Renormalized parameters and perturbation theory for an n-channel Anderson model with Hund’s rule coupling.
We extend the renormalized perturbation theory for the single impurity Anderson model to the -channel model with a Hund’s rule coupling, and show that the exact results for the spin, orbital and charge susceptibilities, as well as the leading low temperature dependence for the resistivity, are obtained by working to second order in the renormalized couplings. A universal relation is obtained between the renormalized parameters, independent of , in the Kondo regime. An expression for the dynamic spin susceptibility is also derived by taking into account repeated quasiparticle scattering, which is asymptotically exact in the low frequency regime and satisfies the Korringa-Shiba relation. The renormalized parameters, including the renormalized Hund’s rule coupling, are deduced from numerical renormalization group calculations for the model for the case . The results confirm explicitly the universal relations between the parameters in the Kondo regime. Using these results we evaluate the spin, orbital and charge susceptibilities, temperature dependence of the low temperature resistivity and dynamic spin susceptibility for the particle-hole symmetric model.
The single impurity Anderson model And61 () has played an important role in understanding many aspects of the behavior of electrons in systems with strong electron correlation. Non-perturbative methods have had to be developed to make predictions for the behavior of the model in the strong interaction regime. Among the most successful have been the seminal and pioneering work of Wilson and associates Wil75 (); KWW80a () based on the numerical renormalization group (NRG), and the exact solutions using the Bethe ansatz for the linear dispersion version of the model AFL83 (); TW83 (). Though the model was originally put forward to describe magnetic impurities in a host metal, it has proved to be applicable to other situations. One main area of application is as a model for strong correlation effects in quantum dots PG04 (). In this application certain parameters of the model, such as the impurity level which determines the electron occupancy on the quantum dot, can be varied by a gate voltage. This makes it possible to sweep through different parameter regimes of the model, which would be difficult to do for real magnetic impurities, and so the predictions of the model can be tested more rigorously. The presence of the narrow many-body resonance in the strong correlation (Kondo) regime at low temperatures can be inferred directly from the measurements of the current through the dot as a function of an applied bias voltage GSMAMK98 (); COK98 ().
Apart from these direct applications of the model, it has also played a role in the calculations of strong correlation effects in lattice models. It is possible to map a class of infinite dimensional lattice models of strong electron correlation onto an effective Anderson impurity model with a self-consistency condition, which determines the density of states of the effective medium GKKR96 (). This mapping requires that the self-energy is a function of frequency only which is the case in the limit of infinite dimensionality, and the mapping is exact in this limit. For many strongly correlated systems it is known that the wave vector dependence of the self-energy is much less important than the frequency dependence so this approach can be used as a good first approximation for systems in three dimensions (dynamical mean field theory (DMFT)). As the assumption of linear dispersion is not valid for the effective impurity model generated in this application, there are no exact Bethe ansatz solutions, so the most reliable non-perturbative approaches, such as the NRG have to be used.
It has not proved possible so far to access the strong correlation regime of the Anderson model by an approach based purely on perturbation theory in powers of the local interaction . However, it has been shown that, if the perturbation theory is reorganized such that the basic parameters of the model are renormalized, then a perturbation theory in the renormalized interaction , taken only to second order gives formally the exact results for the low temperature properties and low frequency dynamics, provided counter terms are taken into account to avoid over-counting Hew93 (); Hew01 (). The renormalized parameters have to be determined but these can be calculated very accurately from an analysis of the low energy excitations of an NRG calculation on the approach to the low energy fixed point HOM04 (). So far this approach has only been developed in detail for the non-degenerate one channel model, but the approach is one that can be applied to a more general class of models including lattice models. Here we extend the calculations to an -channel impurity Anderson model with the inclusion of a Hund’s rule exchange term. The Hamiltonian takes the form,
where , , are creation and annihilation operators for an electron in an impurity state with total angular momentum quantum number , and -component , and spin component . The impurity level in a magnetic field we take as , where () and () and is the chemical potential, and the Bohr magneton. The creation and annihilation operators , are for partial wave conduction electrons with energy . The hybridization matrix element for impurity levels with the conduction electron states is . We denote the hybridization width factor by , which we can take to be a constant in the wide flat band limit. The remaining part of the Hamiltonian, describes the interaction between the electrons in the impurity state, which we take to be of the form,
As well as the direct Coulomb interaction between the electrons, we include a Hund’s rule exchange term between electrons in states with different values. The sign for the exchange term has been chosen so that corresponds to a ferromagnetic interaction. This model can be used to describe transition metal impurities, such as Mn or Fe, in a metallic host in the absence of spin orbit or crystal field splittings. We can interpret the model more generally with as a channel index taking values where is the number of channels. The Hund’s rule term tends to align the electrons on the impurity site such that for large and large the impurity state will correspond to a spin . The model with has also been used to describe capacitively coupled double quantum dotsGLK06 (), where the impurity channels correspond to different dots. In that application, however, the inter-dot interaction will in general differ from the intra-dot interaction , so the case here, with , is a special point with SU(2n) symmetry when .
The structure of this paper will be as follows. In the next section we formulate the renormalized perturbation theory (RPT) for this model in terms of the renormalized parameters, , , and . We then show that the low temperature behavior, as measured by the charge and spin susceptibilities and the low temperature contribution to the resistivity, can be obtained exactly from the RPT taken to second order in powers of and . In the localized or Kondo regime we show that , and can be expressed in terms of a single parameter which we take as the Kondo temperature . This relation is independent of the channel index and hence applies to all values of . Though we cannot calculate , and for the general -model using the NRG we can calculate them for the two channel case . We look at this case in detail and confirm the universal relation between the renormalized parameters in the Kondo regime predicted using the RPT.
Ii Renormalized Perturbation Theory
We start with the Fourier transform of the single particle Green’s function for the impurity -state,
where and and the brackets denote a thermal average.
where is the self-energy. For the zero temperature Green’s function, which will be our main concern, can be replaced by continuous variable , and summations over replaced by integrations over . For the perturbation theory in powers of and it will be convenient to separate the interaction terms in the Hamiltonian into the terms involving interactions between electrons in the same channel and those between electrons in different channels. We rewrite the Hamiltonian from Eq. (2) in the form,
The vertices associated with the three types of interaction terms are illustrated in Fig. 1.
For the renormalized perturbation theory, the Green’s function in Eq. (4) can be re-expressed as , where is the quasiparticle Green’s function given by
and the renormalized parameters, and are given by
where evaluated at . The quasiparticle self-energy is given by
where we have assumed the Luttinger theorem Lut60 (), , so that as . When expressed in this form, the part of the self-energy and its derivative have been absorbed into renormalizing the parameters and , so in setting up the perturbation expansion any further renormalization of these terms must be excluded, or it will result in over-counting. In working with the fully renormalized quasiparticles it is appropriate to use the renormalized or effective interactions between the quasiparticles. In the single channel case, we defined the renormalized interaction in terms of the four vertex in the zero frequency limit Hew93 (). In this case we need to consider the more general four vertex, , which corresponds to the Fourier coefficient of the connected skeleton diagram for the two particle Green’s function,
with the external legs removed. Using the fact that the spin and angular momentum are conserved independently, and taking into account the antisymmetry conditions of the fermion creation and annihilation operators, it was shown by Yoshimori Yos76 () that this vertex at zero frequency can be expressed in terms of two parameters, and , as
To first order in the interaction terms, and , we have and . We generalize this result to specify the renormalized parameters, , and , by the relation,
where the factor arises from the rescaling of the fields to define the quasiparticle Green’s function given in Eq. (6). For this reduces to
which is the definition of used in earlier work Hew93 ().
We can combine these terms to define a quasiparticle Hamiltonian ,
The brackets :: indicate that the operator within the brackets must be normal ordered with respect to the ground state of the interacting system, which plays the role of the vacuum. This is because the interaction terms only come into play when more than one quasiparticle is created from the vacuum.
The renormalized Hamiltonian is not equivalent to the original model, and the relation between the original and renormalized model is best expressed in the Lagrangian formulation, where frequency enters explicitly Hew01 (). For simplicity, we consider the case in the absence of a magnetic field, where the energy levels are independent of and . If the Lagrangian density describes the original model, then by suitably re-arranging the terms we can write
where the remainder part is known as the counter term and takes the form,
where , , and . Though we can express the coefficients , , explicitly in terms of the self-energy terms and vertices at zero frequency, these relations are not useful in carrying out the expansion. We want to work entirely with the renormalized parameters and carry out the expansion in powers of and . We assume that the can be expressed in powers of and , and determine them order by order from the conditions that there should be no further renormalization of quantities taken to be already fully renormalized. These conditions are
and that the renormalized 4-vertex at zero frequency, is such that
In the field theory context these conditions are more commonly known as the renormalization conditions. They follow directly from the definitions of the renormalized self-energy in Eq. (7) and the definitions of the renormalized parameters given in Eq. (10).
The propagator in the RPT is the free quasiparticle Green’s function,
The spectral density of the corresponding retarded Green’s function gives the free quasiparticle density of states, given by
From Fermi liquid theory, the quasiparticle interaction terms do not contribute to the linear specific heat coefficient of the electrons. It follows that the impurity contribution to this coefficient is proportional to the free quasiparticle density of states evaluated at the Fermi level and is given by
In the absence of a magnetic field this reduces to , where is the quasiparticle density of states per single spin and channel.
If we integrate the free quasiparticle density of states in Eq. (19) to the Fermi level then we get at , which is given by
which defines the phase shift in the channel with quantum numbers and . For this model it has been shown by Shiba Shi75 () that , giving a generalization of the Friedel sum rule, so that we have ; the quasiparticle occupation number in each channel is equal to the impurity occupation number in that channel. However, Yoshimori and Zawadowski YZ82 () have shown that this form of the Friedel sum rule does not hold for a more general model in which scattering processes can occur between -states, , such that . They derive a restricted form of the sum rule such that , where . In this more general case, therefore, the quasiparticle number does not equal the occupation number in the same channel but we have the more restricted result, . Using either result, however, we can derive expressions for the zero field spin , orbital and charge susceptibilities. We differentiate the combinations, , with and respectively, with respect to the magnetic field or in the charge case with respect to . To evaluate these expressions we need to calculate the renormalized self-energy. This calculation taken to first order in and proceeds as in the one channel case Hew93 (); Hew01 (), and gives
These results can also be obtained from a mean field theory on the quasiparticle part of the Hamiltonian given in Eq. (13) Hew93c (); Hew93b (). It can be shown using the Ward identities derived by Yoshimori Yos76 (), which are generalizations of the Ward identities derived by Yamada Yam75a (); Yam75b () for the single channel case, that these results are exact. Hence all higher order correction terms in and cancel out.
In the localized regime a large value of suppresses the charge fluctuations on the impurity so . Treating this as an equality, we get a relation between , and ,
When , this reduces to
For the case of half-filling, where and , the non-linear relation between the renormalized parameters in Eq. (25) becomes a linear relation between , and ,
For we get
which agrees with the one channel result for .
When and we are in the localized limit, we have only one energy scale which we can take to be the Kondo temperature, defined for general such that , equivalent to taking . In this limit the result for the Wilson ratio, . This is the same as that for the N-fold degenerate Anderson model used to describe rare earth impurities for . This result could have been anticipated, because the models can be shown to be equivalent by putting the orbital and spin indices into a combined index GGL09 ().
Switching on the interaction () will reduce the local orbital fluctuations, as the configuration with the spins aligned will be favored. For we can expect the orbital fluctuations to be almost fully suppressed so that which, as an equality, gives a further relation between , and ,
At half-filling this gives another linear relation between , and ,
An equivalent condition to that in Eq. (30) can be obtained using the argument of Nozières and Blandin NB80 () that the occupation number in a channel should be independent of any small change in the chemical potential in a channel in this regime. When both the local charge and orbital fluctuations are suppressed, the renormalized parameters can be expressed in terms of the Kondo temperature . From Eq. (25) and (29) we deduce
for the particle-hole symmetric case we have , so then we have
which was conjectured earlier on the basis of a phenomenological mean field approach Hew93b (); Hew93c (). A notable feature of this result is that there is no explicit dependence on . In this regime from Eq. (22) we have for the spin susceptibility,
Yoshimori Yos76 () has also derived an exact result for the low temperature impurity contribution to the resistivity in the particle-hole symmetric case and . In terms of the renormalized parameters, the result is
where is given by
This result can be derived in the RPT from a calculation of the renormalized self-energy to second order in and . With the Hund’s rule interaction term, there are several types of second order scattering diagrams which are illustrated in Fig. 2. The vertices are of the same type as shown in Fig. 1 but are weighted by the renormalized interaction terms. The calculations follow along similar lines to those for the single channel case Hew93 (); Hew01 (). The first order diagrams and the terms linear in are canceled by the counter terms to this order, and there are no corrections from the counter terms to the vertices to second order for the case with particle-hole symmetry. The contributions to from diagrams of the types (i) to (iv) respectively in units of are: ; ; ; -; which give the result in Eq. (35).
In the localized regime at half-filling the result in Eq. (34) simplifies to give
which agrees with the result derived by Nozières Noz74 () and Yamada Yam75a (); Yam75b () for the case . Thus all the exact Fermi liquid relations can be derived from the RPT taken to second order only.
It was shown in earlier work Hew06 () that the RPT approach can provide a description of the dynamic spin susceptibility for the model in the low frequency regime. The calculation takes account of the repeated quasiparticle scattering, giving results which are exact in the low frequency limit , and in remarkably good agreement with the results from a direct NRG calculation. We extend the calculation to the -channel model given in Eq. (1) and (2). We consider the Fourier transform of the transverse spin susceptibility,
where and , (). We consider the scattering of a spin up quasiparticle with a spin down quasihole both in channel , in the absence of a magnetic field. This particle-hole pair can scatter into a particle-hole pair in the same channel or a different channel . We consider the scattering into the same channel first of all. The matrix element for this process is , except we must allow for the fact that already takes into account these processes for so, to prevent over-counting, we must use , where is the corresponding counter term. It will be convenient to use the notation for . Just taking this type of repeated scattering into account gives us a result which has the same form as in the single channel case Hew06 (),
where we have analytically continued to real frequency . The free quasiparticle-quasihole propagator in a single channel, , is independent of the channel index in the absence of a magnetic field, and is given by
for . We must also take into account that the quasiparticle-quasihole pair being created in channel can scatter into a different channel , and also be finally annihilated in a channel with . The matrix element for this type of scattering is , corresponding to the diagram in Fig. 1 (ii), but again, to avoid over-counting, we replace it by . In the absence of a magnetic field, the quasiparticle-quasihole propagator is independent of the channel index , so the summation over the states introduces a factor . The result of taking these scattering processes into account is that the pair propagator in Eq. (38) is replaced by
which leads to the result,
We need to determine the combination . We can do this by requiring that this expression gives in the zero frequency limit, which is equivalent to the requirement that these scattering processes contribute to the four vertex at zero frequency are not over-counted. This condition gives
In the Kondo regime this condition simplifies to , which gives the one channel result for .
By rewriting Eq. (41) in the form,
and taking the imaginary part, it is straight forward to show that the expression for satisfies the exact Korringa-Shiba relation,
So far we have not discussed how one can calculate the renormalized parameters , , and . In the Kondo regime these reduce to a single parameter , so one possibility is to deduce its value from experiment by fitting the predictions to the measurements of a physical quantity in the low temperature regime, say the impurity susceptibility or resistivity. Outside the Kondo regime we have four parameters to determine, and to calculate all four from experiment one loses much of the predictive power of the approach. However, it was shown earlier for the single channel Anderson model how the parameters, , and , can be calculated in terms of the bare parameters, , and , from the many-body low energy excitations of an NRG calculation HOM04 (). There are problems in carrying out this procedure for the general -channel model, due to the truncation of states which has to be carried out in an NRG calculation to reach the very low energy scales. Truncation means that only a fraction states can be retained at each NRG iteration. It is possible, however, for the case to compensate for the lower percentage by increasing the number of states kept at each iteration as the matrices do not get so large. In the next section we present for calculations of , , and , in terms of , , and , for the model.
Iii NRG Calculation of the Renormalized Parameters for n=2
The two-channel model the Hamiltonian given in Eq. (2) can be re-expressed in the form,
with a ferromagnetic Heisenberg exchange coupling between the electrons in the different channels, and . Our calculations will be restricted to the particle-hole symmetric model so we take in the one-electron part of the Hamiltonian given in Eq. (1). The energy of the two electron triplet state of the isolated impurity with particle-hole symmetry is and that of the 4-electron or 0-electron state is 0, so if we are interested in the case when the triplet state is the ground state configuration, we need to consider the regime .
For the NRG calculations the model is recast in a form such that the impurity is coupled via a hybridization to two tight binding chains which describe the conduction electron states, one chain for each channel. The conduction electron band is discretized with a discretization parameter , such that the couplings decrease along the chains as for large , where is the th site along the chain from the impurity. The calculations are then carried out iteratively by direct diagonalization, starting at the impurity site and adding one further site to each chain at each iteration step. The number of basis states used has to be truncated when the matrices get too large for diagonalization on a practical timescale, which can occur after only a few iterative steps. When truncation is applied a fixed number of states is retained at each step. For the model considered here, we take 3600 states, which is a factor of 3 to 4 more than for the non-degenerate model () and a discretization factor . We can check the expected accuracy of our calculations by using this value for to calculate and for the single channel model and compare with the values deduced indirectly from the exact Bethe ansatz results for the specific heat coefficient and the zero temperature spin susceptibility HOM04 (). For , , keeping states, we get the values, and , which can be compared with those deduced from the Bethe ansatz, and . This gives an accuracy of better than 0.3%. For further details on setting up the NRG calculations, we refer to the original papers Wil75 (); KWW80a () and the recent review article BCP08 ().
With this discrete spectrum the Green’s function in Eq. (4) takes the form,
where is the Green’s function for the first site for the isolated conduction band chain.
The connection between the NRG approach and the renormalized perturbation theory is based on identifying the quasiparticle Hamiltonian, given in Eq. (12) and (13), as the low energy fixed point of the NRG together with the leading irrelevant terms Hew05 (). The lowest single-particle excitations from the NRG ground state should correspond to a quasiparticle excitation described by the one-body part of the quasiparticle Hamiltonian as given in Eq. (12). For the calculation of the interaction terms, and , from the NRG we have to consider the difference between two-body excitations from the NRG ground state and the two corresponding one-body excitations.
The low energy single-particle excitations are given by the poles of the non-interacting quasiparticle Green’s function when analytically continued to real frequency . The equation for these poles is the same as that for the non-interacting model but with a renormalized hybridization and energy level . Therefore, the lowest energy single particle and hole excitations, and , from the interacting ground state should be solutions of the equation,
If we substitute the excitations energies, and , as calculated in the NRG for a finite chain length , into Eq. (47) then we can deduce corresponding -dependent renormalized parameters and . Only if and become independent of for large , do the low energy one-particle energy levels of the interacting system correspond to those of a renormalized non-interacting model. If this is the case, then the asymptotic values for large define the renormalized parameters (and hence ) and .
To calculate the renormalized interaction terms, we first have to diagonalize the non-interacting impurity model with the renormalized parameters, which describes the quasiparticles. The interaction terms are then added to the quasiparticle Hamiltonian and expressed using the diagonalized single quasiparticle states as a basis. The energy difference between the lowest two-particle state and the sum of the corresponding two quasiparticle states is equal to the expectation value of interaction terms in the quasiparticle Hamiltonian. The interaction parameter can be calculated from the NRG results for the lowest two-particle excitation in the same channel which will be independent of . For a finite length chain the value will depend upon , and for this to correspond to a low energy quasiparticle Hamiltonian should become independent of for large . The asymptotic values of for large