many-body Kondo resonance. This is the picture of the Kondo effect provided by slave-boson mean-field theory coleman1984 (), and is an exact description of the SU() Kondo effect in the large limit newns1987 (). Note that for finite , one must also deal with true two-particle scattering off the singlet, something that will never be captured by the RLM; we thus only use it to obtain insight into the elastic scattering properties. In the RLM picture, the first two terms in the phase shift in Eq. (6) are attributed to a Lorentzian scattering resonance centered at with a width vitu2008 (). In the SU() case, one thus finds that the Kondo resonance is centered at a distance above the Fermi energy, giving a heuristic explanation for the fact that the low-energy transmission coefficient through the dot is only . The fact that the Kondo resonance sits above the Fermi energy is indeed seen in exact NRG calculations of the impurity spectral density choi2005 (); choi2006 ().

The low energy expansion of the RLM phase shift also gives the form Eq. (6) with . Note that there is no apriori reason that this relation must hold for the expansion of the true phase shift, as the correspondence to a non-interacting resonant level is not exact. Despite this caveat, one finds that the prediction from the RLM picture is quite good even at a quantitative level. The exact relation between and is extracted mora2009b () from the Bethe ansatz solution bazhanov2003 () and reads

(7)

where is given Eq. (5). In the SU(2) case, or more generally for a half-filled dot with , , corresponding to a Kondo resonance centered at the Fermi level. This is expected for a model where particle-hole symmetry is not broken. In the SU(4) case, Eq. (7) gives mora2009 () instead of in the RLM. As expected, the agreement becomes even better at larger , and RLM result is indeed the limit of Eq. (7).

The phase shift in Eq. (6) completely characterizes the low-energy elastic scattering off the Kondo singlet. For further calculations, it is useful to describe it using a Hamiltonian formulation. The free Hamiltonian describing purely elastic scattering is given by

(8)

Decoupled from the outset, the variables are the same as in the original model. In contrast, the variables have been modified to now include the elastic phase shift in Eq. (6). This point will be expanded on in Sec. III when we discuss the calculation of the current.

We turn now to inelastic effects, which arise from quasiparticle interactions in the Fermi liquid theory. These interactions can be written in a Hamiltonian form mora2009b ()

(9)

where denotes normal ordering and is the density of state for 1D fermions moving along one direction. To summarize, the Fermi liquid theory is generated by the Hamiltonian , given by Eqs. (8) and (9), with the elastic phase shift (6). In fact, Eqs. (6) and (9) correspond to a systematic expansion of the energy nozieres1974a+nozieres1978 (); mora2009b (), compatible with the SU(N) symmetry and the Pauli principle. It includes all first and second order terms in the low energy coupling strength .

The great advantage of the Fermi liquid approach is that it can also be applied to non-equilibrium situations. Note that the Fermi level appears twice in the above equations: it defines the reference for energies in the elastic phase shift (6) and also for the normal ordering in Eq. (9). When the system is put out-of-equilibrium, for instance when each lead has its own Fermi level, loses its meaning as a Fermi level and becomes merely an absolute energy reference. This can be used to relate mora2009b () the coefficients as we shall show below.

ii.3 Kondo floating and perturbation theory

To make progress in calculating physical observables at low energies, we will treat the interacting part (c.f. Eq. (9)) of the Fermi-liquid Hamiltonian perturbatively. Among the various diagrams built from Eq. (9), it is convenient to separate the trivial Hartree contributions to the electron self-energy from the more complicated diagrams. The former are obtained by keeping an incoming and an outgoing line and by closing all other external lines to form loops as shown Fig. 1.

Figure 1: Examples of Hartree diagrams for the self-energy built from Eq. (9). The full dots (resp. black and grey) indicate vertices with four or six external lines. , and denote spins.

The resulting diagrams are then in correspondence with the diagrams describing scattering by a local potential. Therefore they can be included in the elastic phase shift,

(10)

where we have defined and . is the actual quasiparticle distribution () relative to the ground state with Fermi energy . We see again that sets the reference in Eq. (10) for both and . Including Hartree diagrams is essentially tantamount to a mean-field treatment of the interaction term Eq. (9). On a physical level, these Hartree terms can be interpreted as a mean-field energy shift of the Kondo resonance arising from a finite quasiparticle population and their interactions. We shall see that in the case of an SU() Kondo quantum dot, these Hartree terms play a significant role in determining the non-linear conductance; this is not the case in the more conventional SU() Kondo effect.

While the idea of perturbatively treating is straightforward enough, a possible weakness of the Fermi liquid approach is the number of seemingly undetermined parameters in Eqs. (6) and (9). The standard Fermi liquid treatment of the Kondo effect allows one to relate the coefficients and via the so-called ‘floating’ of the Kondo resonance (to be discussed below); these coefficients correspond to leading-order Fermi liquid corrections. However, for transport quantities in the general SU() Kondo case, we will see that the remaining coefficients, corresponding to higher-order corrections, are also important. Luckily, these too can be related to one another using a novel and powerful extension of the Kondo floating recently proposed in Ref. mora2009b (). It allows to relate the different phenomenological coefficients of Eqs. (6) and (9); we describe the basic reasoning involved in what follows.

The Kondo resonance is a many-body phenomenon that results from the sharpness of the Fermi sea boundary kondo1964 (). Physically, conduction electrons build their own resonance. The structure of this resonance is therefore changing with the conduction electron occupation numbers, as Eq. (10) shows explicitly. However it can not depend on , which is a fixed energy reference. This idea is implemented by shifting the Fermi level by while keeping the absolute energy and the absolute occupation numbers fixed in Eq. (10). As a result and . Imposing invariance of the phase shift leads to the following Fermi liquid identities

(11a)
(11b)

where the first relation (11a) was initially derived for the general SU() case by Nozières and Blandin nozieres1980 ().

Note that an alternative way to derive Eqs. (11) is to insist that the entire structure of the Kondo resonance simply translates in energy when we dope the system with quasiparticles in a way that corresponds to a simple increase of the Fermi energy mora2009b (). Nozières’ original derivation of Eq.(11a) in the SU() case nozieres1974a+nozieres1978 () also used this idea, but restricted attention to an initial state with no quasiparticles. Eqs.(11b) follow when we apply the same reasoning to an initial state having some finite number of quasiparticles. Note that for SU(2), or a half-filled dot (), from Eq. (7) so that and . The next-to-leading order corrections all vanish in agreement with previous works on the ordinary SU(2) case nozieres1974a+nozieres1978 (); meir2002 (); golub2006 (); sela2006 ().

It is worth mentioning that the second generation of Fermi liquid terms can also be derived in the framework of conformal field theory. In Ref. mora2009b (), a single cubic Casimir operator is given, which reproduces the three terms corresponding to the coefficients , and . The identities (11b) are then automatically satisfied.

The floating of the Kondo resonance (and resulting conditions) also has an important consequence for calculations of observables in the presence of a voltage: the results will not depend on where one decided to place the dot Fermi energy within the energy window defined by the chemical potentials of the leads. On a technical level, this is because, by virtue of Eqs. (11), any shift of the dot Fermi energy will be completely compensated by a corresponding shift in the Hartree contributions arising from the quasiparticle interactions. This invariance is explained in detail in Fig. 2. Note also that this invariance has physical consequences as well: it implies, for example, that the current is not affected by the capacitive coupling to the leads (in the Kondo limit).

Figure 2: Diagrammatic construction for the independence of observables in . Crosses correspond to elastic scattering. Two- and three-particle interactions are represented by, respectively, black and grey full circles. Many diagrams in the perturbative expansion in  (9) exhibit a dependence in . Nevertheless, it is possible to gather and combine those diagrams to produce -invariant forms. The combination (a) that appears in the irreducible self-energy does not depend on as a result of Eq. (11a). Combination (b) is a second invariant, thanks to Eqs. (11b), contributing to the irreducible self-energy. (a) and (b) together imply the phase shift invariance discussed in the text. Finally, the four-particle vertices of (c) can always be combined to cancel the dependence in thanks to Eq. (11b). Apart from (a), (b) and (c), all other diagrammatic parts involve energy differences in which the reference naturally disappears. The combinations (a), (b) and (c) can be understood as emerging from Ward identities related to the U(1) gauge symmetry. For example, Eq. (11a) has been shown hewson1993 (); hewson1994 () to derive from a Ward identity with a vanishing charge susceptibility.

Given the above invariance, it is convenient for calculations to choose the Fermi level such that

(12)

so that any closed fermionic loop built from an energy-independent vertex vanishes. For this choice of position, vanishes which greatly simplifies the phase shift expression (10). Moreover, the vertex in Eq. (9) does not contribute to the current and the noise when the perturbative calculation is stopped at second order. The reason for that is that the vertex is already second order and can only appear once. Its six legs are connected to at most two current vertices so that at least two of these legs must connect to form a closed loop implying a vanishing contribution. In contrast to these simplifications, in Eq. (10) remains generally different from zero due to the energy dependence of the vertex in Eq. (9).

On may wonder whether the physical argument of the floating of the Kondo resonance, as presented in Ref. mora2009b () and repeated in this paper, is sufficient to extend the results of this paper to higher orders Fermi liquid corrections. Applying the floating argument to the next (third) order, one obtains an incomplete set of relations between the coefficients such that some of them remain undetermined. In the language of conformal field theory, it means that more than one operator is involved at each (higher) order. How to relate the coefficients of those operators is a rather difficult problem. In the SU(2) case, a solution was given by Lesage and Saleur lesage1999a+lesage1999b ().

We finally turn to the discussion of the Fermi liquid model renormalization. Treated naively, the model leads to divergences in physical quantities. It is regularized affleck1993 () by introducing an energy cutoff (different from the original band width of the model) larger than typical energies of the problem but smaller than . Energies in Eq.(8) are therefore restricted to the window . The dependence of observables in is then removed by adding counterterms in the Hamiltonian. It is strictly equivalent to the introduction of cutoff dependence in the coupling constants mora2003 () (, , etc). The corresponding counterterms are discussed in Appendix A.

Iii Current calculation

We now outline the calculation of the current using the Fermi liquid theory described in previous sections. Again, the complete Hamiltonian is (c.f. Eqs. (8,9)), corresponding to respectively to elastic and inelastic scattering; the approach will be to treat as a perturbation. Slightly abusing terminology, we will include all Hartree contributions arising in perturbation theory in the free Hamiltonian ; will thus correspond to the elastic phase shift given in Eq. (10). Contributions to the current which only involve (thus defined) will be referred to as the ‘elastic current’. is then added perturbatively, without Hartree diagrams, in order to compute the corrections due to inelastic scattering.

iii.1 The current operator

The current operator at is generally given by

(13)

where is the electron mass. Various expressions can be obtained for the current depending on which basis it is expanded. It is convenient mora2008 () in our case to choose the basis of scattering states that includes completely elastic (and Hartree terms) scattering, i.e. the phase shift (10), and that correspond to eigenstates of the single-particle scattering matrix. Such states will have waves incident from both the left and right leads. This is in contrast to another standard choice vitu2008 (), which is to use scattering states which either have an incident wave from the left lead, or from the right. We refer to such states as the ‘left/right’ states.

We first discuss our scattering states in first quantization. Eigenfunctions corresponding to the variables do not see the dot or the Kondo effect. Using Eq. (2), they read

(14)

where measures the asymmetry of the coupling to the leads, see Eq. (2), and the eigenenergies are measured from the Fermi level . The situation is more complicated for the variables. The associated eigenfunctions at small , close to the dot, depend on the complex ground state wavefunction of the Kondo problem. They are not known, and in fact it can not even be reduced to a one-particle problem. However, we can write the eigenfunctions far from the dot,

(15)

where the matrix is related to the phase shift (10), at eigenenergy . The eigenstates (14) and (15) have the same energy. They can be combined to give the left and right scattering states with the energy-dependent transmission . In the SU(2) case (or generally particle-hole symmetric case), and the system is closed to unitarity for symmetric leads coupling.

We come back to second quantization and project the electron operator over the eigenstates (14) and (15). Conservation of the current implies that does not depend on . We choose an arbitrary far from the dot, is the current at and at . If denotes the conserved current, . The combination leads to the compact expression

(16)

with and . Physically, operators taken at () correspond to incoming (outgoing) states blanter2000 (). The second line in Eq. (16) turns out not to contribute to the mean current, the noise or any moment of the current.

Before proceeding with the calculation, it is worth noting that in the SU(2) case, the proximity to the unitary situation allows a simpler treatment kaminski2000 (). The current is written with . All quantum or thermal fluctuations are included in the backscattering current which can be written in terms of and operators golub2006 (); gogolin2006b (). However, the range of application of this approach is restricted to the SU(2) case with a completely symmetric leads coupling. In any other situations neglecting fluctuations in is incorrect mora2008 (); vitu2008 () and Eq. (16) becomes necessary.

iii.2 Elastic contribution to the current

We are now in a position to compute the mean value of the current in an out-of-equilibrium situation. A dc bias is applied between the two electrodes imposing . Left and right scattering states, corresponding to and operators, are in thermal equilibrium with chemical potentials and . Hence, using Eq. (2), we obtain the populations

(17a)
(17b)
(17c)
(17d)

for all spins . Eq. (12), that implies a vanishing Hartree diagram, is satisfied with and . is the Fermi distribution.

The average current is obtained from Eq. (16) and reproduces the Landauer-Büttiker formula blanter2000 ()

(18)

with the transmission

(19)

and the phase shift

(20)

where we have used the identity (11b), . Here, the phase shift has an extra dependence due to mean-field (Hartree) interaction contributions (cf. Eq. (10)). Within the heuristic resonant-level picture, we can interpret this as the voltage inducing a quasiparticle population, whose interactions in turn yield a mean-field upward energy shift of the Kondo resonance. Note that the relevant interactions here are not the leading-order Fermi liquid interactions described by , but rather the next-leading-order interaction described by .

At zero temperature, the current can be expanded to second order in . The asymmetry and the zero-energy transmission are characterized by

(21)

with in the symmetric case. The current takes the form

(22)

iii.3 Inelastic contribution to the current

The Keldysh framework kamenev2005 () is well-suited to estimate interaction corrections (9) to the current. The mean current takes the form

(23)

where the Keldysh contour runs along the forward time direction on the branch followed by a backward evolution on the branch . is the corresponding time ordering operator. Time evolution of and is in the interaction representation with the unperturbed Hamitonian  (8). Mean values are also taken with respect to  (8) with bias voltage, see Eqs. (17). Note that the time in Eq. (23) is arbitrary for our steady-state situation. Finally, in order to maintain the original order of operators in , we take left (creation) operators on the branch and right (annihilation) one on the branch.

A perturbative study of Eq. (23) is possible by expansion in and use of Wick’s theorem. This leads to usual diagrammatics where one should keep track of the Keldysh branch index. The lowest order recovers the results of Sec. III.2 describing elastic scattering. The next first order gives only Hartree terms already included in Eq. (22). gives rise in general to three vertices with coefficients , and where the last two are already second order in . Thus it is consistent to keep only in the second order expansion in . A typical Green’s function is defined by . For clarity, spin indices are omitted here and below since all noninteracting Green’s functions are spin diagonal. Noninteracting Green’s functions are matrices in Keldysh space given in momentum-energy space by

(24a)
(24b)
(24c)

with the Pauli matrix , , and , as given Eqs. (17). We wish to compute the second order correction from Eq. (23). It involves the self-energy contribution shown Fig.3 and defined by

(25)
Figure 3: Second order diagram describing the interaction correction to the current from Hamiltonian Eq. (9). The open circle represents a current vertex while filled black dots correspond to interaction vertices. are spin degrees of freedom. The self-energy term is formed by the three lines connecting the two interaction vertices.

The causality identity, for ,

(26)

is derived by writing the explicit time dependence in Eq. (25). It leads to various cancellations, in particular for terms where the lines external to the self-energy (25) bear no dependence. The lines that join the current vertex to the self-energy in Fig.3 travels from (or ) to (the dot) and the opposite. Thus, using the Green’s function (24a) in real space (with )

(27)

and the identity (26), one shows that the terms with operators taken at in Eq. (16) give a vanishing contribution to the current. This is merely a consequence of causality: interaction, which takes place at , can only affect outgoing current and not the incoming part. We are left with the current correction

(28)

The summation over and gives two terms: (i) one includes the combination . It gives a contribution proportional to exactly cancelled by a counterterm. Details are given in Appendix A. (ii) the second term involves the combination and remains finite in the limit . It reads

(29)

with given by Eq. (21) and .

We proceed further and restrict ourselves to the zero-temperature case. The left and right Fermi step functions are introduced by going to frequency space for Eq. (25), and then by using Eqs. (24a) and (17). The result involves a sum of terms with products of and . Two distinct integrals,

(30a)
(30b)

corresponding respectively to one- and two-particles transfer, appear with the following combination

(31)

With and , we obtain the current correction

(32)

This result can be given a quite simple physical interpretation along the line of Ref. sela2006 (). The term in the interaction part of the Hamiltonian (9) can be decomposed on the left/right operators basis using Eq. (2). It then describes processes where , or electrons are transfered from one scattering state to the other. Using Fermi’s golden rule and , the total rate of one-electron transfer is evaluated to be where

(33)

From , the total rate for two-electron transfer is where . For one- and two-electron transfers, and are interpreted as the corresponding charge transfered between leads vitu2008 (). Writing the current correction as

we recover Eq. (32).

iii.4 Current for SU(2) and SU(4)

The results of Secs. III.2III.3 can be extended to finite temperature as explained in Appendix B. We detail results for the total current in the case and cases.

For SU(2), a single electron is trapped on the dot, and . The current takes the form

(34)

where . In the particle-hole SU(4) symmetric case with two electrons, and . The current reads

(35)

where .

Turning now to the SU() case with one electron on the dot, one finds that the inelastic contribution to the current vanishes identically (c.f. Eq. (32)), as the ‘effective charges’ associated with interaction-induced scattering events are proportional to and hence identically zero vitu2008 (). The only contribution is thus from the elastic channel (c.f. Eq. (22)), yielding:

(36)

where . There is no temperature correction up to this order of the low energy expansion. The case with three electrons and SU(4) symmetry is related to the one-electron case by particle-hole symmetry. The Kondo resonance is thus changed from above to below the Fermi energy. The result for the current is then the same as Eq. (36), but with an opposite sign for the asymmetry (, ), i.e. the roles of left (L) and right (R) leads are exchanged for hole transport.

The differential conductance obtained from Eq. (36) gives an asymmetric curve whenever . Consider the first the strongly asymmetric case, where becomes sizeable. In this case, the asymmetric linear correction in Eq. (36) dominates even at low bias voltage. For strong asymmetry , the conductance measures the density of states of the Kondo resonance glazman2005 () at . The asymmetric linear term thus follows the side of the Kondo resonance and reveals that the resonance peak is located away from the Fermi level lehur2007 (). This behaviour is in fact generic to the SU(N) case when the occupation of the dot is away from half-filling. In the SU(2) case or generally for a half-filled dot (), the resonance peak is located at the Fermi level which suppresses the asymmetric linear term, see Eqs. (34) and (35).

Turning now to the case of a symmetric dot-lead coupling (), we see that as expected, the differential conductance is symmetric in at all dot fillings; hence, it exhibits a quadratic behaviour at low bias. In the SU() case, the conductance obtained from Eq. (36) is predicted to be maximum at , in agreement with results obtained from slave boson mean field theory delattre2009 (). Within the Fermi liquid approach, and for one electron on the dot, this behaviour is at first glance rather puzzling. As we have already indicated, in the SU() case, the conductance is completely due to the elastic transport channel. Using the heuristic picture provide by the resonant level picture (i.e. elastic scattering due to a Lorentzian Kondo resonance sitting above the Fermi energy), one would expect that the differential conductance should increase with increasing voltage, due to the positive curvature of the expected (Lorentzian) transmission coefficient. This picture is in fact incorrect, as it neglects the important Hartree contributions discussed in Sec. III.2. Heuristically, as the voltage is increased, quasiparticle interactions lead to a mean-field upward energy shift of the position of the Kondo resonance. Because of the relation , this energy-shift effect dominates, and causes the conductance to decrease; without this mean-field energy shift, the conductance would indeed exhibit a quadratic increase at small voltages. Note that an incorrect upturn in the conductance was reported in previous works: Ref. vitu2008 () neglected the higher-order Fermi liquid interaction parameter and the resulting mean-field energy shift, while Ref. mora2008 () treated it incorrectly (corrected in mora2009 ()). Note also that the results for the conductance presented in Ref. lehur2007 () only apply to a system with a strongly asymmetric dot-lead coupling.

Iv Current noise

Fluctuations in the current are almost as important as the current itself. In particular, the shot noise (at zero temperature) carries information about charge transfer in the mesoscopic system. The purpose of this section is to detail the calculation of the zero-frequency current noise,

(37)

with the current fluctuation , see Eq. (16) for the current operator expression.

Insight can be gained by first examining the strong coupling fixed point at zero temperature, with so that . Quantum expectations in Eq. (37) are evaluated with the free Hamiltonian (8). The shot noise,

(38)

is pure partition noise like a coherent scatterer blanter2000 (). This result implies a vanishing noise in the particle-hole symmetric case, like standard SU(2), with symmetric leads coupling ( and ). In this specific case, the shot noise is only determined by the vicinity of the Kondo strong coupling fixed point, that is by the inelastic Hamiltonian (9) and the corrections to in the elastic phase shift (6). The shot noise is therefore highly non-linear with at low bias voltage. Since the corresponding current is close to unitarity, an effective charge has been extracted from the ratio of the noise to the backscattering current sela2006 (). should however not be confused with a fractional charge. It emerges as an average charge during additional and independent Poissonian processes involving one and two charges transfer as shown by the calculation of the full couting statistics gogolin2006b (). Nevertheless, this charge is universal and characterizes the vicinity of the Kondo strong coupling fixed point. It can be seen as an out-of-equilibrium equivalent of the Wilson ratio.

In asymmetric situations ( or ), the linear part (38) of the noise does not vanish and even dominates at low bias voltage. For instance in the SU(4) case, so that . This property is quite relevant for experiments and may be used to discriminate SU(2) and SU(4) symmetries for which the current gives essentially the same answer delattre2009 (). In a way similar to the symmetric SU(2) case, we can define an effective charge from the ratio of the non-linear parts () in the noise and the current vitu2008 (); mora2008 (). This is however less straightforward to measure experimentally since it requires a proper subtraction of the linear terms.

iv.1 Elastic contribution to the noise

Inserting the current operator (16) in Eq. (37), the elastic Hamiltonian (8) gives a gaussian measure which allows to use Wick’s theorem, and thus Eqs. (17). Like for the current, we obtain a Landauer-Büttiker formula blanter2000 () for the noise with the same transmission (19) and phase shift (20). At zero temperature, it reads

(39)

An expansion to second order in yields the elastic (non-linear) correction to the noise (38),

(40)

with coefficients,

(41a)
(41b)

and the total elastic noise reads . The first order correction (41a) gives an asymmetric part to the noise for . In a way similar to the current case, particle-hole transformation (, ) reverts the sign of the asymmetry (41a) which indicates that the Kondo resonance is centered off the Fermi level.

iv.2 Inelastic contribution to the noise

We follow the same procedure as for the interaction correction to the current established in Sec. III.3. The mean value in Eq. (