The strong-coupling limit of a Kondo spin coupled to a mesoscopic quantum dot: effective Hamiltonian in the presence of exchange correlations
We consider a Kondo spin that is coupled antiferromagnetically to a large chaotic quantum dot. Such a dot is described by the so-called universal Hamiltonian and its electrons are interacting via a ferromagnetic exchange interaction. We derive an effective Hamiltonian in the limit of strong Kondo coupling, where the screened Kondo spin effectively removes one electron from the dot. We find that the exchange coupling constant in this reduced dot (with one less electron) is renormalized and that new interaction terms appear beyond the conventional terms of the strong-coupling limit. The eigenenergies of this effective Hamiltonian are found to be in excellent agreement with exact numerical results of the original model in the limit of strong Kondo coupling.
pacs:72.15.Qm, 72.10.Fk, 73.21.La, 73.23.Hk,
The Kondo resonance, which emerges when a localized impurity spin interacts antiferromagnetically with a delocalized electron gas, has generated considerable interest over several decades by now. (1); (2) The observation that the Kondo resonance can be realized in the mesoscopic regime of quantum dots, in which many of the system parameters are experimentally tunable has led to much renewed interest over the last decade.(4); (3); (5); (6); (7); (8); (9); (10); (11); (12); (13) This experimental work has been accompanied by much theoretical progress on the mesoscopic aspects of the Kondo problem.(14); (15); (16); (17); (18); (21); (19); (20); (23); (22); (24); (25); (26); (27)
In the mesoscopic regime, the spin-1/2 Kondo impurity is typically represented by a small quantum dot with an odd number of electrons, while the delocalized electron gas is realized by electrons in leads or in a large quantum dot. In this work we focus on the latter case, assuming a small and a large quantum dots that are coupled antiferromegnetically as in Fig. 1a (see Ref. (9) for an experimental realization of such a setup).
There are certain features that distinguishes the mesoscopic Kondo regime from the bulk limit. While the conventional Kondo theory assumes a continuum band of energy levels in the electron gas, the single-particle energy levels in the large quantum dot are discrete. The discreteness and the dot-specific realization of these energy levels become important when the Kondo temperature , the characteristic energy scale of the correlated Kondo resonance, is of the same order of magnitude or smaller than the average level spacing .(17); (19); (23); (24); (27) In the conventional bulk Kondo model the electron-electron interactions in the electron gas are often neglected. However, for the present double-dot setup, electron-electron interactions in the large dot can play an important role. In the following we assume the single-particle dynamics in the large quantum dot to be chaotic,(28); (29); (30); (31) in which case the dot is described by the so-called “universal Hamiltonian”. (32) This Hamiltonian describes the low energy physics in a Thouless energy interval around the dot’s Fermi energy. For a semiconductor quantum dot with a fixed number of electrons and in the limit of a large Thouless conductance, the electron-electron interaction is dominated by a ferromagnetic exchange interaction that is proportional to the square of the total dot spin. Despite its conceptual simplicity, this universal Hamiltonian description was shown to yield a quantitative agreement(33) with experimental results measuring the statistics of the Coulomb blockade peak heights and spacings.(34)
The effect of ferromagnetic exchange correlations on the Kondo resonance was first addressed analytically in the bulk limit,(35) and, more recently, mean-field studies were carried out in the mesoscopic regime.(23) In a recent work, we studied this problem numerically and provided analytical results for the weak and strong Kondo coupling limits.(27) We found that for weak Kondo coupling, the Kondo spin acts like an external magnetic field, assisting the ferromagnetic polarization of electrons in the large dot. In the case of strong Kondo coupling, the Kondo spin effectively removes one of the electrons of the large dot. We showed that this “reduced” dot with one less level and one less electron can again be described by a universal Hamiltonian but with a renormalized exchange constant.
A central issue that was not addressed in our previous work concerns the nature of residual interactions in the reduced dot beyond the renormalization of its exchange interaction term. From the work of Nozières(36) we know that a non-interacting electron gas turns into a Fermi-liquid when strongly coupled to a Kondo spin. The dominant effective interaction between the quasi-particles in this Fermi liquid is a repulsive interaction between spins of opposite orientation that are in close proximity to the Kondo spin. In the present case, the finite exchange interaction in the large dot leads to new effective interaction terms in the strong-coupling limit. To identify these new interaction terms, we follow a strategy that is similar to the one used by Nozières,(36) i.e., we perform an explicit perturbative expansion of the effective Hamiltonian of the reduced dot in the strong-coupling limit. In the presence of exchange interaction, this strong-coupling expansion is significantly more involved. However, the resulting effective quasi-particle interaction contains only a few new terms. The analytical expressions we derived for these effective exchange-like interactions are validated by a comparison with a full numerical diagonalization of the original two coupled dot model.
The outline of this paper is as follows: In section II we present the model of a spin-1/2 quantum dot that is Kondo-coupled to a large quantum dot (described by the universal Hamiltonian), and discuss the transformation from the single-particle orbital basis of the large dot to a chain site basis, commonly employed in the strong-coupling limit. In Section III we discuss the limit of strong Kondo coupling and use a projection method to derive an effective Hamiltonian for the reduced large quantum dot with one less electron. In section IV we describe the evaluation of the eigenenergies of this effective Hamiltonian, and in section V we compare the results derived from this effective Hamiltonian with an exact numerical solution of the original model. In section VI we conclude with a summary and discussion.
We consider a chaotic quantum dot that is coupled antiferromagnetically to a Kondo spin as realized, e.g., by a small quantum dot with an odd number of electrons. A schematic illustration of such a system is shown in Fig. 1a. In the following we will refer to the large quantum dot as the “dot” and to the small dot as the “Kondo spin.”
In the limit of a large Thouless conductance, a quantum dot whose single-particle dynamics are chaotic is described by the universal Hamiltonian(32)
Here creates an electron with spin up/down () in level in an orbital single-particle level with energy . We assume spin-degenerate single-particle levels spanning a bandwidth of ( is the mean level spacing). The second term on the r.h.s. of Eq. (1) represents a ferromagnetic exchange interaction () where ( are Pauli matrices) is the total spin of the dot. In Eq. (1) we have ignored a constant charging energy term and a repulsive Cooper channel term.
The dot is coupled antiferromagnetically to a Kondo spin ()(23)
where () is the Kondo coupling constant and is the spin density of the dot at the tunneling position . The dot spin density at position is given by
where creates an electron with spin at position . In terms of the single-particle orbital wave functions , the field operator is given by and the local density of states of the dot is given by(17); (23); (24) , with an average value of .
ii.2 Chain site basis
The strong-coupling limit of the system in Fig. 1a is more clearly described when the Hamiltonian in Eq. (2) is rewritten in a different basis, known as the chain site basis. This new basis is obtained by a unitary transformation of the orbital basis(2)
such that site corresponds to the tunneling position , and the new one-body site Hamiltonian of the dot is tridiagonal, i.e., each site is coupled to its two nearest neighbors. Such a transformation is constructed by choosing and carrying out a Gram-Schmidt orthogonalization procedure.(2)
The chain site single-particle energies are given by the diagonal elements of when the latter is rewritten in the site basis. The off-diagonal matrix elements and describe the hopping amplitudes between neighboring sites. A spin can be associated with each site, where the spin of site is equal to the spin density at the tunneling position, i.e., . In the site basis, the Kondo spin couples only to a single site . The Hamiltonian in Eq. (2) is now given by
where the total spin of the dot is . Here is the one-body Hamiltonian of the dot in the site basis
with being the hopping Hamiltonian
The site basis formulation is particularly advantageous for the strong-coupling limit when the site effectively decouples from the rest of the chain. Accordingly, we decompose the Hamiltonian in Eq. (5) into three terms (see Fig. 1b for a schematic illustration)
where describes the Hamiltonian of the Kondo spin plus site , is the Hamiltonian of a “reduced” dot with sites and contains the remaining coupling terms. Writing , where is the spin of the reduced dot, we have
Here and in the following, operators in the reduced dot space of chain sites are denoted by primed quantities. For such operators, the summation over sites starts from rather than from .
ii.3 Site basis with good spin quantum numbers
The Hamiltonian in Eq. (8) is invariant under spin rotations and therefore conserves the total spin of the system (Kondo spin plus dot spin) . To take advantage of this symmetry, it is convenient to use a basis for which both the total spin and the corresponding magnetic quantum number are good quantum numbers.
There are different ways to construct a basis with good total spin, but one of them is particulary useful in the strong-coupling limit . To zeroth order in and , we can ignore the coupling term , in which case the subsystem of Kondo spin plus site 0 decouples from the reduced dot. The Hamiltonian is easily diagonalized by coupling the spins and to and by using .
If site is singly occupied, i.e., , this spin coupling will lead to either a singlet (lowest energy) or a triplet (highest energy). However, if site is empty or doubly occupied , the spin at site 0 and the corresponding Kondo coupling term vanish. This results in an unscreened Kondo spin in a doublet state (), the energy of which is intermediate between the singlet and triplet states.
We now construct a basis of good total spin that also reflects the division into singlet/doulet/triplet manifolds. The eigenstates of are characterized by with being the magnetic quantum number of . The eigenstates of the reduced dot Hamiltonian with electrons are characterized by , where are the spin and spin projection, respectively, of the reduced dot and denotes all other quantum numbers distinguishing between states of the same We then couple the above eigenstates of with the eigenstates of the reduced dot to form states with good total spin and spin projection quantum numbers . This basis of the coupled system is given by . To keep the notation simple, we omitted the quantum numbers and .
Spin selection rules determine the allowed values of the reduced dot spin for a given value of the total spin . We have for the singlet subspace, for the doublet subspace, and for the triplet subspace.
Iii Strong-Coupling Hamiltonian
The strong-coupling limit is defined by . Since , this limit corresponds to , where is the average single-particle level density per site. In the strong-coupling limit, the lowest eigenstates of are dominated by the singlet manifold. The bare singlet Hamiltonian (in the limit when is ignored) is given by the reduced dot Hamiltonian with electrons (except for a constant shift). However, virtual transitions between the singlet and doublet/triplet manifolds add correction terms to this Hamiltonian. Our goal is to determine the resulting effective Hamiltonian for the reduced dot in the strong-coupling limit.
iii.1 Projection technique
In the limit of strong but finite Kondo coupling, the above three manifolds (singlet, doublet and triplet) are coupled to each other. Specifically, the exchange coupling term in couples the singlet and triplet manifolds, while the hopping term between sites 0 and 1 couples each of the singlet and triplet manifolds to the doublet manifold. To account for these couplings we define projection operators on the corresponding singlet/doublet/triplet subspaces () and decompose the wave function accordingly.(37) The Schrödinger equation for the coupled system can then be written as
where each of the two indices assumes any of three values and .
In the strong-coupling limit, our system is described to zeroth order by the singlet Hamiltonian , which contains the bare reduced dot Hamiltonian (for electrons) and (which assumes a constant value), completely decoupled from each other. Higher order corrections come from the coupling terms in which lead to an effective “dressed” Hamiltonian of the reduced dot. This effective Hamiltonian is formally determined by eliminating and in Eqs. (12) and by writing a single equation in the singlet manifold , where is given by
The diagonal components in the above equation are determined by evaluating [Eq. (9)] in each of the three subspaces . The coupling terms in [Eq. (11)] do not contribute to these diagonal components with the exception of which contributes to only. We find
Contributions to off-diagonal components with originate in . The hopping term in changes the spin by and can only couple the doublet to each of the singlet and triplet manifolds, while the exchange term in only couples the singlet and triplet manifolds.
iii.2 Expansion in the strong-coupling limit
The effective Hamiltonian and the construction of a good spin basis in the previous subsection are exact, in that no approximations were made beyond the original Hamiltonian in Eq. (2). However, the form (13) of is not very useful for practical calculations. In the strong-coupling limit . Since the exchange constant is typically below , the condition is automatically satisfied in the strong-coupling limit. We can then expand in the two small dimensionless parameters and . We will do so up to fourth order in these parameters, where the expansion terms are measured in units of (this energy unit is set by the energy of the singlet).
The starting point for this expansion is the unperturbed singlet Hamiltonian , the eigenbasis of which is given by . The corresponding eigenvalues are
where are the eigenvalues of . These unperturbed eigenvalues, , are the limiting solutions to which the eigenvalues of the full Hamiltonian in Eq. (13) converge for . The differences between and at large but finite values of are induced by the virtual transitions from the singlet to the doublet or triplet subspaces. These virtual excitations, in turn, give rise to effective interaction terms in the reduced dot, denoted by . The full effective Hamiltonian in the singlet manifod is then given by .
The effective interaction terms in must be consistent with charge and spin conservation.(2) In particular, must be a scalar operator in spin space (i.e., invariant under rotations in spin space) and invariant under time reversal. This restricts the possible terms that appear in the effective Hamiltonian in the strong-coupling limit.
Scalar one-body terms, i.e., and lead to a renormalization of the one-body part of the reduced dot Hamiltonian . Two-body scalars that can be constructed from the spin at site and the total spin of the reduced dot are , and . The first is the Nozières term known from the conventional Kondo problem(36) (in the absence of exchange, ) but the other two terms are new. The scalar triple product (the imaginary is necessary for time-reversal invariance) does not lead to a new term since , while a fourth order invariant is given by . Other invariants such as , and are allowed but, as we shall see, they cancel out in the effective Hamiltonian.
We rewrite the effective Hamiltonian in (13) as where
In the terms and above we have replaced by (the neglected term gives contributions that are higher than fourth order in the expansion parameters).
We next expand each to fourth order in the parameters and . Since the energy is of the order , the fractions appearing in each term can be brought to a form with being small in the expansion parameters. We then approximate . In the following we summarize the explicit calculation of each term.
For we find
In Eq. (19), we omitted the product terms since their contribution is higher than fourth order, while the contribution of can be shown to vanish identically.
To keep track of the various contributions for each of the , we label them in the following by . These terms are understood to act only in the space of the reduced dot while the Kondo spin and the spin on site 0 are locked into a singlet. The operators in the reduced dot space are obtained by taking a partial expectation value in the singlet state. The corresponding operators in the full space are given, respectively, by . In Appendix A we list several relations that are useful in calculating the expectation values of various operators in the singlet space.
The most dominant contribution to Eq. (19) arises from the unity operator term (in the round brackets). We find
where we have substituted by the hopping Hamiltonian between sites and ,
and used Eq. (A-7). Alternatively, describes the spin transitions illustrated in Fig. 2a, and the result in Eq. (23) can be derived by using Table 1 in Appendix A to sum up all the corresponding transition pathways.
Except for an additional constant shift of , this is a one-body operator that can be incorporated into the unperturbed singlet basis by simply redefining the site energy and hopping amplitude in the unperturbed Hamiltonian as follows(37)
The term involving in Eq. (19) contributes only for a finite exchange interaction (). We have
We note that is a spin invariant in the reduced dot space.
The term involving in Eq. (19) is given by
This term appears in the conventional Kondo problem (where ) and is known as the Nozières term. Nozières found(36) that this term yields an effective interaction in the singlet-space that repels opposite spins on site 1. This term is induced by virtual transitions of the type singlet–doublet–triplet–doublet–singlet. Once we insert a triplet projection in the r.h.s. of Eq. (30), i.e., we write the corresponding singlet expectation value as , we can replace both and by the hopping Hamiltonian between sites and [see Eq. (24)]. Using , we have
The Nozières term (32) vanishes when site is either empty () or doubly occupied () but is negative for , thus favoring a singly occupied site .
We next turn to the singlet-triplet transitions as described by in Eq. (III.2). The corresponding expression for is given by
In Eq. (35) we omitted the terms and , which can be shown to vanish.