Anderson model on a lattice
at particle-hole symmetry
We study the non-interacting two-impurity Anderson model on a lattice using the Green function equation-of-motion method. A case of particular interest is the RKKY limit that is characterized by a small hybridization between impurities and host electrons and the absence of a direct coupling between the impurities. In contrast to the low-density case, at half band-filling and particle-hole symmetry, the RKKY interaction decays asthe inverse square of the impurity distance along the axis of a simple cubic lattice. In the RKKY limit, for the spectral function we generically observe a small splitting of the single-impurity resonance into two peaks. For a vanishing density-density correlation function of the host electrons, we find only a broadened single peak in the local density of states.
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Impurities diluted in a metallic host pose a fundamental problem in solid-state theory . A famous example is the formation of a ‘Kondo cloud’ around a magnetic impurity in a metallic host that leads to a complete screening of a spin-1/2 impurity at zero temperature and to a very narrow Abrikosov-Suhl resonance in the single-particle density of states . The formation of a magnetic impurity and its screening by the host electrons is contained in the single-impurity Anderson model ; for a review, see Ref. , and references therein.
A single (magnetic) impurity induces distortions in the host electrons’ charge and spin density (‘Friedel oscillations’) [5, 6]. These distortions can be sensed by a second (magnetic) impurity so that the host electrons generate an effective interaction between the impurities, known as RKKY interaction, named after Ruderman and Kittel , Kasuya , and Yosida ; for a concise derivation from perturbation theory, see appendix I of .
The combined physics of the Kondo effect and of the RKKY interaction is contained in the two-impurity Anderson model (TIAM) . It describes two impurities embedded in a metallic host at lattice sites and . When the local Hubbard interaction on the impurities is strong, the model covers the local Kondo physics and the RKKY interaction between magnetic impurities. However, the TIAM is a true many-particle problem that cannot be solved in general; for a recent investigation using an extended non-crossing approximation, see Ref. , and references therein.
The TIAM is frequently invoked in studies of coupled quantum dots where each dot represents an impurity, see Ref.  for a recent study. However, the quantum dots have individual leads, i.e., there are two independent host metals so that the indirect exchange interaction is different from the solid-state case for which the TIAM was designed originally. Moreover, the direct coupling between the quantum dots is generically large and dominates over the RKKY interaction.
In the present work, we consider the non-interacting two-impurity Anderson model that can be solved exactly using the Green function equation-of-motion method . In contrast to the perturbative RKKY derivation, the results include the full impurity-host hybridization and multiple scattering events of the host electrons off the impurities to all orders. The formulae provide the basis of a Gutzwiller approach to the TIAM which permits a variational analysis of the competition between the single-impurity Kondo effect and the two-impurity RKKY interaction . For this reason, we are particularly interested in the case of particle-hole symmetry at half band-filling.
To study the competition between single-impurity and two-impurity physics, we focus on the RKKY limit of a small, local hybridization in the absence of a direct electron transfer between the impurities. This case was not worked out in detail in Ref.  where the impurities were dominantly coupled by a direct electron transfer. Although we cannot study the formation of a local moments, we determine the effective RKKY interaction of the (non-interacting) impurities as a function of their separation fully analytically. We find that the RKKY interaction generically leads to two peaks in the single-particle density of states, i.e., the host metal generates a small but finite transfer matrix element between the impurities.
Our work is organized as follows. We formulate the two-impurity Anderson model in section 2. In section 3 we solve the non-interacting model using the equation-of-motion method for the retarded Green functions . We introduce and utilize particle-hole symmetry in section 4, and work out the impurity properties for tight-binding host electrons on a simple-cubic lattice in detail in section 5. Short conclusions, section 6, end our presentation. Some technical details are deferred to the appendix.
2 Two-impurity Anderson model
We start our investigations with the definition of the Hamiltonian. Then, we rephrase the problem in terms of a single-site two-orbital model.
Two impurities in a metallic host on a lattice are modeled by the Hamiltonian 
Here, is the kinetic energy of the non-interacting spin-1/2 host electrons (),
where the electrons tunnel between the sites and of the lattice with amplitude . The kinetic energy is diagonal in Fourier space. For from the first Brillouin zone we define
where is the (even) number of lattice sites. With
where is the dispersion relation.
With we also permit a direct electron transfer with amplitude between the impurity orbitals at sites and ,
Next, represents the Hubbard interaction to model the Coulomb repulsion on the impurities,
where counts the number of impurity electrons ().
Lastly, describes the hybridization between impurity and host electron states,
The model (1) poses a difficult many-particle problem that cannot be solved in general.
2.2 Single-site two-orbital model
As a second step, we map the two-impurity model onto an asymmetric two-orbital model.
2.2.1 Kinetic energy of d-electrons
We introduce the -basis for the impurity electrons using the unitary transformation
The inverse transformation reads
Then, , eq. (6), is diagonal in the -basis,
In this representation, has the form of a splitting of two impurity levels on the same site.
For a unitary transformation we have
Therefore, the average number of -electrons equals the average number of -electrons.
In the -basis, the hybridization , see eq. (8), takes the form
The two impurity levels hybridize with the conduction electrons with the matrix elements
We do not elaborate in the new basis because we restrict ourselves to the non-interacting model in the following.
3 Green functions for the non-interacting model
For in (1), the single-particle Green functions can be calculated exactly using the equation-of-motion method. From the Green functions, all ground-state properties and the single-particle density of states can be derived.
The retarded Green functions for the host electrons and the impurity electrons read
where is the ground state for , is the Heaviside step function, is the anti-commutator, and
describes the time evolution of a Schrödinger operator in the Heisenberg picture; we set for convenience.
3.2 Equations of motion
The time derivatives of the single-electron Heisenberg operators read
Moreover, we define the Fourier transformation for a retarded Green function (, )
With these equations it is possible to derive a closed set of algebraic equations for the Green functions. For later use, we define the bare Green functions
that appear in the absence of the hybridization, .
3.2.1 Host electrons
The first set of equations involves the host electrons,
where we dropped the spin index for convenience. After Fourier transformation, we find that
so that the host Green function obeys
When we insert this expression into eq. (27) we find that
where we introduced the notation and , and the hybridization matrix functions
The resulting matrix problem (29) for fixed frequency and fixed Bloch momentum is readily solved to give
The trace over all -states results in the average host Green function
with . We need the average host Green function for the calculation of the ground-state energy and the impurity contribution to the density of states.
3.2.2 Impurity electrons
After Fourier transformation the equations of motion for the impurity Green functions read
We insert the second equation into the first to obtain
The solution of this matrix problem for fixed gives
For our further investigations, the diagonal impurity Green functions in eq. (38) are sufficient.
3.3 Ground-state expectation values
Lastly, we use the Green functions to calculate the impurity contribution to the single-particle density of states. In addition, we derive the ground-state energy, impurity density, and hybridization energy.
3.3.1 Density of states and ground-state energy
We introduce the single-particle density of states
where are the energies of the single-particle levels. We introduce the exact single-particle and single-hole excitations of the ground state ,
where is the energy of the ground state . Then, the single-particle density of states can be written as
The sum on all single-particle excitations is equivalent to the trace over the subspace of all single-particle excitations,
We can equally use the excitations and to perform the trace over the single-particle excitations of the ground state. Therefore, we may write
for the contribution to the density of states that originates from the impurities and their hybridization to the host electrons.
The change in the ground-state energy contribution due to the hybridization between host electrons and impurities is given by
where () are the single-particle energies in the presence (absence) of the hybridization , and the factor two accounts for the spin degeneracy. With the help of the density of states it is readily calculated from
3.3.2 Particle density and hybridization energy
In general, for the retarded Green function
we obtain its Fourier transformation as
The corresponding density of states is given by
The first term is finite only if because for the eigenenergies of . Therefore,
for the ground-state expectation value of the operator product .
As an example, we give explicit expressions for the impurity occupancies,
For the hybridization matrix element we find
see eq. (32). The hybridization energy reads
4 Particle-hole symmetry at half band-filling
We are interested in the case where there is on average one electron on each of the impurities. This can be assured for a particle-hole symmetric Hamiltonian (1) at half band-filling.
We consider a bipartite lattice. We assume that there exists half a reciprocal lattice vector for which
We also assume inversion symmetry, ; recall that . For the electron transfer matrix elements between two sites at distance this implies
where MBZ is the reduced (or ‘magnetic’) Brillouin zone that contains only half of the vectors of the Brillouin zone.
Consequently, the electron transfer matrix elements between sites on the same sublattice (-lattice) ought to be imaginary, and those between sites on different sublattices (-lattice) must be real,
However, in a solid in the absence of spin-orbit coupling, the tunnel amplitudes for electrons are real so that particle-hole symmetry actually implies . In our conceptual study we do not impose this constraint.
We demand that the transfer element in eq. (6) has the same properties as . Therefore, we assume to be imaginary when and are on the same sublattice, and real when they are on different sublattices. This can be cast into the relation
in eq. (10).
4.2 Particle-hole transformation
as particle-hole transformation so that . The unitary operator that generates the particle-hole transformation is provided in appendix A.
Using eqs. (60) and (63) it is readily shown that and are particle-hole symmetric, i.e., and . Moreover, eqs. (57) and (63) ensure that maps onto itself under . Lastly, the hybridization is seen to be invariant when we use eqs. (62) and (63). Therefore, the particle-hole transformation maps onto itself,
The operator for the total particle number is given by
Under the particle-hole transformation it transforms as
Therefore, the particle-hole transformation maps systems at and above half filling to those at and below half filling, and vice versa.
4.3 Half-filled bands
In the following we consider paramagnetic bands at half filling where the number of electrons equals the (even) number of orbitals, , and . Note that there are lattices sites for the host electrons and two additional impurity orbitals on the lattice sites and .
At half band-filling, the non-degenerate ground state maps onto itself under the particle-hole transformation, . Therefore, we find
i.e., each impurity level is exactly half filled for all hybridizations and interaction strengths,
Moreover, it is readily shown that the bare density of states is symmetric,
so that the Fermi energy is at at half band-filling.
When the Hamiltonian is expressed in the -basis, we note that
at half band-filling. Moreover, the particle-hole transformation implies
where we used the notation , . Therefore, particle-hole symmetry at half band-filling leads to
so that there is no hybridization between the -orbitals at half band-filling,