A Empty-orbital regime

Fermi-liquid theory for the single-impurity Anderson model

Abstract

We generalize Nozières’ Fermi-liquid theory for the low-energy behavior of the Kondo model to that of the single-impurity Anderson model. In addition to the electrons’ phase shift at the Fermi energy, the low-energy Fermi-liquid theory is characterized by four Fermi-liquid parameters: the two given by Nozières that enter to first order in the excitation energy, and two additional ones that enter to second order and are needed away from particle-hole symmetry. We express all four parameters in terms of zero-temperature physical observables, namely the local charge and spin susceptibilities and their derivatives w.r.t. the local level position. We determine these in terms of the bare parameters of the Anderson model using Bethe Ansatz and Numerical Renormalization Group (NRG) calculations. Our low-energy Fermi-liquid theory applies throughout the crossover from the strong-coupling Kondo regime via the mixed-valence regime to the empty-orbital regime. From the Fermi-liquid theory, we determine the conductance through a quantum dot symmetrically coupled to two leads in the regime of small magnetic field, low temperature and small bias voltage, and compute the coefficients of the , , and terms exactly in terms of the Fermi-liquid parameters. The coefficients of , and are found to change sign during the Kondo to empty-orbital crossover. The crossover becomes universal in the limit that the local interaction is much larger than the level width. For completeness, we also compute the shot noise and discuss the resulting Fano factor.

pacs:
71.10.Ay, 73.63.Kv, 72.15.Qm

I Introduction and Summary

i.1 Introduction

The single-impurity Anderson model, originally introduced to describe d-level impurities such as Fe or Mn in metallic alloys (1); Tsvelick and Wiegmann (1983); Hewson (1993b), may well be one of the most intensely studied models in condensed matter physics, since it covers a rich variety of behaviors and non-perturbative effects, including spin formation, mixed-valence physics, and Kondo screening. Indeed, various extensions of the Anderson model underlie our understanding of correlated metals and superconductors, Mott insulators (4), non-Fermi-liquid systems (5), and heavy fermion materials (6).

The Anderson model has also emerged as a standard tool to describe Coulomb blockade in electron transport through quantum dot nanodevices (7); ?; (9). Since quantum dots can experimentally be probed under nonequilibrium conditions, this opened a new chapter in the study of the Anderson model, involving its properties in the context of nonequilibrium transport. This raised novel questions, not relevant for impurities in bulk systems, involving the behavior of the nonlinear conductance through a quantum dot as a function of source-drain bias. To date, no exact results are available for the nonlinear conductance through a quantum dot described by an Anderson model away from its electron-hole symmetrical point.

In the present paper, we fill this gap, albeit only at low energies, by developing a Fermi-liquid (FL) theory for the low-energy behavior of the asymmetric Anderson model. The theory is similar in spirit to the FL theory developed by Nozières for the Kondo model, but employs two additional FL parameters, whose form had not been established up to now. While these parameters do not influence quantities such as the Wilson ratio, they are necessary to determine non-equilibrium transport properties such as shot noise or the non-linear conductance discussed here. We show how to express all FL parameters of our theory in terms of the zero-temperature, equilibrium values of physical quantities such as the charge and spin susceptibilities and the linear conductance. Such a Fermi-liquid theory is useful, because it offers an exact description of the system’s low-energy excitations, induced, e.g., by a small temperature or a nonequilibrium steady-state transport due to a small source-drain voltage. In this way, knowledge of ground state properties can be elegantly used to make exact predictions about low-lying excitations.

i.2 Anderson model basics

In its simplest form, the Anderson model consists of a single spinful interacting level of energy and occupation , described by the simple Hamiltonian

 Hd=εd^nd+U2^n2d, (1)

which is coupled by a tunneling rate to the Fermi sea of spinful conduction electrons. In the presence of a local magnetic field, the level is Zeeman-split by an additional term (we use units where the Lande factor times Bohr magneton give ). In the non-equilibrium context of nano-devices, – also discussed here, – the level may be coupled to several leads characterized by different tunneling rates and Fermi energies. As mentioned before, this simple model exhibits a surprisingly rich behavior. In particular, in the limit of small and a single electron on the level, i.e. an average charge , a local magnetic moment is formed on the level. In this “Kondo limit”, formally achieved for 1

 εd=−U/2,U/Δ≫1, (2)

the Anderson model maps onto the Kondo model at small energies (11) and accounts for the Kondo effect (12); Hewson (1993b), i.e. the dynamical screening of the spin of this localized electron at low temperatures.

Despite being the realm of strong correlations, the low-energy structure of the screened Kondo state can be captured by simple means. Following Wilson’s solution of the Kondo model by the numerical renormalization group (13), Nozières realized that the low temperature behavior of the Kondo model can be described as a local Fermi liquid, and can be understood in terms of weakly interacting quasiparticles. He formulated an effective Fermi-liquid theory for these, in terms of the phase shift that a quasipaticle incurs when scattering off the screened singlet (14). This phase shift, say , depends not only on the kinetic energy and spin of the quasiparticle, but also on the entire distribution function of the quasiparticles with which it interacts. Nozières expanded this phase shift to leading order in and the deviation of the quasiparticle distribution function from its ground-state form, and viewed the two expansion coefficients as phenomenological parameters, and , called Fermi-liquid parameters. These parameters can be viewed as coupling constants in an effective Fermi-liquid Hamiltonian, which, when treated in the Hartree approximation, generates the phase shifts. The parameters and can be expressed in terms of zero-temperature physical observables by exploiting the fact that the phase shifts determine, via the Friedel sum rule, the local charge and magnetization at zero temperature. In this way, both and are found to be proportional to the zero-temperature impurity spin susceptibility, , whose inverse defines the Kondo temperature, , the characteristic low-energy scale of the Kondo model.

Using the resulting quasiparticle Fermi-liquid (quasiparticle FL) theory, Nozières (14); ?; ? was able to reconstruct all essential low temperature characteristics of the Kondo model, such as the value of the anomalous Wilson ratio (the dimensionless ratio of the impurity’s contribution to the susceptibility and to the linear specific heat coefficient),  (see Ref. (13)), or the quadratic temperature and magnetic field dependence of the resistivity.

Independently, Yamada and Yoshida developed a diagrammatic Fermi-liquid theory (17); ?; ?; ?: they reproduced the above-mentioned features within the Anderson model by means of a perturbative approach and demonstrated by using Ward identities that they hold up to infinite order in .

Both the quasiparticle and the diagrammatic Fermi-liquid approaches proved to be extremely useful. The diagrammatic FL approach has been extended to orbitally degenerate versions of the Anderson model (21); (22); (23); Hanl et al. (2014), see also the interaction between two impurities (25), and to out of equilibrium (26), and led to the construction of the renormalized perturbation theory Hewson (1993a); ?; ?; Hewson (1993b); (30); ?; ?; ?; ?; ?; ? (see also Ref. (37)) and its application to various extensions of the Anderson model (38); (39); (40). Nozières’ quasiparticle FL approach has been widely used to study non-equilibrium transport in correlated nano-structures described by the Kondo model or generalizations thereof (41); (42); (43); (44); (45); Mora (2009); (47); (48). In particular, the effective Fermi-liquid Hamiltonian of the Kondo model was used to calculate the leading dependence of the conductance on temperature, bias voltage and magnetic field, and to determine the coefficients of the leading , and terms, say , and . These Fermi-liquid transport coefficients turn out to be universal numbers, because for the Kondo model the zero-energy phase shift, , has a universal value, .

Surprisingly, Nozières’ quasiparticle Fermi-liquid theory has not yet been extended to the case of the Anderson model (except for the special case of electron-hole symmetry (49)), although this model has a Fermi-liquid ground state in all parameter regimes (50); (51); ?. The reason has probably been that such a theory requires additional Fermi-liquid parameters, called and below, and no strategy was known to relate these to physical observables. In this work, we fill this gap and develop a comprehensive Fermi-liquid approach to the Anderson model, applicable also away from particle-hole symmetry (53); (54). Our strategy is a natural generalization of that used by Nozières for the Kondo model. We develop an effective quasiparticle theory characterized by four Fermi liquid parameters (, , and ), and use these to expand the phase shifts of the quasiparticles systematically as a function of the quasiparticles’ energy and distribution. Using the Friedel sum rule, we express these Fermi-liquid parameters in terms of four zero-temperature physical parameters, namely the local charge and spin susceptibilities, and , and their derivatives and w.r.t. the local level position . We then use the resulting Fermi-liquid Hamiltonian for the Anderson model to calculate the conductance to quadratic order in temperature, bias voltage and magnetic field, in a similar manner as for the Fermi-liquid Hamiltonian for the Kondo model. However, the Fermi-liquid transport coefficients , and are no longer universal, but depend on , , , and the zero-energy phase shift , all of which are functions of . For completeness, we also compute the current noise to third order in the voltage. We calculate these functions explicitly by using Bethe Ansatz and NRG(13); (51); ?. We thus obtain explicit results for the dependence of , , and the current noise throughout the entire crossover from the strong-coupling Kondo regime () via the mixed-valence regime () to the empty-orbital regime ().

i.3 Summary and overview of main results

In this subsection, we gather the main ideas of our approach and its main results in the form of an executive summary. Details of their derivation are presented in subsequent sections.

We shall focus on the quantum dot configuration connected symmetrically to two lead reservoirs. In this case, the level on the dot couples only to the ‘symmetrical’ combination of electronic states in the leads. Correspondingly, the Fermi-liquid theory can be constructed in terms of quasiparticles in ‘even’ and ‘odd’ channels, and , respectively (47). Since the ‘odd’ quasiparticles do not hybridize with the -level, the effective low-energy Fermi-liquid Hamiltonian can be constructed solely from the ‘even’ quasiparticles, and is given to leading and subleading order by

 HFL = ∑σ∫ε(ε−σB/2)b†εσbεσ+Hα+Hϕ+… (3) Hα = −∑σ∫ε1,ε2[α12π(ε1+ε2)+α24π(ε1+ε2)2]b†ε1σbε2σ Hϕ = ∫ε1,…,ε4[ϕ1π+ϕ24π(4∑i=1εi)]:b†ε1↑bε2↑b†ε3↓bε4↓:,

where is the magnetic field. Here , , and are the four Fermi-liquid parameters. The form of Eq. (3) can be justified rigorously using Conformal Field Theory arguments as discussed in the Supplemental Material 2. The operators here create incoming single-particle scattering states of kinetic energy and spin , and incorporate already the zero-temperature phase shift experienced by electrons at the Fermi energy, . The term in this expansion accounts for energy dependent elastic scattering, while the terms in describe local interactions between the quasiparticles. In the Kondo model, charge fluctuations are suppressed, and the low-energy theory exhibits electron-hole symmetry under the transformation . In the presence of such symmetry, the parameters and must vanish, since their presence would violate electron-hole symmetry. Furthermore, as shown by Nozières (14); ?; ?, the parameters and are equal in the Kondo model. Therefore the Kondo model’s effective FL theory (3) is characterized by a single Fermi-liquid scale, , defined as

 E∗≡π4α1, (4)

and identified as the Kondo temperature, . We use units in which . In contrast, in the generic Anderson model, three of the four Fermi-liquid parameters are independent (more precisely, each of them is a function of three variables, , and the dimensionless ratios and ), and therefore the low-energy behavior cannot be characterized by a single Fermi-liquid scale. Nevertheless, we shall still use Eq. (4) to define the characteristic energy scale and express physical quantities in terms of it. We emphasize that whereas the calculation of Nozières accounted only for local spin excitations, our approach includes both spin and charge fluctuations and allows us to capture the mixed-valence regime and smoothly interpolate between the Kondo and Coulomb blockade regions.

To make use of the Fermi-liquid theory in its full power, we shall determine the Fermi-liquid parameters in Eq. (3) in terms of the bare parameters of the Anderson model, , , and . To this end, we shall first demonstrate that the four FL parameters of the Anderson model are directly related to zero-temperature physical observables, and can be expressed solely in terms of the local charge () and spin () susceptibilities of the Anderson model and their derivatives ( and ) with respect to ,

 Missing dimension or its units for \hskip (5a) Missing dimension or its units for \hskip (5b)

The expressions for and were known Yamada (1975); Yosida and Yamada (1975); Hewson (1993a, b) (see Sec. S-I in (55)) , those for and are central results of this work. We then determine the FL parameters from these relations, by computing the susceptibilities and from NRG (13); (51); ? and, complementarily, by computing the Bethe Ansatz solution to the Anderson model Kawakami and Okiji (1982); (57).

Typical results of our computations are shown in Fig. 1, where we display the four Fermi-liquid parameters for moderately strong interactions, , as a function of the level’s position. In agreement with the discussion above, the parameters and vanish at the electron-hole symmetrical point, , and are antisymmetrical with respect to it, while the Fermi-liquid parameters and display a symmetrical behavior. In the local-moment regime, , charge fluctuations are suppressed, and the charge susceptibility can be neglected in the expression of the Fermi-liquid parameters. Here we can derive an analytical approximation for them [Eqs. (26) and (27)] by making use of the Bethe Ansatz expression for the spin susceptibility in the local-moment regime, . Although Eqs. (26) and (27) are expected to be valid only for , even for the moderate interaction of Fig. 1, surprisingly good agreement with the complete solution is found for . In the opposite limit of an almost empty orbital, , interactions are negligible, and transport is well described by a non-interacting resonant level model. The crossover from the local-moment to the empty-orbital regime becomes universal for large values of , for which the dimensionless Fermi-liquid parameters, , , , and can be expressed as universal functions of .

Equipped with our Fermi-liquid theory and with the four Fermi-liquid parameters, we then study a quantum dot device, coupled symmetrically to two leads 3, and derive exact results for the FL transport coefficients, , , and , characterizing the conductance at low bias voltage, temperature and magnetic field,

 G(V,T,B)−G0≈−2e2/h(E∗)2(cTT2+cV(eV)2+cBB2), (6)

with denoting the linear conductance of the quantum dot at zero temperature and zero magnetic field. In terms of the Fermi-liquid parameters, the coefficient can be expressed, e.g., as

 Invalid decimal number (7)

The other two coefficients and are expressed by similarly complex expressions, given by Eqs. (50) and (51) in Section IV.2. The value of these coefficients can be trivially determined in the empty-orbital regime, where the following asymptotic values are obtained,

Moving to the Kondo regime, the coefficients and change sign and their ratio changes by a factor of 2 as compared to the empty-orbital regime,

 cKT=π416≃6.009,cKV=3π232≃0.925, (9)

reflecting the emergence of strong correlations in the Kondo regime. In hindsight, this sign change may be not very surprising: in the Kondo regime, the perfect conductance through the Kondo resonance is reduced by a finite temperature (bias), destroying Kondo coherence, while in the empty-orbital regime a gradual lifting of the Coulomb blockade is expected as the temperature or bias voltage is increased.

also changes sign and its ratio with increases by a factor in the Kondo regime, where

 cKB=π216≃0.617. (10)

The evolution of the normalized coefficients , , and is shown in Fig. 2(a) for as a function of the level’s position, , using Bethe ansatz computations. Susceptibilities can also be computed from NRG and Fig. 2(b) illustrates the excellent agreement between Bethe ansatz and NRG on one transport coefficient. Importantly, all three transport coefficients can be, in principle, extracted from transport measurements, and thus the predictions of this Fermi-liquid theory can be verified by straightforward transport measurements (59).

In addition, we also compute the zero frequency current noise at low voltage. It is characterized by a generalized Fano factor  (47), see Eq. (53) in Sec. IV.3, defined as the ratio of the leading corrections to the noise and current with respect to the strong coupling fixed point values. We find for the Fano factor

 F=cos4δ(α21+5ϕ21)+4ϕ21+sin4δ0(α2/2−3ϕ2/8)cos2δ0(α21+5ϕ21)+sin2δ0(α2−3ϕ2/4), (11)

displayed in Fig. 3 for different . At particle-hole symmetry (in agreement with Ref. (49)), varies between in the non-interacting case , corresponding to Poissonian statistics for the backscattered current, to at large , emphasizing the role of interactions and two-particle backscattering processes (42); (44); (47). As increases towards the empty orbital regime, the Fano factor interpolates to the non-interacting Poissonian result . The sign change as is varied indicates that describes a backscattering current at but transmitted electrons at large .

The rest of this paper is organized as follows. In Sec. II, we construct the basic Fermi-liquid theory for the Anderson model and relate the Fermi-liquid parameters of the effective Hamiltonian to physical observables [(5)]. In Sec. III we construct the current operator and set the framework for non-equilibrium calculations, which we then use to compute the expectation value of the current and noise perturbatively. The final form of the transport coefficients and Fano factor is presented in Sec. IV. Sec. V concludes and offers an outlook. The empty-orbital limit is discussed in Appendix A. Technical details regarding the Bethe ansatz equations and their integral solutions, a Conformal Field Theory approach to the strong coupling fixed point and the calculation of the T-matrix, are left to the Supplemental Material (55). In addition, the SM also contains detailed numerical results for the FL transport coefficients, and a comparison to previous works for the Wilson ratio.

Ii Fermi-liquid theory

In this section, we present our Fermi-liquid theory for the Anderson model. The Fermi-liquid theory is by essence a perturbative approach. It gives the expansion of observables at bias voltages and temperatures smaller than the Kondo temperature . We begin in Sec. II.1 by a reminder of the Fermi-liquid approach to the Kondo model, as introduced by Nozières (14); (16); (15); Lesage and Saleur (1999a); ?, and explain in detail how the model’s invariance, in the wide-band limit 4, under a global energy shift can be used to relate the different Fermi-liquid parameters. In Sec. II.2, we extend this approach to the Anderson model. In Sec. II.3, we take advantage of the Friedel sum rule to express all Fermi-liquid parameters in terms of the spin and charge susceptibilities, see Eqs. (5), a result of considerable practical importance. The spin and charge susceptibilities are simple ground state observables – and can be computed semi-analytically by Bethe Ansatz – while the Fermi-liquid theory is able to deal with more complicated situations, such as finite temperature or out-of-equilibrium settings. Analytical expressions of the Fermi-liquid parameters are obtained in the Kondo and empty-orbital limits in Sec. II.4. Finally, the effective Fermi-liquid Hamiltonian, applicable at low energy and already advertised in Eq. (3), is discussed in Sec. II.5.

ii.1 Kondo model

We begin by briefly reviewing Nozières’ local Fermi-liquid theory for the Kondo model. The main ideas are well established – for details we refer to the seminal papers of Nozières (14); (16); (15) or to Refs. Hewson (1993a); (41); Mora (2009). Our goal here is to phrase the arguments in such a way that they will generalize naturally to the case of the Anderson model, discussed in the next subsection.

For energies well below the Kondo temperature, the reduction of phase-space for inelastic processes implies that elastic scattering dominates, due to the same phase-space argument (63); ?; (65) as in conventional bulk Fermi liquids. The system can then be characterized by the phase shift, , acquired by a quasiparticle with kinetic energy and spin that scatters off the screened Kondo singlet (the form of this phase shift can be derived explicitly from the effective Fermi-liquid Hamiltonian Eq. (3) [with ], as explained in Sec. II.5 below). Since the singlet has a many-body origin, depends not only on but also on the quasiparticle distribution functions and . Our goal is to find a simple description of this phase shift function, valid for small excitation energies relative to the ground state.

In equilibrium and at zero temperature and magnetic field, the quasi-particle ground state is characterized by a well-defined zero-temperature chemical potential . Let be an arbitrary reference energy, different from , which serves as the chemical potential of a reference ground state with distribution function . We then Taylor-expand the phase shift around this reference state as

 δσ(ε,nσ′)=δ0+α1(ε−ε0)−ϕ1∫ε′δn¯σ,ε0(ε′), (12)

with . The last term accounts for local interactions with other quasiparticles, and denotes the spin opposite to , since by the Pauli principle local interactions can involve only quasiparticles of opposite spins. We should stress that the distributions can have arbitrary shapes (depending on chemical potential, temperature, magnetic field and, for out-of-equilibrium distributions, source-drain voltage), as long as the expansion variables and in Eq. (12) are small compared to the Fermi-liquid scale  5. The Taylor coefficients , and serve as the Fermi-liquid parameters of the theory. Their dependence on drops out in the wide-band limit considered here, and they are universal coefficients.

Now, the key point is to realize that the function is of course independent of the reference energy used for its Taylor expansion. Differentiating Eq. (12) w.r.t. (and noting that depends also on ) one thus obtains , or

 α1=ϕ1. (13)

This relation constitutes one of Nozières’ central Fermi-liquid identities for the Kondo model.

As can be checked easily, Eq. (13) guarantees that for any distribution with a well-defined chemical potential, e.g.  for nonzero temperature, the phase shift , depends on energy and chemical potential only through the combination . In other words, if is changed to , e.g. by doping the system to increase the electron density, then the new phase shift at equals the old one at ,

 δσ(ε+δμ,nμ+δμ)=δσ(ε,nμ), (14)

as illustrated in Fig. 4. [In fact, an alternative way to derive Eq. (13) is to impose Eq. (14), with the same on both sides of the equation, as condition on the general phase shift expansion Eq. (12) for ; the calculations are simplest if done at zero temperature, i.e. with .] Since at the energy dependence of the phase shift determines that of the Kondo resonance in the impurity spectral function, , the latter, too, is invariant under a simultaneous shift of and . Pictorially speaking, the “Kondo resonance floats on the Fermi sea” (14); Mora (2009): if the Fermi surface rises, the Kondo resonance rises with it, and if the Fermi see is deep enough (wide-band limit), the Kondo resonance does not change its shape while rising.

The next step is to express and in terms of physical quantities, such as the local charge and the local spin susceptibility . This can be done by calculating the latter quantities via the Friedel sum rule, evaluating the ground state phase shift in a small magnetic field. We discuss this in detail in the next section, in the more general context of the Anderson model. Here we just quote the results: for the Kondo model, one finds , and, since , from Eq. (4), for the Fermi-liquid energy scale controlling the expansion Eq. (12).

Before proceeding further with the Anderson model, we wish to emphasize two important points:
(i) We have restricted our attention to elastic scattering processes. As pointed out in Ref. (47), inelastic processes involve the difference between the energies of incoming and outgoing electrons and are therefore invariant under a global shift of all energies by .
(ii) Eq. (12) corresponds to the first few terms of a general expansion of in powers of and . In the calculation of the conductance, for example at finite temperature, the and terms give a vanishing linear contribution and must therefore be taken into account up to second order. To be consistent, one then needs to include the next subleading terms in the expansion of . This has been worked out explicitly for the SU() case with  (45); (67); (48); (47); Mora (2009). These subleading terms, however, turn out to vanish identically in the SU(2) Kondo model, as a result of electron-hole symmetry. This is no longer the case for the asymmetric Anderson model, as we will see below.

ii.2 Anderson model

The Anderson model is described by a low-energy Fermi-liquid fixed point for all regimes of parameters, hence we now seek to generalize the above approach to this model, too. The main complication compared to the Kondo model is that the Anderson model involves an additional energy scale, namely the impurity level , and its physics depends in an essential way on the distance between its impurity energy level and the chemical potential. We again Taylor expand the phase shift w.r.t. to a reference energy , as in Eq. (12), but now include the next order in excitation energies Mora (2009):

 δσ(ε,nσ′) =δ0,εd−ε0+α1,εd−ε0(ε−ε0) (15) .−ϕ1,εd−ε0∫ε′δn¯σ,ε0(ε′)+α2,εd−ε0(ε−ε0)2 .−12ϕ2,εd−ε0∫ε′(ε+ε′−2ε0)δn¯σ,ε0(ε′)+…

, , , and are the Taylor coefficients of this expansion. In contrast to the case of the Kondo model, they now do depend explicitly on the reference energy , and since we are in the wide-band limit, this dependence can arise only via the difference . For notational simplicity, we will suppress this subscript below, taking this dependence to be understood. In the Kondo limit of Eq. (2), the dependence on drops out, and the coefficients , , , and become universal, as seen in the previous section for , and .

Similarly to Sec. II.1, the Taylor coefficients are not all independent as a result of the phase shift invariance under a change in . Differentiating Eq. (15) w.r.t. , and equating the coefficients of the various terms in the expansion (cst, , to zero, we therefore obtain the following three relations 6:

 −δ′0−α1+ϕ1 = 0, (16a) −α′1−2α2+ϕ2/2 = 0, (16b) ϕ′1+ϕ2 = 0. (16c)

Here a prime denotes a derivative with respect to the energy argument, e.g. .

As can be checked easily, Eqs. (16) guarantee that for any distribution with a well-defined chemical potential, e.g. , the phase shift (where the subscript indicates the dependence of its Fermi-liquid parameters) remains invariant if , and are all shifted by the same amount:

 δσ,εd+δμ(ε+δμ,nμ+δμ)=δσ,εd(ε,nμ). (17)

Conversely, an alternative way to derive Eqs. (16) is to impose Eq. (17) as a condition on the Taylor expansion (15) for .

Collecting results, the first order Fermi-liquid parameters, and , are related to each other through

 ϕ1−α1=δ′0, (18)

while the second-order Fermi-liquid parameters, and , can be expressed via Eqs. (16) in terms of derivatives of lower-order ones:

 α2=−δ′′04−3α′14,ϕ2=−ϕ′1. (19)

Having established the above relations between the Fermi-liquid parameters, we henceforth choose the reference energy at the zero-temperature chemical potential, . Moreover, since the choice of is arbitrary in the wide-band limit, we henceforth set . Hence, the energy argument of the Fermi-liquid parameters is henceforth understood to be , i.e.  stands for , etc.

ii.3 Charge and spin static susceptibilities

Our next task is to express the Fermi-liquid parameters in terms of physical quantities. This can be done using the Friedel sum rule. To this end, consider a zero-temperature system in a small nonzero magnetic field, , with distribution and spin-split chemical potentials, , as illustrated in Fig. 5. Using this distribution for in Eq. (15), with and , we find:

 δσ(ε,n0μσ′) = δ0+α1ε−ϕ12¯σB+α2ε2 (20) Missing or unrecognized delimiter for \right

Now evoke the Friedel sum rule (69). For given spin it relates the average charge bound by the impurity at , , to the ground state phase shift at the chemical potential, i.e. at :

 πndσ = δσ(μσ,n0μσ′) (21a) = δ0+σ2(α1+ϕ1)B+14(α2+ϕ2/4)B2. (21b)

Thus, the average local charge and average magnetization of the local level can be expressed as:

 nd = nd↑+nd↓=2δ0π+12π(α2+ϕ2/4)B2, (22a) md = (nd↑−nd↓)/2=B2π(α1+ϕ1). (22b)

In the strong-coupling Kondo regime we have at zero field, implying . In general, however, is a function of . From Eqs. (22), the local charge and spin susceptibilities at zero field are given by

 χc = −∂nd∂εd∣∣∣B=0=−2δ′0π=2π(α1−ϕ1), (23a) χs = −∂md∂B∣∣∣B=0=12π(α1+ϕ1). (23b)

Using Eqs. (23a) and (23b), the Fermi-liquid parameters can be written in terms of the charge and spin susceptilibities and , and their derivatives w.r.t. to , denoted by and . The result is given in Eq. (5) in the introduction. As a consistency check, we note from Eq. (5) that , thus Eqs. (22) imply

 ∂nd∂B=−∂md∂εd, (24)

which is a standard thermodynamic identity.

For the Anderson model, , , and their derivatives w.r.t.  can all be computed using the Bethe Ansatz, as detailed in the SM (55). This allows us to explicitly determine how the Fermi-liquid parameters depend on . A corresponding plot is shown in Fig. 1 for .

The Anderson model has a particle-hole symmetry, which manifests itself as an invariance under the replacements for the impurity single-particle energy and for the impurity charge. The particle-hole symmetric point therefore corresponds to and . Moreover, and are symmetric with respect to particle-hole symmetry, while and are antisymmetric. Consequently, Eqs. (5) show that and are symmetric while and are antisymmetric, a feature already pointed out in the introduction. As a result, and identically vanish at the particle-hole symmetric point . At this point, our result for the current will therefore agree with those of Refs. (14); (26); (38); (39). In the Kondo limit of Eq. (2), charge fluctuations are suppressed such that , and Eq. (23a) reproduces the Fermi-liquid identity Eq. (13) of the Kondo model.

As discussed Section S-1 of the Supplemental Material (55), our approach reproduces the known FL relation between susceptibilities and the linear specific heat coefficient, and the corresponding Wilson ratio.

So far in this section, we have not used the specific form of the Anderson model. The only ingredients that we have used are the presence of a single-particle energy for the impurity and the assumption of Fermi-liquid behavior. This emphasizes the generality of our Fermi-liquid approach, which is also applicable, for instance, to other impurity models such as the interacting resonant model (70).

ii.4 Analytical expressions

In order to better understand the dependence of the Fermi-liquid parameters on , it is instructive to consider certain limiting cases where analytical expressions can be derived. In the Kondo regime, and , spin excitations dominate and the charge susceptibility can be neglected (, ), so that [from Eqs. (5)]

 α1≃ϕ1≃πχs,4α2/3≃ϕ2≃−πχ′s. (25)

The spin susceptibility is given with a very good accuracy by the asymptotical expression

 χs=12√2UΔeπ(U/8Δ−Δ/2U)e−x2, (26)

where we introduced the distance to the particle-hole symmetric point . Eq. (26) agrees with the well-known formula (50), up to an extra factor , which was neglected in (50) because the limit is implicit there. Differentiating Eq. (26) w.r.t. , we find

 −χ′s=π1/22ΔUeπ(U/8Δ−Δ/2U)xe−x2. (27)

Eqs. (25) to (27) together largely explain the shape of all the curves in Fig. 1, namely approximately Gaussian for and , or the derivative of a Gaussian for and .

The other limit in which analytical expressions can be derived is the empty-orbital regime, for . The results are detailed in Appendix A. Together with Eqs. (26) and (27), they give us a good analytical understanding of the dependence of the Fermi-liquid parameters. In the Kondo regime, and follow the spin susceptibility (or the inverse Kondo temperature) and decrease with increasing (for ) while crossing over into the mixed-valence regime. Finally, in the empty-orbital regime , hence still follows the spin susceptibility, but with a factor , , while becomes negligible.

It is interesting to consider the ratios and which measure the importance of the second generation of Fermi parameters compared to the first one. In the Kondo region but far enough from particle-hole symmetry, [the precise formula is implied by Eq. (27)] so that . The two ratios are small but increase with and towards the mixed-valence region where they reach values of order . Above, in the empty-orbital region, , for but is negligible for , while continues to increase with [see Eqs. (59) to (61)].

ii.5 Hamiltonian form

The analysis carried out so far may seem abstract. It is based on the elastic phase shift alone and it is not clear how transport quantities and other observables can be computed. We thus need to write an explicit low-energy Hamiltonian reproducing the phase shift of Eq. (15). The leading order, or strong coupling Hamiltonian, is simply given by the first term of Eq. (3),

 H0=∑σ∫dε(ε−σB/2)b†εσbεσ, (28)

where the quasiparticle operators , defined in the introduction, satisfy the fermionic anticommutation relations

 {bεσ,b†ε′σ′}=δσ,σ′δ(ε−ε′),{bεσ,bε′σ′}=0. (29)

The low-energy Hamiltonian admits an expansion in correspondence with the phase shift expansion 7 of Eq. (15), the increasing orders being increasingly irrelevant in the renormalization group sense Lesage and Saleur (1999a); ?. The first two terms of this expansion are given in Eq. (3). A more formal but complete justification of the form of the Hamiltonian, using conformal field theory arguments, is given in the SM (55).

The computation of the elastic phase shift with involves all processes stemming from and , in addition to the Hartree diagrams inherited from . Using , it is straightforward to check that Eq. (15) is reproduced, as required.

The low energy expansion of Eq. (15) is valid as long as typical energies (, or ) are smaller than a certain energy scale depending on . At large , this energy scale is in the Kondo regime. It crosses over to in the mixed-valence regime where physical quantities are universal when energies are measured in units of , see Sec. S-II in (55). In the empty-orbital regime, a resonant level model centered around emerges, see appendix A, and this energy scale crosses over to .

To summarize this section, Eq. (3) constitute a rigorous and exact low-energy Hamiltonian for the Anderson model (or for other similar models), and a basis for computing the low-energy quadratic behavior of observables. We shall use it in the next section to compute the conductance and the noise. The introduction of the elastic phase shift was mainly aimed at determining the expressions of the Fermi-liquid parameters given in Eq. (5).

Iii Current and noise calculations

The Fermi liquid theory developed so far is very general, and applies to many quantum impurity systems with a Fermi liquid ground state and a single relevant channel of spinful electrons attached to it. We now turn to the concrete case of the Anderson model and calculate the current and the noise through a quantum dot using the Fermi-liquid theory described in the previous section. For this purpose, the geometry of lead-dot coupling becomes important and scattering state wave functions have to be introduced in the spirit of Landauer’s approach. Similar calculations can be found in Refs. (45); (47); (23). Sec. III.1 introduces the Anderson model and the corresponding Fermi-liquid Hamiltonian valid at low energy, already outlined in the Introduction. The current operator is given in Sec. III.2 and expanded over the convenient basis of quasiparticle states. The perturbative calculations of the current and noise current are then separated into an elastic part in Sec. III.3 and an inelastic part in Sec. III.4.

iii.1 Hamiltonians

Anderson model

We consider the model of a single-level dot symmetrically coupled to right and left leads with the Hamiltonian , with and

 HAM=∑σ∫dεε~b†εσ~bεσ+εd∑σnσ+U^nd↑^nd↓+√ν0t∑σ∫dε(~b†εσdσ+d†σ~bεσ), (30)

where, instead of the original left and right operators, and , we use the symmetric and antisymmetric combinations

 (~bεσaεσ)=1√2(111−1)(cL,εσcR,εσ). (31)

These satisfy the same anticommutation relations as in Eq. (29). The leads are approximated, as usual(13); Tsvelick and Wiegmann (1983), by a linear spectrum with a constant density of states per spin species, otherwise the results would not be universal. is the electron operator of the dot and the corresponding density for spin . denotes the charging energy, the single-particle energy on the dot and the tunneling matrix element from the dot to the symmetric combination of leads. The antisymmetric combination , associated with the wavefunction

 ψakσ(x)=(ei(kF+k)x−e−i(kF+k)x)/√2 (32)

for all , decouples from the dot variables. Here describes the left lead and the right lead, energies and wavevectors are related through . For simplicity, the whole system is assumed to be one-dimensional. Being odd in , this wavefunction vanishes at the origin and is therefore not affected by the Anderson impurity. We define the hybridization for later use.

Effective low-energy Hamiltonian

At low energy, screening takes place and the Anderson model flows to a Fermi-liquid fixed point for all values of , and . The Hamiltonian describing the low-energy physics of Eq. (30) is then given by , with the Fermi-liquid Hamiltonian for the even channel given by Eq. (3).

The difference between the original operators associated with symmetric combinations of lead states and the corresponding quasiparticle operators is the zero-energy phase shift , i.e. the phase shift that arises for . Hence is associated with the scattering state

 ψbkσ(x)={(ei(kF+k)x−S0e−i(kF+k)x)/√2x<0,(e−i(kF+k)x−S0ei(kF+k)x)/√2x>0, (33)

with the S-matrix . In contrast, for the antisymmetric combination of lead states described by -operators, which decouple from the dot variables, the corresponding S-matrix is trivially equal to 1, i.e. the corresponding scattering phase is zero.

iii.2 Current operator

In a one-dimensional geometry, the local current operator is given by

 ^I(x)=eℏ2mi∑σ(ψ†σ(x)∂xψσ(x)−∂xψ†σ(x)ψσ(x)) (34)

where is the electron mass. Various expressions for the current can be derived depending on which basis of states it is expanded in. Here we choose a basis adapted to the low-energy model, namely we expand over the zero-energy scattering states

 ψσ(x)=∫dε√ν0[ψakσ(x)aεσ+ψbkσ(x)bεσ]. (35)

with the density of states of incoming quasiparticles.

A voltage bias applied between the two leads, , drives a current through the quantum dot. In a stationary situation, the current is conserved along the one-dimensional space. We thus define the symmetric current operator as , where is arbitrary, corresponding to the average of the left and right currents. Inserting the expansion Eq. (35) in Eq. (34), one finds the Landauer-Buttiker (72) type current expression

 ^I=e2h∑σ∫ε,ε′a†εσbε′σ(ei(k′−k)x−S0e−i(k′−k)x)+h.c., (36)

with . A more compact expression can be obtained with the definition , namely

 ^I=e2h∑σ(a†σ(x)bσ(x)−a†σ(−x)(S0bσ)(−x)+h.c.). (37)

Physically, operators taken at () correspond here to incoming (outgoing) states.

Fluctuations in the current are characterized by the zero frequency current noise

 S=2∫dt⟨Δ^I(t)Δ^I(0)⟩ (38)

where .

iii.3 Elastic scattering

We study the average current through the dot in the presence of a voltage bias. We include in this section only the elastic and Hartree contributions, the inelastic terms will be considered in the next Sec. III.4.

Strong coupling fixed point

We start by considering the strong coupling fixed point, i.e. without the Fermi-liquid corrections and , where we have a free gas of quasiparticles. The Hamiltonian is and and create eigenstates of the model. The left and right scattering states, which are even and odd combinations of and , are in thermal equilibrium with spin-dependent chemical potentials and . Hence, we have

 ⟨a†εσaε′σ′⟩=⟨b†εσbε′σ′⟩=δσ,σ′δ(ε−ε′)fLσ(ε)+fRσ(ε)2⟨a†ε