Higher-order Fermi-liquid corrections for an Anderson impurity away from half-filling
Abstract
We study the higher-order Fermi-liquid relations of Kondo systems for arbitrary impurity-electron fillings, extending the many-body quantum theoretical approach of Yamada-Yosida. It includes partly a microscopic clarification of the related achievements based on Nozières’ phenomenological description: Filippone, Moca, von Delft, and Mora [Phys. Rev. B 95, 165404 (2017) ]. In our formulation, the Fermi-liquid parameters such as the quasi-particle energy, damping, and transport coefficients are related to each other through the total vertex , which may be regarded as a generalized Landau quasi-particle interaction. We obtain exactly this function up to linear order with respect to the frequencies and using the anti-symmetry and analytic properties. The coefficients acquire additional contributions of three-body fluctuations away from half-filling through the non-linear susceptibilities. We also apply the formulation to non-equilibrium transport through a quantum dot, and clarify how the zero-bias peak evolves in a magnetic field.
pacs:
71.10.Ay, 71.27.+a, 72.15.QmIntroduction.— Universal low-energy behavior of interacting Fermi systems has been one of the most fascinating properties in condensed matter physics. Landau’s Fermi liquid theory Landau (1957); Abrikosov et al. (1965); Landau et al. (1991) phenomenologically explains transport properties of electrons in a wide class of metals and normal liquid He successfully Leggett (1975), and may also be applied to exotic systems such as neutron stars and ultra-cold Fermi gases Bloch et al. (2012). It starts with an expansion of the energy with respect to the deviation of the momentum distribution function from the ground state,
(1) |
The single quasi-particle energy and the interaction between quasi-particles can microscopically be related to the four-point vertex function, defined explicitly in the many-body quantum theory Abrikosov et al. (1965); Landau et al. (1991). The field theoretic description has advantages over the phenomenologic approach: the transport equations can be derived directly using the Green’s function without relying on empirical assumptions nor the collision integral with the Boltzmann equation Eliashberg (1962a, b). For instance, through a microscopic consideration about the anti-symmetry properties of the vertex function Landau et al. (1991), sufficient conditions for the collective zero sound mode to exist have been derived Mermin (1967).
Nozières extended the phenomenological Fermi-liquid description to Kondo systems Nozières (1974), expanding the scattering phase shift with respect to a deviation of the occupation number of the impurity level in a way analogous to Eq. (1). Fully microscopic description was constructed by Yamada-Yosida, Shiba, and Yoshimori Yamada (1975a, b); Shiba (1975); Yoshimori (1976), and has also been extended to out-of-equilibrium quantum dots driven by a bias voltage Hershfield et al. (1992); Oguri (2001). The two different types of descriptions complement each other and explain the universal behavior at temperatures much lower than the Kondo energy scale . It is successful especially in the particle-hole symmetric case, i.e. at half-filling, where the phase shift is locked at and the quadratic , and corrections emerge only through the quasi-particle damping.
Away from half-filling, however, the Kondo resonance peak deviates from the Fermi energy , and as a consequence, the quadratic corrections emerge also through the real part of the self-energy due to the Coulomb interaction Yoshimori (1976); Horvatić and Zlatić (1982). It makes the problem difficult, and such corrections have not been fully understood for a long time. Recently, there has been a significant breakthrough which shed light on this problem by extending Nozières’ phenomenological description Mora et al. (2015); Filippone et al. (2017). Specifically, Filippone, Moca, von Delft and Mora (FMvDM) determined especially the quadratic coefficients of the self-energy away from half-filling ^{1}^{1}1 Parameter correspondence: , , , and ..
In this Letter, we provide a microscopic Fermi-liquid description for the non-equilibrium Anderson impurity Anderson (1961) away from half-filling. One of the most pronounced merits of this formulation is that the real and imaginary parts of the transport coefficients are derived together from an explicit expression of the total vertex at low frequencies. It gives a clear answer to the long-standing problem. Specifically, an asymptotically exact expression is obtained, up to linear order in and , using the anti-symmetry and analytic properties with the Ward identities. The low-energy Fermi-liquid behavior is characterized by the expansion coefficients which are shown to be expressed in terms of the linear and non-linear susceptibilities.
These susceptibilities can be calculated using methods such as the numerical normalization group (NRG) Krishna-murthy et al. (1980) and the Bethe ansatz solution Kawakami and Okiji (1982); Wiegmann and Tsvelick (1983). We apply the microscopic formulation to non-equilibrium current through a quantum dot in the Kondo regime, and calculate the coefficients using the NRG. The result shows that the zero-bias peak of splits at a magnetic field of the order of , and resolves a controversial issue about the splitting Filippone et al. (2017). There are other numerical methods which work efficiently at different energy scales, such as the quantum Monte Carlo Gull et al. (2011), time-dependent NRG Anders and Schiller (2006) and density-matrix renormalization group Kirino et al. (2008). Our approach has a numerical advantage at low energies as both the linear and non-linear susceptibilities can be deduced from the flow of energy eigenvalues near the fixed point of NRG Hewson et al. (2004).
The microscopic theory gives exact relations between different response functions and has given theoretical support for the universal scaling observed in the nonequilibrium currents through quantum dots in the Kondo regime Grobis et al. (2008); Scott et al. (2009). Furthermore, recent ultra-sensitive current noise measurements have successfully determined the Fermi-liquid parameters Ferrier et al. (2016) i.e., the Wilson ratio and the renormalization factor of quasi-particles. However, such comparisons so far have relied on the theoretical predictions at half-filling. The exact formula of transport coefficients, presented in Eqs. (22) and (23), overcomes this restriction and can be applied to quantum dots for arbitrary electron fillings. Our formulation also has potential application for a wide class of Kondo systems such as dilute magnetic alloys and quantum impurities with various kinds of internal degrees of freedom.
Non-linear 3-body susceptibilities for impurity levels.— We consider the single Anderson impurity coupled to two noninteracting leads ();
(2) |
Here, creates an impurity electron with spin and . Conduction electrons in each lead are normalized such that . In a magnetic field , the impurity level is given by , where (-1) for () spin. The hybridization between and impurity electrons broadens the impurity level: with and . We consider the parameter region, where the half band-width is much greater than the other energy scales, .
We use the causal impurity Green’s function and self-energy defined at :
(3) |
The phase shift , or the density of states at , is a primary parameter which characterizes the Fermi-liquid ground state. The Friedel sum rule relates to the occupation number which can also be given by the first derivative of the free energy ,
(4) |
The leading Fermi-liquid corrections are determined by the static susceptibilities Yamada (1975a),
(5) |
It can also be expressed as , and is an enhancement factor similar to the Stoner factor. The usual spin and charge susceptibilities, and , are given by linear combinations of ^{2}^{2}2 and . and is an even function of . . These susceptibilities also determine the characteristic energy scale and the Wilson ratio which corresponds to a dimensionless quasi-particles interaction Krishna-murthy et al. (1980); Hewson (2001).
Away from half-filling, the third derivatives of the free energy also contribute to the next leading Fermi-liquid corrections, as we will show later
(6) |
It can also be expressed as a static thee-point function of the impurity occupation ,
(7) |
Higher-order Fermi-liquid corrections at .— The Ward identity, which reflects the current conservation for each spin component , plays a central role Yoshimori (1976),
(8) |
Here, the total vertex includes all contributions of multiple scattering, and Fig. 1 shows the assignment of arguments. The anti-symmetry properties of the total vertex also impose strong restrictions on the low-energy behavior as a consequence of the exclusion principle Abrikosov et al. (1965); Landau et al. (1991); Mermin (1967); Rohringer et al. (2012),
(9) |
For instance, at zero frequencies the parallel-spin component vanishes , and the leading Fermi-liquid relations ^{3}^{3}3 . follow from Eq. (8).
Another important clue is the analytic property. Non-analytic part of the vertex function is accompanied by the “” functions and is pure imaginary, while the analytic part is real. Thus, the low-frequency expansion of the real part of starts with a homogeneous polynomial of degree one. However, such a homogeneous polynomial of linear form cannot satisfy the anti-symmetry property Eq. (9) provided . Therefore, the parallel-spin component does not have an analytic part of linear order. Thus, for ,
(10) |
To our knowledges, this property has not explicitly been recognized so far. We have also calculated the skeleton diagrams for up to order and have confirmed Eq. (10) perturbatively ^{4}^{4}4Details will be given elsewhere: A. Oguri and A. C. Hewson, Phys. Rev. B 97, 035435 (2018); Phys. Rev. B 97, 045406 (2018). . In the linear order, the non-analytic part shows the dependence Shiba (1975) with a coefficient determined by Yamada-Yosida Yamada (1975b):
(11) |
A series of higher-order Fermi-liquid relations follow from this property of the total vertex for parallel spins.
We obtain an identity between the double derivatives of the real part of the self-energy using Eqs. (8) and (10),
(12) |
Note that by definition, and Eq. (12) agrees with FMvDM’s result given in Eq. (B8b) of Ref. Filippone et al., 2017. Furthermore, using Eqs. (8) and (12), the total vertex for anti-parallel spins can be calculated exactly up to terms of order ,
(13) |
Note that the -linear contribution is real and analytic.
We see in Eqs. (12) and (13) that expansion coefficients depends on which includes contributions from three-body fluctuations . The three-body correlations vanish in the particle-hole symmetric case since the spin (charge) susceptibility takes a maximum (minimum): and at and . We also find that the term of Eq. (13) involves four-body fluctuations in the real part through which remains finite even in the particle-hole symmetric case. The four-body fluctuations will also contribute to higher-order terms of the parallel-spin vertex.
We have also calculated the total vertex for two independent frequencies up to linear order in and :
(14) |
(15) |
The analytic real part can be deduced from Eqs. (11) and (13) using the anti-symmetry properties Eq. (9). The non-analytic part has been obtained through an additional consideration about the singular Green’s-function products Eliashberg (1962b); Yoshimori (1976); Oguri (2001). Specifically, the and contributions emerge from the intermediate particle-hole and particle-particle pair excitations, respectively. We note that the total vertex, Eqs. (14) and (15), can be regarded as a quantum-impurity analogue of Landau’s phemomenological interaction , and can also be compared with Nozières’ function Landau (1957); Nozières (1974). One of the advantages of the microscopic formulation to the phenomenological descriptions is that the real and imaginary parts, which contribute to the energy-shift and damping of quasi-particles, are described in a unified way with clearly defined correlation functions.
The and self-energy corrections.— The correction of the retarded self-energy can be deduced from the derivative of with respect to using the formula Yamada (1975b); Abrikosov et al. (1965); Note4 ,
(16) | |||
(17) |
Substituting Eqs. (14) and (15) into Eq. (17) ^{5}^{5}5Here, , in the retarded form. , we obtain
(18) |
Here, the real part, , emerges from the analytic part of the total vertex for anti-parallel spins.
In a previous work, we have diagrammatically shown that the low-bias self-energy can be calculated taking a variational derivative of the equilibrium self-energy with respect to the internal Green’s functions Oguri (2001, 2005). Revisiting the details of the calculation, we find exactly the same quantum-mechanical intermediate states, which consequently lead to Eq. (18), determine both the and corrections ^{6}^{6}6 Eq. (17) of this paper is identical to Eq. (14) of Ref. Oguri, 2001 i.e., with defined in Ref. Oguri, 2001 . The relation between these two corrections has been first pointed out by FMvDM using Nozières’ description Filippone et al. (2017). Our result provides a complete proof for this observation.
Using the above results, low-energy behavior of the retarded self-energy is exactly determined up to terms of order , , and . To be specific, the bias voltage is applied through the chemical potentials of the left and right leads, and , with additional parameters satisfying . Thus, the self-energy generally depends not only on but also Oguri (2001). The asymptotically exact imaginary and real parts of the retarded self-energy are given by
(19) |
(20) |
We note that Eq. (20) is consistent with the previous result of ours Oguri (2001), derived for general electron-fillings without the knowledge of Eq. (12) ^{7}^{7}7 Substituting the real part given in Eq. (12) into Eq. (19) of Ref. Oguri, 2001, we obtain the expression which agrees with Eq. (20) at . . Equation (20) is a generalized formula of the real part, which also extends FMvDM’s result Filippone et al. (2017) to asymmetric junctions Note4 .
Non-equilibrium magneto transport.— We next consider the current flowing through the Anderson impurity ^{8}^{8}8, where and . See also Note5 , using the Meir-Wingreen formula Meir and Wingreen (1992) with Eqs. (19) and (20). Specifically, we examine a symmetric junction with and , for which the conductance can be expressed in the form
(21) |
(22) | |||
(23) |
Here, contributions of the three-body fluctuations enter through the derivatives of susceptibilities with respect to or , which are accompanied by the factor . For the magneto conductance in the Kondo regime, there is a controversial issue Filippone et al. (2017): whether or not the zero-bias peak of splits at a magnetic field of the order of the Kondo energy scale . We demonstrate in the following that calculations with the exact conductance formula, Eqs. (22) and (23), resolve the problem Note4 .
We have calculated the phase shift and the enhancement factor as functions of at using the NRG Hewson et al. (2006, 2011). The dimensionless coefficients and have been determined substituting the NRG results into Eqs. (22) and (23). The result is shown in Fig. 2 as a function of , using defined at for each case of () ^{9}^{9}9 We used for the discretization parameter and keep states per iteration Krishna-murthy et al. (1980) . We see that both and show the universal Kondo behavior. This is consistent with the behavior of the Wilson ratio which is almost saturated to the strong-coupling value for Hewson et al. (2006). Furthermore, and change sign at of order : at very close magnetic-field values . This means that the zero-bias peak does split for because increases from the zero-bias value as increases. These observations are consistent with the previous second-order renormalized perturbation result Hewson et al. (2005).
Conclusion.— We have provided a many-body quantum theoretical description of the Fermi-liquid state in the particle-hole asymmetric case. The Fermi-liquid corrections away from half-filling are characterized by additional contributions of the three-body fluctuations which enter through the non-linear response function . The asymptotically exact expression of the total vertex describes low-energy properties in a unified way: this function and its derivatives with respect to or determine the quasi-particle interaction, energy shift, damping, and transport coefficients can be generated systematically up to order , , and , with the Ward identities given in Eqs. (8) and (17). Furthermore, the non-equilibrium self-energy Eq. (20) is applicable to the asymmetric tunneling couplings, and has potential application for real quantum dots Grobis et al. (2008); Scott et al. (2009); Ferrier et al. (2016). We have also demonstrated an application to the non-linear magnetoconductance through a quantum dot in the Kondo regime, and have shown that the zero-bias peak of splits naturally at a magnetic finite field of order . Our description can be extended, and may be used, to explore a wide class of Kondo systems and more general quantum impurities.
We wish to thank J. Bauer and R. Sakano for valuable discussions, and C. Mora and J. von Delft for sending us Ref. Filippone et al., 2017 prior to publication. This work was supported by JSPS KAKENHI (No. 26400319) and a Grant-in-Aid for Scientific Research (S) (No. 26220711).
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