Mesoscopic Anderson Box: Connecting Weak to Strong Coupling
We study the Anderson impurity problem in a mesoscopic setting, namely, the “Anderson box” in which the impurity is coupled to finite reservoir having a discrete spectrum and large sample-to-sample mesoscopic fluctuations. Note that both the weakly coupled and strong coupling Anderson impurity problems are characterized by a Fermi-liquid theory with weakly interacting quasiparticles. We study how the statistical fluctuations in these two problems are connected, using random matrix theory and the slave boson mean field approximation (SBMFA). First, for a resonant level model such as results from the SBMFA, we find the joint distribution of energy levels with and without the resonant level present. Second, if only energy levels within the Kondo resonance are considered, the distributions of perturbed levels collapse to universal forms for both orthogonal and unitary ensembles for all values of the coupling. These universal curves are described well by a simple Wigner-surmise-type toy model. Third, we study the fluctuations of the mean field parameters in the SBMFA, finding that they are small. Finally, the change in the intensity of an eigenfunction at an arbitrary point is studied, such as is relevant in conductance measurements. We find that the introduction of the strongly-coupled impurity considerably changes the wave function but that a substantial correlation remains.
The Kondo problem Kondo (1964); Hewson (1993), namely the physics of a magnetic impurity weakly coupled to a sea of otherwise non-interacting electrons, is one of the most thoroughly studied questions of many-body solid state physics. One reason for this ongoing interest is that the Kondo problem is a deceptively simple model system which nevertheless displays very non-trivial behavior and so requires the use of a large variety of theoretical tools to be thoroughly understood, including exact approaches (the numerical renormalization group Wilson (1975); Bulla et al. (2008), Bethe ansatz techniques Andrei (1980); Wiegmann (1980), and bosonization Giamarchi (2004); Schotte and Schotte (1969); Schotte (1970); Blume et al. (1970)) as well as various approximation schemes (perturbative renormalization Anderson (1970); Fowler and Zawadowzki (1971) and mean field theories Lee et al. (2006); Lacroix and Cyrot (1979); Coleman (1983); Read et al. (1984)).
In its original form, the Kondo problem refers to a dilute set of real magnetic impurities (e.g. Fe) in some macroscopic metallic host (say Au). In such circumstances, the density of states of the metallic host can be considered as flat and featureless within the energy scale at which the Kondo physics takes place. Modeling that case with a simple impurity model such as either the - model or the Anderson impurity model Hewson (1993), one finds that a single energy scale, the Kondo temperature , emerges and distinguishes two rather different temperature regimes. For temperatures much larger than , the magnetic impurity behaves as a free moment with an effective coupling which, although renormalized to a larger value than the (bare) microscopic one, remains small. For on the other hand, the magnetic impurity is screened by the electron gas and the system behaves as a Fermi liquid Nozières (1974) characterized by a phase shift and a residual interaction associated with virtual breaking of the Kondo singlet.
That the Kondo effect is in some circumstances relevant to the physics of quantum dots was first theoretically predicted Ng and Lee (1988); Glazman and Raikh (1988) and then considerably later confirmed experimentally Goldhaber-Gordon et al. (1998); Cronenwett et al. (1998); van der Wiel et al. (2000). Indeed, for temperatures much lower than both the mean level spacing and the charging energy, a small quantum dot in the Coulomb blockade regime can be described by the Anderson impurity model, with the dots playing the role of the magnetic impurity and the leads the role of the electron sea. Quantum dots, however, bring the possibility of two novel twists to the traditional Kondo problem. The first follows from the unprecedented control over the shape, parameters, and spatial organization of quantum dots: such control makes it possible to design and study more complex “quantum impurities” such as the two-channel, two impurity, or SU(4) Kondo problems Grobis et al. (2007); Chang and Chen (2009). The second twist, which shall be our main concern here, is that the density of states in the electron sea may have low energy structure and features, in contrast to the flat band typical of the original Kondo effect in metals.
Indeed, the small dot playing the role of the quantum impurity need not be connected to macroscopic leads, but rather may interact instead with a larger dot. The larger dot may itself be large enough to be modeled by a sea of non-interacting electrons (perhaps with a constant charging energy term) but, on the other hand, be small enough to be fully coherent and display finite size effects Kouwenhoven et al. (1997). These finite size effects introduce two additional energy scales into the Kondo problem. The first is simply the existence of a finite mean level spacing, leading to what has been called the “Kondo box” problem by Thimm and coworkers Thimm et al. (1999). The other energy scale introduced by the finite electron sea is the Thouless energy where is the typical time to travel across the “electron-reservoir” dot. When probed with an energy resolution smaller than , both the spectrum and the wave-functions of the electron sea display mesoscopic fluctuations Kouwenhoven et al. (1997), which will affect the Kondo physics and hence lead to what has been called the “mesoscopic Kondo problem” Kaul et al. (2005). Similar studies were also conducted in the context of disordered systems Kettemann (2004); Kettemann and Mucciolo (2006).
Both the Kondo box problem and the high temperature regime of the mesoscopic Kondo problem are by now reasonably well understood. For a finite but constant level spacing in the large dot, various theoretical approaches ranging from the non-crossing approximation Thimm et al. (1999) and slave boson mean field theory Simon and Affleck (2003) to exact quantum Monte Carlo Yoo et al. (2005); Kaul et al. (2006, 2009) and numerical renormalization group methods Cornaglia and Balseiro (2002a, b, 2003) have been used to map out the effect on the spectral function Cornaglia and Balseiro (2002b), persistent current Kang and Shin (2000); Affleck and Simon (2001), conductance Simon and Affleck (2002); Simon et al. (2006); Pereira et al. (2008), and magnetization Cornaglia and Balseiro (2002a); Kaul et al. (2006, 2009). In the same way, a mix of perturbative renormalization group analysis Zaránd and Udvardi (1996); Kaul et al. (2005); Bedrich et al. (2010) and quantum Monte Carlo Kaul et al. (2005) have made it possible to understand the high temperature regime of the mesoscopic problem (see also Refs. Kettemann, 2004; Kettemann and Mucciolo, 2006, 2007; Zhuravlev et al., 2007 for treatment of disordered systems). The picture that emerges is that mesoscopic fluctuations of the density of states translate into mesoscopic fluctuations of the Kondo temperature, but that once this translation has been properly taken into account, the high-temperature physics remains essentially the same as in the flat band case. In particular, physical properties can be written as the same universal function of the ratio as in the bulk flat-band case, as long as is understood as a realization dependent parameter Kaul et al. (2005). In this sense, the Kondo temperature remains a perfectly well defined concept (and quantity) in the mesoscopic regime, as long as it is defined from the high-temperature behavior.
In contrast, the consequences of mesoscopic fluctuations on Kondo physics in the low temperature regime, , remain largely unexplored. A few things are nevertheless known: for instance, using the example of the local susceptibility, exact Monte Carlo calculations have confirmed that below physical quantities do not have the universal character typical of the traditional (flat band) Kondo problem Kaul et al. (2005). This result is not surprising since the mesoscopic fluctuations existing at all scales between the mean level spacing and introduce in some sense a much larger set of parameters in the definition of the problem, leaving no particular reason why all physical quantities should be expressed in terms of . Thus, the low temperature regime of the mesoscopic Kondo problem should display non-trivial but interesting features. On the other hand, it seems reasonably clear that the very low temperature regime should be described by a Nozières-Landau Fermi liquid, as in the original Kondo problem. Indeed, the physical reasoning behind the emergence of Fermi liquid behavior at low temperatures, namely that for energies much lower than the impurity spin has to be completely screened, applies as well in the mesoscopic case as long as .
As a consequence, the mesoscopic Kondo problem provides an interesting example of a system which, as the temperature is lowered, starts as a (nearly) non-interacting electron gas with some mesoscopic fluctuations when , goes through an intrinsically correlated regime for , and then becomes again a non-interacting electron gas (essentially) with a priori different mesoscopic fluctuations as becomes much smaller than . A natural question, then, is to characterize the correlation between the statistical fluctuations of the electron gas corresponding to the two limiting regimes. The goal of this paper is to address this issue (some preliminary results were reported in Ref. Liu et al., 2012). As an exact treatment of the low temperature mesoscopic Kondo problem is not an easy task, we shall tackle this problem here in a simplified framework, namely the one of slave boson/fermion mean field theory, within which a complete understanding can be obtained. We shall furthermore limit our study to the case where the dynamics in the finite “electron sea” reservoir is chaotic, and thus the statistical fluctuations of the high temperature Fermi gas is described by random matrix theory Bohigas (1991).
The structure of this paper is as follows. In Sec. II, we introduce more formally the mesoscopic Kondo model under study and describe the mean field approach on which the analysis is based. Sec. III is devoted to the fluctuations of the mean field parameters. Fluctuations of physical static quantities are analyzed in Sec. IV. We then turn in Sec. V to the study of the spectral fluctuations. For the resonant level model arising from the mean field treatment, we give in particular a derivation of the spectral joint distribution function, as well as a simplified analysis, in the spirit of the Wigner surmise Bohigas (1991), of some correlation functions involving the levels of the low and high temperature regimes. Wave function correlations are then considered in Sec. VI. Finally, Sec. VII contains some discussion and conclusions.
ii.1 Mesoscopic bath + Anderson impurity
We investigate the low temperature properties of a mesoscopic bath of electrons (e.g., a big quantum dot), coupled to a magnetic impurity (e.g. a small quantum dot or a magnetic ion). The Hamiltonian of the system is
where describes the mesoscopic electronic bath and describes the interaction between the bath and the local magnetic impurity. Here, in a particular realization of this general model, the mesoscopic bath is described by the non-interacting (i.e. quadratic) Hamiltonian
where indexes the level, is the spin component, and is the chemical potential. We assume that, in , the local Coulomb interaction between -electrons is such that , so states with two -electrons on the impurity must be projected out. With this constraint implemented, the local impurity term is taken as
where the annihilation and creation operators and act on the states of the impurity (small dot). The state in the reservoir to which the -electrons couple is labeled with the corresponding operator related to the bath eigenstate operators through
where denotes the one-body wave functions of the . The local normalization relation implies that the average intensity is , where denotes the configuration average. Finally, the width of the -state, , because of coupling to the reservoir is given in terms of the mean density of states, , by
where is the bandwidth of the electron bath.
and the local density of states, , should be assumed small: , or equivalently . Indeed, this condition implies that the strength of the second order processes involving an empty-impurity virtual-state is much smaller than the mean level spacing . Furthermore, as we discuss in more detail in Section III, the Kondo regime is characterized by , for which the fluctuations of the number of particles on the impurity is weak. If increases to the point that , one enters the mixed valence regime where these fluctuations become important.
ii.2 Random matrix model
To study the mesoscopic fluctuations of our impurity model, we assume chaotic motion in the reservoir in the classical limit. Random matrix theory (RMT) provides a good model of the quantum energy levels and wave functions in this situation Bohigas (1991); Kouwenhoven et al. (1997): we use the Gaussian orthogonal ensemble (GOE, ) for time reversal symmetric systems and the Gaussian unitary ensemble (GUE, ) for non-symmetric systems Mehta (1991); Bohigas (1991). The joint distribution function of the unperturbed reservoir-dot energy levels is therefore given by Mehta (1991)
(with where is the mean level spacing in the center of the semicircle). The corresponding distribution of values of the wave function at , the site in the reservoir to which the impurity is connected, is the Porter-Thomas distribution,
Furthermore, in the GOE and GUE, the eigenvalues and eigenvectors are uncorrelated.
For the GOE and GUE, the mean density of states follows a semicircular law—a result that is rather unphysical. Except when explicitly specified, we assume either that we consider only the center of that the semicircle or some rectification procedure has been applied, so we effectively work with a flat mean density of states.
ii.3 Slave boson mean-field approximation
Following the standard procedure Coleman (1983); Read et al. (1984); Lee et al. (2006); von Lohneysen et al. (2007); Fulde et al. (2006); Burdin (2009), we introduce auxiliary boson and fermion annihilation (creation) operators, such that , with the constraint
The impurity interaction (3) is rewritten as
The mapping between physical states and auxiliary states of the impurity is
This auxiliary operator representation is exact in the limit as long as the constraint (9) is satisfied and the bosonic term in is treated exactly.
Note that we use here a slave boson formalism with U(1) gauge symmetry. Generalized slave boson fields have been introduced in order to preserve the SU(2) symmetry of the model, as discussed in Refs. Lee et al., 2006 and Affleck et al., 1988. Such a generalized SU(2) slave boson approach would not change crucially the physics of the single impurity mean-field solution, but it may become relevent for models with more than one impurity.
The mean-field treatment of the Anderson box Hamiltonian invokes two complementary approximations: (i) The bosonic operator is considered a complex field, with an amplitude and a phase . Since the Hamiltonian is invariant with respect to the gauge transformation and , the phase is not a physical observable, and we choose :
where is a positive real number. This approximation corresponds to assuming that the bosonic field condenses. (ii) The constraint (9) is satisfied on average, by introducing a static Lagrange multiplier, . The Hamiltonian of Eq. (1) treated within the slave boson mean-field approximation thus reads
The mean-field parameters and must be chosen to minimize the free energy of the system, , yielding the saddle point relations
where the thermal averages have to be computed self-consistently from the mean-field Hamiltonian. 111The chemical potential can be considered as an external tunable parameter, or determined self-consistently if one considers a given electronic occupancy . In this latter case, Eqs. (13)-(14) have to be completed by a third relation: .
ii.4 Method for solving the mean-field equations
In this section, we explain how to solve the self-consistent equations for the effective parameters, and . We start by introducing the imaginary-time equilibrium Green functions
Using the equations of motion from the mean-field Hamiltonian Eq. (12) and after straightforward algebra, we find
Self-consistency therefore can be achieved by iterating successively Eqs. (19)-(II.4), which define the Green functions in terms of the parameters and , and Eqs. (23)-(24), which fix and from the Green functions.
As an example of the output from this procedure, we show in Figs. 1 and 2, as a function of the strength of the coupling , the one-body energy levels that result from a slave-boson mean field theory (SBMFT) treatment of the Anderson box for a particular realization of the box. As we discuss in more detail below (see section IV.2), a non-trivial solution of the SBMFT equations exists only for above some critical value , or equivalently [see Eq. (6)] for larger than a threshold . We thus show the non-interacting levels below that value and break the axis at that point [ (GOE) and (GUE) for the realizations chosen]. Clearly, the levels do indeed shift substantially as a function of coupling strength; notice as well the additional level injected near the Fermi energy. The change in the levels occurs more sharply and for slightly smaller values of in the GOE case than for the GUE. Finally, we observe that, as one follows a level as a function of , little change occurs after some point. The coupling strength at which levels reach their limiting value depends on the distance to the Fermi energy; it corresponds to the point where the Abrikosov-Suhl resonance becomes large enough to include the considered level. These limiting values of the energies are the SBMFT approximation to the single quasi-particle levels of the Nozières Fermi liquid theory.
ii.5 Qualitative behavior
Before entering into the detailed quantitative analysis, we describe here some simple general properties of the mesoscopic Kondo problem within the SBMFT perspective.
We note first that the mean-field equations (23)-(24), have a trivial solution and . This solution is actually the only one in the high temperature regime: the mesoscopic bath is effectively decoupled from the local magnetic impurity which can be considered a free spin-. The onset of a solution defines, in the mean field approach, the Kondo temperature .
Below , the self-consistent mean-field approach results in an effective one-particle problem, specifically a resonant level model with resonant energy and effective coupling . This resonance is interpreted as the Abrikosov-Suhl resonance characterizing the one-particle local energy spectrum of the Kondo problem below . The width of this resonance, , vanishes for , and quickly reaches a value of order when (more detailed analysis is in Sec. III.2). Note that the mesoscopic Kondo problem differs from the bulk case: mesoscopic fluctuations may affect the large but finite number of energy levels that lie within the resonance.
The Anderson box is, however, a many-body problem. Its ground state cannot be described too naively in terms of one-body electronic wave functions, and more generally one should question the validity of the one particle description for each physical quantity under investigation. In this respect however, the configuration we consider, namely the low temperature regime of the Kondo box problem, is particularly favorable. Indeed, the line of argument developed by Nozières Nozières (1974) to show that the low temperature regime of the Kondo problem is a Fermi-liquid applies equally well in the mesoscopic case as in the bulk one for which it was originally devised. Therefore, as long as both the temperature and the mean level spacing are much smaller than the Kondo temperature, we a priori expect the physics of the Kondo box to be described in terms of fermionic quasi-particles. The notions of one particle energies and wavefunction fluctuations in the strong interaction regime, which will be our main concern below, are therefore relevant. We take the point of view that, as in the bulk case Lee et al. (2006); Burdin (2009); Coleman (1983); Read et al. (1984), the mean field approach provides a good approximation for these quasi-particles in this low temperature regime, and therefore for the physical quantities derived from them Ullmo et al. (). As we shall see furthermore, most of the fluctuation properties we shall investigate have universal features that makes them largely independent from possible corrections to this approximation (such as, for instance, corrections on the Kondo temperature), making the approach we are following particularly robust.
Iii Fluctuations of the mean field parameters
To begin our investigations of the low temperature properties of the mesoscopic Kondo problem within SBMFT, we consider the fluctuations of the mean field parameters and appearing in Eq. (12). We shall comment also on the degree to which these fluctuations are connected with those of the Kondo temperature Kettemann (2004); Kaul et al. (2005); Bedrich et al. (2010).
iii.1 Preliminary analysis
We start with a few basic comments about the eigenvalues and eigenstates () of the mean field Hamiltonian Eq. (12). Concerning the latter, we shall be interested in the two quantities,
measures the overlap probability between the eigenstate and the impurity state , and the admixture of this eigensate with and , the electron-bath state connected to the impurity. Note that is a real quantity. In this section we use to denote the additional resonant level added to the original system, and so in the limit , one has and ().
Expressing the Green function of the mean field Hamiltonian as
we can check that are the poles of the Green function . From Eq. (19) we have therefore immediately that the are the solutions of the equations
where we have used the notation
for the center and the width of the resonance, and for the normalized wavefunction probability at . Note first that Eq. (28) implies that there is one and only one in each interval : the two sets of eigenvalues are interleaved and so certainly heavily correlated. Furthermore, are the corresponding residues, so again, from Eq. (19),
Eq. (28) is easily solved outside of the resonance, i.e. when : in that case one contribution dominates the sum on the left hand side. [With our convention where corresponds to the extra level added to the original system, we actually just have .] The solution for the fractional shift in the level, is then given by
If the resonance is small , all states are accounted for in this way, except for which is then such that .
If the resonance is large, , the states within the resonance – those satisfying – must be treated differently. Because the left hand side of Eq. (28) can be neglected in this regime, these states have only a weak dependence on . The typical distance between a and the closest is then of order , and the corresponding wave functions participate approximately equally in the Kondo state,
iii.2 Formation of the resonance
Before considering the fluctuations of the mean field parameters and , let us first discuss the physical mechanisms that determine their value. While this discussion is not specific to the mesoscopic Kondo problem, it is useful to review it briefly before addressing the mesoscopic aspects.
where is the Fermi occupation number. One furthermore has the sum rules and [the latter has been used to generate the 1/2 in (37)].
As mentioned in Sec. II.5, the trivial solution of these mean-field equations () is the only one in the high temperature regime. The Kondo temperature is defined, in the mean field approach, as the highest temperature for which a solution occurs. One obtains an equation for by requiring that the non-trivial solution of the mean-field equations continuously vanishes, , in which case , , and . Eq. (38) then reduces to , implying and so . Using Eq. (35) to simplify Eq. (37) then gives the mesoscopic version Bedrich et al. (2010) of the Nagaoka-Suhl equation Nagaoka (1965); Suhl (1965)
In the bulk limit ( and no fluctuations) and for in the middle of the band, this gives for the Kondo temperature, with as shown in Appendix A. Unless explicitly specified, we will always assume this quantity is large compared to the mean level spacing. In this case, the fluctuations of the Kondo temperature for chaotic dynamics described by the random matrix model in Sec. II.2 has been analyzed in Refs. Kaul et al., 2005-Kettemann, 2004 and more recently using SBMFT in Ref. Bedrich et al., 2010. The main result is that , the fluctuation of the Kondo temperature around the bulk Kondo temperature, scales as
Now consider what happens as decreases further below . Dividing Eq. (37) by , we can write it as
where is one outside the resonance and scales as within the resonance [see Eqs. (35)-(36)]. Eq. (41) has a structure very similar to the equation for , Eq. (39). Indeed, might not be strictly equal to [and thus might differ slightly from ] but its scale will remain the same. Then outside the resonance, and . The main difference in the expression for is that the logarithmic divergence associated with the summation of is cutoff not only by the temperature factor at the scale , but also by the ratio at the scale . As becomes significantly smaller than , the temperature cutoff becomes inoperative. This implies in particular that will rather quickly switch from 0 to its zero temperature limit when goes below . We shall in the following not consider the temperature dependence of but rather focus on its low temperature limit.
We see, then, that both and represent physically the scale at which the logarithmic divergence of should be cut to keep this sum equal to . Thus, as long as we are only interested in energy scales, we can write that for ,
The energy dependence of the cutoff within the resonance, however, differs slightly from that of below . As an exponentiation is involved, the prefactors of and somewhat differ; a discussion of the ratio for the bulk case is given in Appendix A.
At low temperature, is fixed in such a way that is of the scale of the Kondo temperature. The condition Eq. (38) then fixes , which governs the center of the resonance so a proportion of the resonance is below the Fermi energy . In the Kondo regime when , will therefore remain near . In the mixed valence regime will float a bit above for a distance which scales as . The order of magnitude of remains thus [as we have assumed above when discussing Eq. (41)].
iii.3 Fluctuations scale of the mean field parameters
With this physical picture of how the mean field parameters and are fixed, it is now relatively straightforward to evaluate the scale of their fluctuations. For simplicity, we assume so the mean-field equations become
The discussion below generalizes easily to finite as long as it is much smaller than .
The average values of and are well approximated by their “bulk-value” analogues and , obtained with the same global parameters but with the fluctuating wave-function probabilities replaced by and the spacing between successive levels taken constant, . We furthermore denote by the solution of Eqs. (43)-(44) with and replaced by their bulk approximation, by and the fluctuating part of the mean field parameters, and by and the fluctuating parts of the sums appearing in Eqs. (43)-(44).
We start by discussing the Kondo limit , in which case , , and . A calculation in Appendix A shows
Furthermore, as we shall be able to verify below, the leading contribution to the fluctuations of and can be taken independently of each other (i.e. the fluctuations of can be computed assuming constant, and reciprocally).
Subtracting its bulk value from Eq. (44), we have , and thus, by definition of ,
If the fluctuations of and are small, we can furthermore approximate by . We thus have
The two last terms on the right-hand-side of Eq. (47) are proportional to [e.g. see Eq. (100) for the second-to-last term] and so are negligible in the Kondo regime. Computing the variance therefore amounts, up to the constant factor , to computing the variance of .
Now, for , we have , where
is a dimensionless quantity which for is essentially independent of , , or the other parameters of the model. Within the resonance, and for our random matrix model, we therefore can take the to have identical distributions (independent of ) characterized by a variance of order one. Neglecting the correlations between the , and treating the at the edge of the resonance as if they were well within it (which is obviously incorrect but should just affect prefactors that we are in any case not computing), we have
Inserting this into Eq. (47), we finally get
With regard to the limits of validity of this estimate, note that our random matrix model (Sec. II.2) assumes implicitly that the Thouless energy is infinite, and more specifically that . For a chaotic ballistic system with , the are independent only in an interval of size ; thus, Eq. (50) should be replaced by .
For the fluctuations of , we proceed in a similar way, subtracting Eq. (43) from its bulk analog and assuming small fluctuations, and so find
Here, however, it is necessary to split the sum over states in Eq. (43) into two parts: where and are defined in the same way as but over an energy range corresponding, respectively, to the inside and outside of the resonance. One has since the former contains the logarithmic divergence. However, the fluctuations of the two quantities are of the same order [basically because when considering the variance, and thus squared quantities, one transforms a diverging sum into a converging one ]. Indeed, the sum is, up to sub-leading corrections, the same as the one entering into the definition of . Its fluctuations have been evaluated in Refs. Kaul et al., 2005-Kettemann, 2004, leading to
which is consistent with Eq. (40). The variance of can, on the other hand, be evaluated following the same route as for , yielding
This shows, then, that the two contributions and scale in the same way.
which is proportional to as for the first two contributions, but the extra smallness factor makes it negligible in the Kondo limit. Gathering everything together, we therefore obtain
[If , and are reduced by a factor , but not ; thus, Eq. (55) remains unchanged.]
Turning to the mixed-valence regime by releasing the constraint , we see that becomes comparable in size to the other contributions to and has the same parametric dependence. Furthermore, taking the derivative [see Eq. (99)] implies that the left hand side of (47) should be multiplied by a factor , which, however, does not change the scaling of . In the same way, using Eq. (55) the two last terms on the right hand side of Eq. (47), which are proportional to , give a contribution to , as well as the term that should be added to the the left hand side of Eq. (51) from [see Eq. (98)]. Those are negligible in the Kondo regime, but are of the same size and with the same scaling as the contribution due to in the mixed-valence regime. We find, then, that the fluctuations of the mean field parameters scale with system size in the same way in both the Kondo and mixed-valence regimes: the variance of both and is proportional to .
iii.4 Numerical investigations
To illustrate the previous discussion, we have computed numerically the self-consistent parameters and for a large number of realizations of our random matrix ensemble at various values of the parameters defining the Anderson box model (always within our regime of interest, , except when explicitly specified). Fig. 3 shows the distributions of and for a choice of parameters such that (close to but not in the mixed valence regime). We see that these distributions are approximately Gaussian and centered on their values for the bulk flat-band case, though note the slightly non-Gaussian tail on the left side in both cases. The distributions for the GOE and GUE are qualitatively similar, with those for the GUE being, as expected, slightly narrower. As anticipated, the fluctuation of these mean parameters is small: the root-mean-square variation is less than 5% of the mean. Fig. 4 further shows how the variance of and varies with the parameters of the model, confirming the behavior in Eqs. (50) and (55).
Iv Other global physical properties
Beyond and themselves, several interesting global properties of the system follow directly from the solution of the mean-field problem. We briefly discuss two of them here.
iv.1 Wilson number: Comparing and the ground state properties
The “Wilson number” is an important quantity in Kondo physics: it compares with the energy scale contained in the ground state magnetic susceptibility. It is defined as , where is the static susceptibility. is thus the ratio between the characteristic high temperature scale and the characteristic low temperature scale of the strong-coupling regime Burdin (2009).
In the bulk Kondo problem, there is only one scale, of course, and so the Wilson number has a fixed value Hewson (1993), namely (approximated as in the SBMFT). For our mesoscopic Anderson box on the other hand, this will be a fluctuating quantity that has to be computed for each realization of the mesoscopic electron bath. Computing according to Eq. (39) and expressing the static susceptibility as with given by Eq. (19), we obtain the distribution of the Wilson number shown in Fig. 3(d). Note the unusual non-Gaussian form of the distribution, with the long tail for large . As a result, the peak of the distribution is slightly smaller than the bulk flat-band value. The magnitude of the fluctuations in is modest for our choice of parameters (about 30%) but considerably larger than the magnitude of the fluctuations of the mean field parameters in Fig. 3(a)-(b).
iv.2 Critical Kondo coupling
Another interesting global quantity is the critical Kondo coupling defined for a given realization of the electron bath by
Here, exceptionally, we move away from the regime . The discreetness of the spectrum is what is making convergent the sum in the above expression, and thus can be defined only because of the finite size of the electron bath.