Charge-Kondo Effect in Mesoscopic Superconductors Coupled to Normal Metals
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J u l Charge-Kondo Effect in Mesoscopic Superconductors Coupled to Normal Metals
Ion Garate , Department of Physics and Astronomy, The University of British Columbia, Vancouver, BC V6T 1Z1, Canada and Canadian Institute for Advanced Research, Toronto, ON M5G 1Z8, Canada. (Dated: September 7, 2018)We develop a theoretical proposal for the charge-Kondo effect in mesoscopic normal-superconductor-normal heterostructures, where the superconducting gap exceeds the electrostaticcharging energy. Charge-Kondo correlations in these devices alter the conventional temperature-dependence of Andreev reflection and electron cotunneling. We predict typical Kondo temperaturesof & PACS numbers:
I. INTRODUCTION
Five decades after its discovery, the Kondo effectis regarded as an archetype many-body phenomenonthat interconnects strongly correlated systems such asatomic nuclei, heavy fermion compounds, superconduct-ing cuprates and quantum dot devices. The Kondo ef-fect emerges from the interaction of localized, degeneratestates with itinerant degrees of freedom. It is character-ized by a low-temperature infrared divergence in pertur-bative calculations of physical observables such as resis-tivity and magnetic susceptibility. Often the localizeddegrees of freedom originate from electronic spins andtheir degeneracy is tied to the usual spin degeneracy.The ensuing spin-Kondo effect (SKE) is realized when di-lute magnetic impurities are embedded in a non-magnetichost, or when conducting electrodes are coupled to quan-tum dots. Aside from spin, any degenerate two-level system thatis coupled to a fermionic bath is a potential host forthe Kondo effect. In particular the charge counter-part of SKE, known as charge-Kondo effect (CKE),arises when a Fermi sea hybridizes with dilute impuritiescontaining two degenerate charge states. This varianthas received sustained theoretical attention for the lasttwo decades, being investigated in non-superconductingsingle-electron devices as well as in valence-skippingelements with attractive onsite interactions (“negative-U molecules”) .
In the former case the electrostaticenergy can be engineered by gate electrodes to be thesame for two states that differ by one electron, whereasin the latter case chemistry dictates that the chargingenergy be degenerate between states that differ by twoelectrons. Incidentally, CKE in single-electron devicesconstitutes a paradigm of the two-channel Kondo effect,where the channels originate from spin.As opposed to SKE, which has been thoroughlyobserved both in its single-channel and two-channelversions, the experimental detection of CKE remainsat a primitive stage. Experimental challenges aboundfor the implementation of CKE in non-superconductingdevices. On one hand, CKE is sensitive to and washedout by background charge noise, much like SKE is sen- sitive to and washed out by magnetic fields. On theother hand, both the charge-Kondo temperature andthe charging energy of the system are required to belarge compared to the single-particle energy-level spac-ing in the grain, which is difficult to achieve in semi-conductors. Prospects may be better for metallic sys-tems, wherein single-particle energy levels form a near-continuum. Yet metallic devices often involve junc-tions with multiple channels, and unfortunately thecharge-Kondo temperature in non-superconducting sys-tems scales exponentially unfavourably with the num-ber of channels. Some of these problems might be cir-cumvented by using atomic point contacts or else byresorting to resonant tunneling devices. Altogether,attempts to measure fingerprints of the CKE in non-superconducting systems have thus far met with sugges-tive yet inconclusive outcomes.
Recent experimental efforts concerning negative-Umolecules appear to have met with more success: charge-Kondo temperatures of ∼
5K have been reported in PbTedoped with valence-skipping Tl. Unfortunately, real ma-terials with valence-skipping compounds are relativelyrare, more complicated and less tunable than artificialsingle-electron devices.In this work we theoretically demonstrate that CKEcan also occur in artificially fabricated mesoscopicsuperconducting islands that are connected to non-superconducting leads (NSN devices). We concentrateon superconducting grains whose many-particle energy-gap (∆) exceeds the electrostatic charging energy ( E c ).These grains behave somewhat like giant negative-Umolecules, and they are to valence-skipping elementswhat quantum dots are to local magnetic moments: moreeasily manufacturable and more readily controllable sys-tems, albeit at the price of a parametrically lower Kondotemperature.Our proposal is a natural and perhaps obvious exten-sion of Refs. [3,4] and [13], as it combines attractive(phonon mediated) and repulsive (electrostatic) Coulombinteractions with a dense single-particle energy spectrum.Surprisingly, there have been no thorough studies of CKEin superconductors with ∆ > E c . The relevant literatureis limited to some peripheral statements, along witha tacit assumption that CKE in superconducting dotsis conceptually similar to CKE in non-superconductingsingle-electron devices (NNN devices). Contrary to thisview, our results aim to draw the attention of theoristsand experimentalists alike towards the study and searchof charge-Kondo correlations in NSN systems.The rest of this paper is organized as follows. In Sec-tion II we introduce the basic microscopic model fora NSN heterostructure. The island is assumed to bechaotic and smaller than the superconducting coherencelength, thereby allowing for coherence between electronstunneling through different junctions. In Section III wemap the superconducting island onto an artificial spin1/2 describing two charge states. This mapping is well-known and has already been exploited in existing Cooperpair boxes. In Section IV we attach normal metallicleads to the superconducting island, and find that at lowenough energies and near the charge degeneracy pointthis system is described by an anisotropic Kondo model.The degree of anisotropy and the magnitude of the Kondocouplings can be changed by tuning the ratio ∆ /E c .In Section V we provide quantitative estimates for thecharge-Kondo temperature in experimentally realizableNSN devices. In addition, we present a detailed discus-sion on how our proposal for CKE fares in comparisonwith previous proposals concerning negative-U moleculesand non-superconducting devices. In Section VI we cal-culate the fingerprints of CKE in physical observables ofNSN systems. Specifically, we focus on the temperature-dependence of the zero-bias conductance at low tempera-tures. In Section VII we determine the fate of CKE whenthe normal-superconducting junctions are highly trans-parent; the outcome depends on the strength of electron-electron interactions in the normal metallic leads. Sec-tion VIII is devoted to a short summary and conclusions,and the Appendix contains technical calculations thatrigorously justify the considerations of Section IV. II. MICROSCOPIC MODEL
Let us suppose we have a mesoscopic superconductinggrain that is weakly coupled to normal metallic leads. ItsHamiltonian can be expressed as H = H l + H d + H T H l = X αkσ ξ k c † αkσ c αkσ H d = X nσ ǫ n d † nσ d nσ + E c ( ˆ N − N g ) + η ˆ T † ˆ T H T = X αkσn t αkn c † αkσ d nσ + h . c .. (1) H l is the Hamiltonian of the leads. c † αkσ creates an elec-tron with momentum k and spin σ , in a lead labeled by α ( α = L(eft) , R(ight)). ξ k is the energy dispersion of theitinerant fermions measured from the Fermi energy. For now we assume that there is only one conduction chan-nel on each lead; as such, k is the momentum perpen-dicular to the interface between the normal-metal andthe superconductor (NS interface). Generalizations tomulti-channel leads, which are necessary when the lineardimensions of the NS interface exceed the Fermi wave-length, will be discussed in Section V. Likewise we ne-glect electron-electron interactions in the leads, althoughthis assumption will be relaxed in Section VIII. H d is the Hamiltonian of the superconducting quan-tum dot. d † nσ creates an electron in the n -th single-particle level of the island. The corresponding energy(measured with respect to the Fermi energy) can be writ-ten as ǫ n = nδ , where δ is the average single-particle levelspacing in the dot. Hence, in this model every energy-level is non-degenerate, save for the twofold spin degen-eracy. ˆ N = P nσ d † nσ d nσ and ˆ T = P n d n ↓ d n ↑ are thenumber and pairing operators in the island, respectively. η is the effective (phonon mediated) coupling betweenelectrons in the island. Hereafter we adopt the BCSapproximation by introducing an s-wave superconduct-ing gap ∆ ≡ η h ˆ T i . The BCS approximation is accurate provided that ∆ > δ , which constitutes the regime of in-terest in this paper. For reasons exposed below, we take∆ > E c , where E c = e / C Σ is the charging energy of theisland. C Σ = C L + C R + C g is the total capacitance de-vice, C L ( R ) is the capacitance of the left (right) NS junc-tion and C g is the gate capacitance. In mesoscopic NSNheterostructures most of the capacitance emanates fromthe junctions and scales roughly as C Σ ∼ ǫA/d , where ǫ is the dielectric constant of a thin insulating layer thatseparates the normal metal from the superconductor, A is the surface-area of each NS junction and d ≃ N g is the chargeinduced in the island by the gate electrode. We neglectintra-dot exchange interactions, which is adequate giventhe spin-singlet order parameter of the superconductor. H T models the single-particle tunneling between theisland and the leads, t αkn being the tunneling amplitude.Throughout this paper (with the exception of SectionVIII) we treat the tunneling term as a weak perturbation.Hence we require that δ >> Γ, where Γ is the broadeningof the energy-levels of the island due to tunneling betweenthe island and the leads.Our model relies on precepts from random matrix the-ory (RMT) and is valid only for dots without spatialsymmetries and for states in the vicinity of the Fermienergy.
According to RMT, single-particle states with | ǫ n | . E T are non-degenerate except for their spin degen-eracy, where E T is the Thouless energy. E T = ~ D/L for a dirty sample (where D is the diffusion constantand L is the linear dimension of the superconductor) and E T = ~ v F /L for a clean sample (where v F is the Fermivelocity). Anticipating the fact that charge-Kondo cor-relations originate mainly from states with | ǫ n | . ∆, werequest E T & ∆ so that Eq. (1) can capture CKE quan-titatively. This condition is tantamount to L . ζ , where ζ is the superconducting coherence length. In sum, thehierarchy of relevant energy scales reads E T & ∆ > E c > δ > Γ . (2) L . ζ implies electron coherence between the two junc-tions, which in turn gives rise to elastic electron cotun-neling and crossed Andreev reflection (Section VI). Un-mentioned as it is in Eq. (2), we note that the Debyefrequency ω D (such that ∆ = 0 iff | ǫ n | . ω D ) is muchlarger than E T for all cases discussed in this paper. III. ISOLATED SUPERCONDUCTING ISLAND:A PSEUDOSPIN 1/2 QUANTUM IMPURITY
In this section we ignore the leads and simply con-sider an electrostatically gated superconducting island,which contains a definite number of particles N because[ ˆ N , ˆ N ] = [ ˆ T † ˆ T , ˆ N ] = 0. Its ground state depends on N as E ( N ) = E c ( N − N − N g ) + ∆2 (1 − ( − N ) , (3)where N is the number of electrons in the island whenthe gate voltage is zero. We assume without loss of gen-erality that N is even so that N − N and N have thesame parity; herein we absorb N in the definition of N .The last term of Eq. (3) imposes an additional energycost whenever N is odd, because of a single unpairedquasiparticle that must reside outside the superconduct-ing condensate. The lowest energy this quasiparticle canhave equals ∆. If ∆ > E c and if the gate voltage is tunedto N g = 2 M + 1 (for any integer M ), the lowest energyeigenvalues are (ordered from lower to higher energy) E (2 M ) = E (2 M + 2) = E c E (2 M + 1) = ∆ E (2 M −
1) = E (2 M + 3) = 4 E c + ∆ E (2 M + 4) = E (2 M −
2) = 9 E c ..., (4)where the ordering of the states changes if E c < ∆ / E (2 M + 1) < ( E (2 M − , E (2 M + 3)) willbe important for establishing CKE in Section IV.Since the ground state of the island is doubly degen-erate, it behaves as a spin 1/2 system at low enoughtemperatures ( T < T ∗ ). For later purposes we introducea pseudospin label: | M i ≡ | ⇓ i ; | M + 2 i ≡ | ⇑ i , (5)where the subscript 0 simply labels the location of theisland. From now on pseudospin and real spin degreesof freedom will be denoted by double and single arrows,respectively. This qubit is well-separated from a denseforest of many-body states, which have excitation ener-gies greater than ∆ − E c . Due to entropic issues that arise from having an approximately manifold degenerateset of excited states, T ∗ ≃ ∆ − E c ln (cid:0) ∆ − E c δ (cid:1) (6)is smaller than the excitation energy gap ∆ − E c .The mapping to a two-level system holds even when M is macroscopically large, and has been amply corrobo-rated by experiments. The degeneracy between | M i and | M + 2 i may be lifted by tuning the gate voltageaway from the degeneracy point, which is akin to apply-ing a pseudospin magnetic field along ˆ z , or by couplingthe island to a superconducting reservoir, which acts asa pseudo-magnetic field in the xy plane. The charge-Kondo effect discussed below emerges when these pseu-dospin magnetic fields are very weak.For completeness, we note that there is a twofoldcharge-degeneracy point even when ∆ < E c . In this casethe degeneracy occurs between states that differ by a sin-gle electron, and therefore this problem is adiabaticallyconnected to that of Ref. [3]. IV. COUPLING TO THE LEADS: KONDOMODEL
In this section we consider the influence of conductingleads on the superconducting island with charge degener-acy, at temperature
T < T ∗ . We assume all non-thermalperturbations to be small compared to ∆ − E c Then theNSN heterostructure behaves as a localized pseudospin1/2 interacting with a bath of itinerant fermions, andEq. (1) can be truncated on symmetry grounds into aneffective Hamiltonian H eff = H l + X αα ′ λ αα ′ S ˜ s αα ′ + X αα ′ h λ αα ′ || S z ˜ s zαα ′ + λ αα ′ ⊥ (cid:0) S + ˜ s − αα ′ + h . c . (cid:1)i , (7)where 2 S = | ⇑ ih⇑ | + | ⇓ ih⇓ | , 2 S z = | ⇑ ih⇑ |−| ⇓ ih⇓ | and S + = | ⇑ ih⇓ | are the pseudospin operatorscharacterizing the state of the superconducting island,and ˜ s iαα ′ = X kk ′ σσ ′ ˜ c † αkσ τ iσσ ′ ˜ c α ′ k ′ σ ′ (8)denotes the pseudospin density in the leads ( i ∈{ , x, y, z } ). In addition, τ is a vector of Pauli matri-ces ( τ is the identity matrix) and ˜ c αkσ is a pseudospinoperator defined as˜ c αk ↑ = c αk ↑ ; ˜ c αk ↓ = c † α, − k ↓ . (9)Physically, ˜ s is the spin density along ˆ z , ˜ s z is the chargedensity and ˜ s ± is the pairing operator. In Eq. (7), theterm involving S ( S z ) describes normal spin-dependent(spin-independent) scattering and the term ∝ S ± de-scribes Andreev processes whereby the number of Cooperpairs in the island changes by one; the xy symmetryis simply a reflection of gauge invariance. Ignoringreal-spin-flips (which are unimportant at low energies),Eq. (7) is the most general form of exchange interac-tion between the localized and itinerant pseudospins. Wehave omitted perturbations that break charge degener-acy; these can be easily incorporated in Eq. (7) as pseu-dospin Zeeman fields. λ αα ′ || , ⊥ are the Kondo couplings. Their microscopic ex-pressions can be extracted perturbatively from Eq. (1)via h f |H eff | i i ≃ X n h f |H T | n ih n |H T | i i E i − E n , (10)where | i i ( | f i ) is the initial (final) state with energy E i ( E f ) and | n i is an intermediate state with energy E n (seeTable I). It follows that λ αα ′ = X n h⇑ αk ; ⇑ |H T | n ih n |H T | ⇑ α ′ k ′ ; ⇑ i E ⇑ αk ; ⇑ − E n + X n h⇑ αk ; ⇓ |H T | n ih n |H T | ⇑ α ′ k ′ ; ⇓ i E ⇑ αk ; ⇓ − E n λ αα ′ || = X n h⇑ αk ; ⇑ |H T | n ih n |H T | ⇑ α ′ k ′ ; ⇑ i E ⇑ αk ; ⇑ − E n − X n h⇑ αk ; ⇓ |H T | n ih n |H T | ⇑ α ′ k ′ ; ⇓ i E ⇑ αk ; ⇓ − E n λ αα ′ ⊥ = X n h⇑ αk ; ⇓ |H T | n ih n |H T | ⇓ α ′ k ′ ; ⇑ i E ⇑ αk ; ⇓ − E n , (11)where | ⇑ αk i ≡ c † αk ↑ | ø i and | ⇓ αk i ≡ c αk ↓ | ø i representspin-up electrons (spin-down holes) at the Fermi surface( | ø i stands for the filled Fermi sea in the leads). Table Icollects all the intermediate states | n i , along with theirenergies E n .We proceed with the evaluation of the tunneling matrixelements using the BCS formalism. For instance, thematrix elements for an electron or a hole tunneling fromlead α to the n -th level in the island are h⇑ αk ; ⇑ |H T γ † n ↑ | ⇑ i ≃ h⇑ αk ; ⇓ |H T γ † n ↑ | ⇓ i ≃ t αkn u n h⇓ αk ; ⇑ |H T γ † n ↑ | ⇑ i ≃ h⇓ αk ; ⇓ |H T γ † n ↑ | ⇓ i ≃ − t ∗ αkn v n , where γ † nσ = u n d † nσ + σv n d n, − σ is the BCS quasiparti-cle creation operator, u n = (1 + ǫ n / p ǫ n + ∆ ) / v n = 1 − u n are the standard BCS coherence factors (weassume u n and v n to be real without loss of generality be-cause we consider only one superconductor). Other ma-trix elements may be computed similarly. Even thoughintermediate states like γ † nσ | M + 2 i do not have a well-defined number of particles, they are excellent approxi-mations to the true eigenstates of the superconductingisland when ∆ >> δ . The explicit expressions for the couplings can now beread out from Table I: λ αα ′ || = X n t ∗ αnk t α ′ k ′ n − E c − p ǫ n + ∆ − t ∗ αnk t α ′ k ′ n E c − p ǫ n + ∆ ! λ αα ′ ⊥ = − X n t ∗ αnk t ∗ α ′ k ′ n u n v n E c − p ǫ n + ∆ . (12)It is easy to verify that λ = 0 by invoking particle-holesymmetry for the energy spectrum of the superconduct-ing island. Since λ || is associated with normal scattering,it involves virtual transitions that do not flip pseudospins.The sign of λ || will be important in the considerationsbelow. λ ⊥ is associated to Andreev scattering, hence itinvolves virtual transitions that flip the pseudospins ofboth the island and the leads. The sign of λ ⊥ is unim-portant as it may be reversed by a gauge transformation.In principle, evaluating the sums in Eq. (12) requiresa detailed knowledge of the tunneling amplitudes t αnk ,which in turn depend on details of the wavefunctionsat the NS interfaces. In practice, simplifying assump-tions may be quantitatively satisfactory. For instance, t αnk ≃ t αn is a good approximation for NS junctionsthat are atomically thin (recall that k is the momen-tum perpendicular to the NS interface). This renders λ αα ′ || , ⊥ independent of k and k ′ . Similarly, we conceive t ∗ αn t α ′ n ≃ t ∗ α t α ′ and t ∗ αn t ∗ α ′ n ≃ t ∗ α t ∗ α ′ with the understand-ing that the n − dependence is weak for states that arecontained within a very narrow strip around the Fermienergy. This approximation is motivated by the fact thatthe main contributions to λ || and λ ⊥ in Eq. (12) originatefrom states with | ǫ n | . E c < E T and | ǫ n | . ∆ < E T ,respectively.Under the above assumptions, the tunneling ampli-tudes may be taken to be real and the Kondo couplingsread λ αα ′ || , ⊥ = λ LL || , ⊥ (cid:18) t R /t L t R /t L ( t R /t L ) (cid:19) . (13)Eq. (13) may be diagonalized with a unitarytransformation, which converts Eq. (7) into asingle-channel Kondo model: H SCK = H l + (cid:2) λ || S z ˜ s z + λ ⊥ (cid:0) S + ˜ s − + h . c . (cid:1)(cid:3) , (14)where ˜ s i = P k,k ′ ˜ c † kσ τ iσσ ′ ˜ c k ′ σ ′ and˜ c kσ = ( t L ˜ c Lkσ + t R ˜ c Rkσ ) / q t L + t R λ || , ⊥ = λ LL || , ⊥ + λ RR || , ⊥ (15)Thus only one out of the two conduction channels iscoupled to the superconductor. Starting from Eq. (12)and using ǫ n = nδ we derive explicit expressions for theKondo couplings at second order in tunneling amplitude: νλ ⊥ = Γ δ f ⊥ ( x ) ; νλ || = Γ δ f || ( x ) , (16) TABLE I: Virtual elastic processes to second order in single-particle tunneling (the extension to fourth order processes isdeferred to the Appendix). The initial and final states are in the truncated, low-energy Hilbert space whereas the intermediatestates trespass into the high-energy sector. We ignore inelastic processes because they are suppressed at T ≪ T ∗ . The energyof all initial and final states is ≃ E c because the most important tunneling events involve electrons/holes at the Fermi surface( ξ k ≃ ξ k ′ ≃ | i i ) Intermediate States ( | n i ) Final State ( | f i ) E i − E n h f |H T | n ih n |H T | i i c † αk ↑ | ø i| M + 2 i | ø i γ † n ↑ | M + 2 i c † α ′ k ′ ↑ | ø i| M + 2 i E c − (4 E c + √ ǫ n + ∆ ) t ∗ αkn t α ′ k ′ n u n c † α ′ k ′ ↑ c † αk ↑ | ø i γ † n ↓ | M + 2 i E c − √ ǫ n + ∆ − t ∗ αkn t α ′ k ′ n v n c † αk ↑ | ø i| M i | ø i γ † n ↑ | M i c † α ′ k ′ ↑ | ø i| M i E c − √ ǫ n + ∆ t ∗ αkn t α ′ k ′ n u n c † α ′ k ′ ↑ c † αk ↑ | ø i γ † n ↓ | M i E c − (4 E c + √ ǫ n + ∆ ) − t ∗ αkn t α ′ k ′ n v n c † αk ↑ | ø i| M i | ø i γ † n ↑ | M i c α ′ k ′ ↓ | ø i| M + 2 i E c − √ ǫ n + ∆ − t ∗ αkn t ∗ α ′ k ′ n u n v n c † αk ↑ c α ′ k ′ ↓ | ø i γ † n ↓ | M + 2 i E c − √ ǫ n + ∆ − t ∗ αkn t ∗ α ′ k ′ n u n v n where x ≡ E c / ∆, f ⊥ ( x ) ≃ π √ − x tan − r x − xf || ( x ) ≃ xf ⊥ + 1 π x √ − x tan − r − x x , (17) ν is the density of states of the leads at the Fermi energyand Γ = πν ( t L + t R ) denotes the broadening of the en-ergy levels in the dot due to its coupling to the leads. Γis related to a dimensionless parameter representing thetransparency of the tunnel junctions: g T = g L + g R = ( t L + t R ) νδ = Γ πδ ≪ , (18)where g L and g R are the conductances of the left andright tunnel junctions in units of 2 πe / ~ . In the deriva-tion of Eq. (17) we approximated P n by R ∞−∞ dn ; thisis justified in mesoscopic superconducting islands, where δ . − ∆. In Fig. (1) we plot f ⊥ and f || . An essentiallyidentical expression for λ ⊥ was found by Hekking et al. in Ref. [34]; however, these authors did not contemplatethe possibility of a Kondo effect and did not evaluate λ || , whose positive sign is important in order to produceCKE.As demonstrated in the Appendix, Kondo correlationsbecome apparent only when we carry out perturbationtheory to fourth order in tunneling. Indeed at fourthorder both coupling constants λ ⊥ and λ || develop in-frared singularities, in a manner that is consistent withthe renormalization group equations of the anisotropicKondo model: dλ || dl = 2 νλ ⊥ − ν λ || λ ⊥ dλ ⊥ dl = 2 νλ ⊥ λ || − ν λ ⊥ ( λ || + λ ⊥ ) , (19)where l ≡ ln( T ∗ /T ) and by definition λ || , ⊥ ( l = 0) ≡ λ || , ⊥ ( T ∗ ) are the values of the Kondo couplings at an E c / ∆ -6-4-202 f || f ⊥ g ⊥ / f ⊥ FIG. 1: Parameters f ⊥ , f || and g ⊥ as a function of E c / ∆.These parameters (defined in the main text and in the Ap-pendix) determine the values of the Kondo couplings at anenergy scale T ∗ (see Eqs. (6) and (20)). When E c is suffi-ciently close to ∆ (precisely how close is contingent on thevalue of Γ /δ ) the bare couplings become large, which inval-idates the perturbative renormalization group approach em-ployed in this work. For typical mesoscopic superconductingislands E c / ∆ ≃ O (1) and therefore λ || ( T ∗ ) /λ ⊥ ( T ∗ ) ≃ O (1). energy scale T ∗ , below which the Kondo model becomesapplicable. The present analysis is quantitatively reliableprovided that λ || , ⊥ ( T ∗ ) ≪
1. We show in the Appendixthat νλ ⊥ ( T ∗ ) ≃ Γ δ f ⊥ (cid:18) δ g ⊥ f ⊥ (cid:19) + 2 Γ δ f || f ⊥ ln ∆ T ∗ νλ || ( T ∗ ) ≃ Γ δ f || (cid:18) δ g || f || (cid:19) + 2 Γ δ f ⊥ ln ∆ T ∗ , (20)where g ⊥ and g || are dimensionles functions of ∆ /E c (theformer is plotted in Fig. (1)). Eq. (20) is valid providedthat the O (Γ ) terms are smaller than the O (Γ) terms;for Γ /δ → λ || ( T ∗ ) > λ || and λ ⊥ will flowto strong coupling through Eq. (19). The energy scale atwhich max { νλ ⊥ ( l c ) , νλ || ( l c ) } ≃ T K = T ∗ exp( − l c ) , (21)where T ∗ should be replaced by 8 E c when E c < T ∗ / V. DISCUSSION
In this section we derive quantitative estimates for thecharge-Kondo temperature in NSN heterostructures, ex-amine the influence of multi-channel junctions, proposevarious experimental NSN platforms wherein CKE mightbe observed, and compare our proposal of CKE in NSNheterostructures with previous proposals of CKE in othersystems.
A. Estimates for the Charge-Kondo Temperature
The Kondo temperature of our system can be quanti-fied combining Eqs. (16), (17), (19), (20) and (21). Forpedagogical purposes we begin by discussing a few par-ticular cases that can be solved analytically.The first special case corresponds to E c ≃ . λ ⊥ ( T ∗ ) ≃ λ || ( T ∗ ) and T K (∆ → E c ) ≃ T ∗ exp( − / νλ ⊥ ) ≃ ∆ exp( g ⊥ ) exp (cid:18) − δ/ Γ2 f ⊥ (cid:19) , (22)where for simplicity we have neglected third order termsin the RG equations. In the second line of Eq. (22)we have made a Taylor expansion under the assumptionthat (Γ /δ ) g ⊥ /f ⊥ ≪ νλ ⊥ ln(∆ /T ∗ ) ≪
1. Formesoscopic superconductors, ln(∆ /T ∗ ) . /δ < .
1. A typicalKondo temperature in this case (using f ⊥ ≃ .
4) reads T K ∼ − ∆ ∼ − T ∗ , the prefactor ofthe Kondo temperature in Eq. (22) is effectively shiftedfrom T ∗ to ∆. An analog situation occurs between E c and δ in SKE. Another simple limit is that of E c ≪ ∆, in which case λ || ( T ∗ ) ≪ λ ⊥ ( T ∗ ) and accordingly T K (∆ ≫ E c ) ≃ T ∗ exp( − π/ νλ ⊥ ) , (23)where once again we have neglected third order terms inEq. (19). The exponent of Eq. (23) is a factor π/ E c / ∆ T K / ∆ Γ/δ
FIG. 2: Kondo temperature (Eq. (21)) (in units of ∆) as afunction of E c / ∆, for Γ /δ ≃ .
08. We choose ∆ /δ ≃ /δ . For thisset of parameters, the Kondo temperature can reach ∼ E c / ∆ in the appropriate range. The decreasein T ∗ as E c / ∆ increases is overcompensated by the gain inthe bare Kondo coupling. In our estimates we have ignoredboth g ⊥ and g || . The perturbative analysis breaks down when νλ || , ⊥ ( T ∗ ) ∼ O (1). For Γ /δ ≃ .
08 this breakdown is appar-ent at E c / ∆ & . Inset:
Kondo temperature (in units of ∆)as a function of Γ /δ , for E c = 0 . negative than the exponent of Eq. (22), and f ⊥ ( E c ≪ ∆) ≃ < f ⊥ ( E c ≃ . /δ , T K (∆ ≫ E c ) ≪ T K (∆ ≃ E c ).A third special case consists of λ || ( T ∗ ) >> λ ⊥ ( T ∗ ).However, in our model this regime does not arise for anyvalue of E c / ∆ and thus we disregard it.We now solve more general cases numerically. Fig. (2)displays the charge-Kondo temperature for a genericrange of parameters. For fixed Γ /δ , T K increases rapidlywith E c / ∆ (at least up until E c / ∆ ≃ . T K ≃ T K ≃ E c / ∆ →
1, higher order tunneling processes (en-coded in g || , ⊥ ) become important, νλ ⊥ , || ( T ∗ ) is no longersmall and correspondingly our perturbative calculationof T K becomes unreliable. The divergence of g ⊥ towards negative values (Fig. 1) casts some doubt on whether theKondo temperature will continue to increase as E c / ∆ →
1. In the inset of Fig. (2) we illustrate the dependenceof T K on Γ /δ . For a given value of E c / ∆, T K increasesexponentially with Γ /δ . Only small values of Γ /δ areallowed in our calculation, which treats tunneling as aweak perturbation. In the strong tunneling regime (Γ ≫ δ ) the charge of the superconducting island fluctuatesstrongly and it is a priori unclear what the fate of CKEwill be. This issue will be addressed in Section VII. B. Influence of Multichannel Leads
Thus far we have been considering only one conductionchannel per lead. This is appropriate for atomic pointcontacts, wherein itinerant electrons are fully character-ized by their momentum in the direction perpendicularto the NS interface. Nonetheless, due to the small Fermiwavelength of metals, most NS contacts contain multi-ple channels that describe e.g. transverse momenta ofelectrons in the lead. Multichannel leads can be read-ily incorporated into Eq. (1) by rewriting the tunnelingHamiltonian and the Hamiltonian for the leads: H l = X kσi ξ k c † kσi c kσi H T = X kσni t kni c † kσi d nσ + h . c ., (24)where i = 1 , ..., N is a channel index that includes thewhich-lead label. In wide junctions the tunneling am-plitude depends on the channel index i . In contrast, wehave taken the kinetic energy of the itinerant fermions tobe independent of i ; this can always be ensured by properrescaling of the itinerant fermion operators. In addition,we have assumed that each energy level in the supercon-ducting dot remains only twofold degenerate, due to thechaotic motion of electrons in the dot.In principle it is possible that only one linear com-bination of the channels couple to the superconductor.This is the case when any one of the following assump-tions (from more to less stringent) is satisfied at energies | ǫ n | . ∆: (i) t kni ≃ t ki (energy-independent magnitudeand sign of the tunneling amplitude), (ii) t kni ≃ a n b ki (separable tunneling amplitude), (iii) t kni t k ′ nj ≃ t ki t kj (energy-independent magnitude of the tunneling ampli-tude, and coherent tunneling through different junctionsdue to L < ζ ). With this proviso one can always make aunitary trasformation in the spirit of Eq. (15) and recoveran effective one-channel Kondo model.When the assumptions (i)-(iii) above fail, a multichan-nel Kondo effect invariably follows. In NNN devices T K drops exponentially when the number of channels in thenormal leads and in the normal metallic grain is muchlarger than one. This argument does not apply to NSNdevices insofar as the superconducting island is describ-able by random matrix theory and L . ζ . However,even in our case T K is generically lower for the multi-channel than for the single-channel Kondo model. Sup-pose for instance that there are N ≫ ∼ /N , i.e. each Kondo coupling must be very small.Eq. (19) then dictates that the Kondo temperature willbe very low. This state of affair changes whenever a uni-tary transformation can map multichannel leads onto aneffective single-channel lead, because in that case the onlynonzero Kondo coupling in the transformed basis scalesas the sum of all the original couplings (see Eq. (15) fora N = 2 example of this). In order to arrive at a single-channel Kondo problem inpresence of wide NS junctions it is helpful that the Kondocouplings be of second order in the tunneling amplitude,because condition (iii) above is more realistic than ei-ther (i) or (ii). In case of CKE in non-superconductingdots the xy -component of the Kondo coupling is ∼ O ( t ),whereas the z − component is ∼ O ( t ). In such situationone is generally left with a multichannel Kondo problem.This is a potentially important difference between CKEin NSN and NNN heterostructures. C. Possible Experimental Setups (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) N point contact SG (a) (b)SG NWNW FIG. 3: Two possible arrangements for the experimen-tal observation of the charge-Kondo effect in NSN sys-tems. S=superconducting island, G=gate, N=normal metal,NW=semiconducting nanowire. (a) Metallic-semiconductinghybrid setup (top view). The width of the semiconduct-ing nanowires is comparable to their Fermi wavelength,which guarantees the emergence of a single-channel Kondomodel at low energies. Local gates placed at the nanowire-superconductor contacts can tune the channel transparency.The nanowires are long enough so that their energy-level spac-ing is negligible. (b) All-metallic setup (side view). Theatomic point contact fits a single conduction channel, whereasthe wider NS junction may contain multiple channels. When L ≫ ζ , T K may be maximized by increasing the tunnelingamplitude at the atomic point contact relative to that at thewide junction. Can the charge Kondo effect proposed in this pa-per be measured in experimentally realizable NSN de-vices? In a preceding subsection we have estimatedcharge Kondo temperatures of ≃ − /δ and E c . This temperature is close to the lowest tem-peratures achieved in current dilution fridge refrigera-tors ( . T K may in principle be madehigher by designing more transparent junctions, tuning∆ /E c closer to one or choosing materials with larger val-ues of ∆. Thus while it will be difficult to experimentallyaccess the unitary regime T ≪ T K , measurements con-ducted in the weak coupling regime T & T K may bewithin reach.Yet in spite of the extensive experimental work onmesoscopic NSN systems, there is no report of any CKE.Some trivial reasons for this include (i) the effective tem-perature of the device being too high, (ii) the gate voltagenot being tuned with sufficient precission to the chargedegeneracy point, and (iii) purposely breaking the chargedegeneracy by coupling the island to a superconductingreservoir. A more fundamental reason might be that theNS junctions fabricated in typical experiments lead tomultichannel Kondo problems with concomitantly lower T K .Next we discuss the experimental platform and re-quirements to observe CKE in NSN heterostructures.First and foremost we need ∆ > E c , which precludesultrasmall superconducting islands due to their domi-nant energy. Second, the superconducting island mustbe placed in the vicinity of a gate electrode. In orderto measure CKE, the resolution of the gate voltage mustbe such that N g − N (0) g = C g ( V g − V (0) g ) /e . T K /E c ,where N (0) g = 2 M + 1 is the charge degeneracy point(for any integer M ), and V (0) g = eN (0) g /C g is the cor-responding gate voltage. For typical values of the gatecapacitances ( C g ≃ T K /E c ≃ − ), it follows that the gate voltage resolu-tion needs to be better than 10 µ V. This is achievable incurrent experimental devices. If N g − N (0) g & T K /E c , theKondo RG equations (Eq. (19)) are cutoff at energy scales ∼ E c ( N g − N (0) g ) & T K and the charge Kondo effect willnot be fully developed. However, even in this case thereshould be observable fingerprints of Kondo correlations(these are discussed in Section VI).Another desirable trait of the superconducting islandis that it have no spatial symmetries and that its lineardimensions L be smaller than the superconducting coher-ence length ζ . This ensures having only one conductionchannel in the dot at any given energy, which increasesthe likelihood of producing a single-channel Kondo modelat low energies. Moreover, smaller island sizes mean asmaller ratio ∆ /δ and thus a larger value of T ∗ , which en-hances the ultraviolet energy cutoff for the Kondo prob-lem. In principle, having a small-sized superconductormight conflict with the requirement that ∆ > E c . Inpractice, there appears to be enough room in parameterspace to ensure that ∆ > E c and L < ζ are satisfiedsimultaneously. For instance, the best developed ma-terial is aluminum, whose BCS coherence length in theballistic limit is ζ = ~ v F / ( π ∆) ≃ . µ m. For a meanfree path of l ≃ ζ = ( ~ D/ ∆) / ≃ ( ζ l ) / ≃ L ≃ A ≃ (80nm) and ǫ ≃
12 (dielectric constantof AlO x ) yields E c ≃ . < ∆ ≃ . ζ ≃ x ( ǫ ≃
40) can pro-duce sufficiently small charging energies even for L ≃ A ≃ (25nm) ( E c ≃ . < ∆ ≃ . ζ ≃ ζ ≃ l ≃ L ≃ A ≃ (30nm) and ǫ ≃
25 (dielectric constantof PbO x ) we get E c ≃ . < ∆ ≃ A ≫ λ F , where λ F is the Fermiwavelength of the metal. In Fig. 3a we sketch an al-ternative setup that combines a superconducting grainwith semiconducting nanowires. This setup is moti-vated by recent success in manufacturing semiconduct-ing/superconducting nanocontacts. The large Fermiwavelength in semiconductors facilitates single-channelNS contacts without making the charging energy of theisland larger than the superconducting gap. In addition,the possibility of tuning Schottky barriers in semiconduc-tor/metal interfaces by local gates creates prospects formeasurable charge-Kondo temperatures.Fig. 3b constitutes yet another all-metallic experimen-tal setup, which allows to reach sizeable T K in NSN de-vices with L ≫ ζ (where electrons cannot coherentlypropagate from one junction to another). One of thejunctions is made from an atomic point contact, whilethe other junction is wide enough to ensure that thecharging energy of the device remains less than the su-perconducting gap. If the single channel at the pointcontact is much more strongly coupled to the supercon-ductor than each of the multiple channels at the widejunction, then the charge-Kondo temperature is close tothat of the single-channel Kondo model. A theoreticallystraightforward way to achieve atomic point contacts isto use scanning tunneling microscopes (STM). STM tipsoperated in the point contact mode can capture An-dreev tunneling (which is altered by charge-Kondo corre-lations), and recent STM experiments have succeededmeasuring the density of states of ultrathin supercon-ducting Pb grains. An added challenge here is that theSTM would have to operate at milliKelvin temperatures.Alongside milliKelvin STMs, mechanically controllablesingle-channel quantum point contacts are being devel-oped in superconducting systems. D. Comparison to Other Charge-Kondo Effects
Part of our motivation for exploring NSN systems isto establish a “quantum-dot counterpart” for the charge-Kondo effect that is believed to arise in compounds dopedwith negative-U molecules (“NU” systems). In this sec-tion we discuss similarities and differences, as well as ad-vantages and disadvantages of CKE in NU, NNN andNSN settings. Some of the highlights are summarized inTable II.The ultraviolet (UV) cutoff Λ is the energy scale belowwhich the Kondo model applies. It is lowest in NSN sys-tems, where T ∗ ≪ ∆ due to entropic arguments. Thisis partly why materials doped with negative-U centersreportedly show T K ≃ T K ∼ TABLE II: Comparison between different charge-Kondo effects (CKE) and the spin-Kondo effect (SKE). QD stands for semicon-ducting quantum dots. ω is the largest of a variety of extrinsic energy scales such as temperature, bias voltage, pseudo-magneticfields and the frequency of an external perturbation.CKE in NSN CKE in NU CKE in NNN SKE in QDTwo-level system N or N + 1 Cooper pairs N or N + 2 electrons N or N + 1 electrons ↑ or ↓ electronUltraviolet cutoff (Λ) T ∗ ( δ ≪ T ∗ < E c ) | U | E c ( ≫ δ ) δ ( ≪ E c )Infrared cutoff ω ω max { ω, δ } ωνλ ⊥ (Λ) ∼ (Γ /δ ) f ⊥ ∼ Γ / | U | ∼ (Γ /δ ) / ∼ Γ /E c νλ || (Λ) ∼ (Γ /δ ) f || = νλ ⊥ (Λ) 0 = νλ ⊥ (Λ) T K /δ unrestricted ≃ ≥ ≥ ≥ NNN systems does not suffer from entropic issues be-cause the ground state and the lowest excited state areboth surrounded by similarly dense sets of single-particlestates.The infrared (IR) cutoff ω is the energy scale at whichthe renormalization of the Kondo couplings stops. InNSN and NU systems ω is the largest amongst thetemperature T , bias voltage V , and pseudo-magneticfields h z , h x . The full Kondo effect is observable onlyif T K > ω . In NNN systems, Kondo-like IR divergencesoriginate from sums involving single-particle states in thedot . Consequently δ too plays the role of an IR cutoffand the Kondo effect in NNN systems is observable onlyif T K > max { δ, ω } . T K > δ is difficult to satisfy in semi-conducting dots.Concerning the Kondo couplings, in NU systems theyare isotropic in pseudospin space and originate from a sin-gle energy level. In NSN and NNN systems, the Kondocouplings are anisotropic and sensitive to the multilevelenergy spectrum of the dot. While the anisotropy be-comes gradually less pronounced at lower energies, it stillplays an essential role in the quantification of T K . More-over, multilevel effects can become interesting on theirown (see Appendix). A unique feature of NSN systemsis that the bare Kondo couplings are strongly depen-dent on ∆ /E c , which can be tuned by magnetic fields.In SKE systems, the bare Kondo coupling is tipicallysmaller than in NSN systems because it contains a prod-uct of two small parameters: Γ /δ and δ/E c .NNN systems can never have less than two Kondochannels, due to real spin degrees of freedom that are de-coupled from the charge pseudospin. In contrast, in NSNdevices spin degrees of freedom are interwoven with thecharge pseudospin (because the order parameter couplesspin-up electrons with spin-down holes) and therefore theone-channel Kondo effect is in principle possible. UnlikeNU and SKE systems, most NNN and NSN systems carrywide junctions with multichannel leads. However, multi-ple channels of a NSN system can be effectively mappedonto a one-channel Kondo model e.g. when the tunnel-ing matrix amplitude is independent of energy. Reachinga two-channel Kondo model in NNN systems with wide junctions is much less likely.In sum, CKE in NSN systems ought to be experimen-tally measurable much like ordinary SKE is observablein semiconducting quantum dots, provided that single-channel Kondo models can be engineered. VI. ZERO-BIAS CONDUCTANCE
In previous sections we have described the charge-Kondo model that arises in mesoscopic NSN heterostru-cures at sufficiently low energies. In this section we in-vestigate its fingerprints in the low-temperature electri-cal transport. We focus on the elastic contribution to thezero-bias conductance, which is dominant at
T < T ∗ be-cause the superconducting gap freezes out inelastic quasi-particle excitations. Fingerprints of CKE can also getmanifested in the low-temperature capacitance of super-conducting islands coupled to a single normal metalliclead. The electrical conductance of mesoscopic NSNheterostructures has been widely discussed in theliterature; nonetheless, there appears tobe no report on many-body anomalies in the low-temperature elastic conductance. When bias voltages areweak, the current-voltage characteristics is encoded in I α = g αα ′ V α ( α ∈ { L, R } ) , (25)where V α and I α = h ˆ I α i = ie sgn( α ) h [ H eff , ˆ N α ] i are thevoltage drop and the current across the α -th junction,respectively. ˆ N α denotes the electron number operator inthe α -th lead and H eff is given by Eq. (7). Also, sgn( α ) ≡ −
1) for α = L ( R ).The zero-bias conductance matrix in Eq. (25) can beexpressed in terms of the Kubo formula: g αα ′ = lim ω → iω Z β dτ e iωτ hT τ ˆ I α ( τ ) ˆ I α ′ (0) i , (26)where β = 1 / ( k B T ) and T τ is the time-ordering operator.0It is straightforward to obtain ˆ I α = ˆ I α single + ˆ I α pair , where:ˆ I α single = − ie sgn( α ) X α ′ = α λ αα ′ || X k , k ′ ,σ S z c † α k σ c α ′ k ′ σ + h . c . ˆ I α pair = ie sgn( α ) X α ′ λ αα ′ ⊥ X k , k ′ ,σ S + σc α k σ c α ′ k ′ − σ + h . c . (27) h ˆ I single i and h ˆ I pair i are the single-particle and Cooper-pair tunneling contributions to the current, respectively. α = α ′ in the expression for ˆ I single simply recognizes thefact that local ( α = α ′ ) single-particle processes are nor-mal reflections that do not contribute to the current. Wetreat the leads as non-superconducting ( h c α k ↑ c α k ′ ↓ i =0), non-magnetic ( h c † α k ↑ c α k ′ ↓ i = 0) and infinitely long( h c † α k σ c α k ′ σ i ∝ δ k , k ′ ) conductors. With this in mind, thecombination of Eqs. (26) and (27) yields g αα ′ = (cid:18) G LA + G EC + G CA G EC − G CA G EC − G CA G RA + G EC + G CA (cid:19) , (28)where G L ( R ) A = 2 G Z dE ( − ∂ E f )2[ νλ LL ( RR ) ⊥ ( E )] G CA = G Z dE ( − ∂ E f )2[ νλ LR ⊥ ( E )] G EC = G Z dE ( − ∂ E f )[ νλ LR || ( E )] , (29) G = πe / ~ and f ( E ) = (exp( βE ) + 1) − . G L ( R ) A is thelocal Andreev conductance across the left (right) junc-tion, G CA is the non-local or crossed Andreev conduc-tance, and G EC is the elastic, single-particle co-tunnelingconductance. The probability for an Andreev re-flection process is proportional to ( λ αα ′ x ) + ( λ αα ′ y ) =2( λ αα ′ ⊥ ) , whereas the probability for a co-tunneling eventis ∝ ( λ LR || ) . The derivation of Eq. (28) is standard except for the following two details. First, λ || , ⊥ ( E ) isenergy-dependent via the Kondo RG flow of Eq. (19),and diverges as E →
0. Second, the current oper-ator in Eq. (27) contains S z and S ± ; we have used hT τ S z ( τ ) S z (0) i = 1 / hT τ S + ( τ ) S − (0) i = 1 / τ > g αα ′ in Eq. (28) is simple tointerpret. Regarding diagonal matrix elements, each lo-cal Andreev reflection contributes twice as much as eachco-tunneling or crossed Andreev reflection event becausethe former involves two electrons crossing the same junc-tion while in the latter processes the two electrons crossdifferent junctions. This fact is reflected by the extrafactor of two in the first line of Eq. (29). Concerningthe off-diagonal matrix elements, electron co-tunnelingand crossed Andreev reflection contribute with oppositesigns because in the later case it is a hole (rather than aelectron) that tunnels out of the superconducting island. Even though Eq. (28) formally agrees with expressionsderived in previous studies of NSN heterostructures, there are conceptual novelties when the superconductingisland behaves as a pseudospin 1/2 quantum impurity.For instance, G EC /G CA → / I R = I L ≡ I and the electrical responseto the total voltage drop V R + V L = V across the devicecan be evaluated by inverting Eq. (28): G ≡ dIdV = G EC + ( G LA + G RA ) G CA + G LA G RA G LA + G RA + 4 G CA . (30)Kondo correlations become apparent when we evaluateEq. (30) for T < T ∗ . Let us suppose that the two NSjunctions can be mapped onto a single-channel Kondomodel. Combining Eqs. (13), (15) and (30) it followsthat G = G t L t R ( t L + t R ) Z dE ( − ∂ E f ) (cid:0) ( νλ || ) + 2( νλ ⊥ ) (cid:1) , (31)where λ || , ⊥ ( E ) obeys Eq. (19).When λ ⊥ ( T ∗ ) = λ || ( T ∗ ) (i.e. E c ≃ . T ∗ ≫ T ≫ T K , the integral in Eq. (31) yields G = G t L t R t L + t R ) ( T /T K ) . (32)For T < T K , the Kubo formula expression breaksdown and the conductance converges to G =2( e /h )4 t L t R / ( t L + t R ) . The same results arise alsoin negative-U molecules with charge degeneracy, innormal-metallic quantum dots with charge degeneracy and even in semiconducting quantum dots with spindegeneracy. The conductance for λ ⊥ ( T ∗ ) = λ || ( T ∗ ) at T ∗ ≫ T ≫ T K is displayed in in Fig. (4). G ( T ) increases with E c / ∆, as expected from Fig. (1). Since the anisotropy ofthe bare Kondo couplings disappears at sufficiently lowtemperatures, G ( T ≪ T K ) is the same irrespective of λ ⊥ ( T ∗ ) /λ || ( T ∗ ).The conductance curves of Fig. (4) have been neitherpredicted nor measured in previous studies of NSN sys-tems. For instance, Ref. [34] neglected the inter-junctionelectron coherence (i.e. G CA = G EC = 0 in Eq. (30))and concluded that the zero-bias conductance throughthe superconducting grain will be independent of tem-perature, as long as T ≪ T ∗ and the gate voltage istuned to the charge degeneracy point. Contemporaryexperiments appeared to agree with this theoreticalprediction, although the temperature dependence of thezero bias peak was not analyzed in detail. Yet accord-ing to our theory, the zero-bias conductance could dis-play traces of a two-channel Kondo effect in islands with1 T/T* G ( T ) / G ( T * ) E c =0.5 ∆ E c =0.4 ∆ E c =0.6 ∆ G(T)/G(T*) [ln(T*/T K )/ln(T/T K )] FIG. 4: Low-temperature (elastic) zero-bias conductance(Eq. (31)) across the NSN heterostructure, in the single-channel Kondo model. The conductance is normalized toits value at T = T ∗ . Curves with red squares, black cir-cles and blue triangles have T K = 0 . T ∗ , T K ≃ . T ∗ and T K = 0 . T ∗ , respectively. For simplicity we ignored g ⊥ , || in Eq. (20) and evaluated T K analytically neglecting thirdorder terms in Eq. (19). We chose ∆ = 1000 δ . Inset:
Thesolid line represents G ( T ) /G ( T ∗ ) for E c = 0 . E c = 0 .
5∆ yields the isotropic Kondomodel ( λ ⊥ ( T ∗ ) = λ || ( T ∗ )), the solid line matches well withEq. (32) (dashed line). L ≫ ζ . This is the case in the experimental setup ofFig. 3b, which should host a low-temperature enhance-ment of the Andreev reflection at the point contact, ac-companied by a reduction of the Andreev reflection inthe wide junction. VII. CHARGE-KONDO EFFECT IN THESTRONG TUNNELING REGIME
Thus far all our calculations have concentrated on theweak tunneling regime, where Γ ≪ δ . In this sectionwe turn to highly transparent NS junctions (Γ ≫ δ )and determine the fate of CKE when charge in the su-perconducting island is strongly fluctuating. On onehand, in the weak tunneling regime the charge-Kondotemperature is enhanced by more transparent normal-superconducting junctions (Section V). On the otherhand, when charge in the dot fluctuates strongly it isno longer appropriate to regard the superconductor as atwo-level system. Which of these two conflicting trendsprevails? It turns out that the answer to this ques-tion depends on electron-electron interactions in the lead,which we have ignored up until now. Strong tunnelingand electron-electron interactions render bosonization asthe most convenient method to employ in this section.In particular we combine and suitably modify the ap-proaches from Refs. [50] and [51]. We consider a one-dimensional normal metallic wireconnected to a superconductor (which need not be one-dimensional). The lead occupies the negative half-axis( x < x = 0, and the supercon-ductor is located at x >
0. We write the low-energy andlong-wavelength Hamiltonian of this system as H = H lead + H C + H B , (33)where H lead = iv F X σ Z −∞ dx (cid:16) Ψ † Lσ ∂ x Ψ Lσ − Ψ † Rσ ∂ x Ψ Rσ (cid:17) (34)is the Hamiltonian corresponding to the lead, H C = E c "Z −∞ dx X σ (cid:16) Ψ † Lσ Ψ Lσ + Ψ † Rσ Ψ Rσ (cid:17) + N g (35)is the charging energy and H B = aV B X σ Ψ † σ (0)Ψ σ (0) + a ∆ B [Ψ ↑ (0)Ψ ↓ (0) + h . c . ](36)is a boundary interaction ( a is the lattice spacing). InEqs. (34), (35) and (36) we have expanded the orig-inal fermion operators as Ψ σ ( x ) ≃ e − ik F x Ψ Lσ ( x ) + e ik F x Ψ Rσ ( x ), where k F is the Fermi wavevector and Ψ Lσ (Ψ Rσ ) describes left-moving (right-moving) fermions. InEq. (35) we have exploited total charge conservation inthe dot-plus-lead system so as to rewrite the chargingenergy in terms of the charge density in the lead (in-stead of in the dot). Eq. (36) follows from integrating outthe superconductor at energy scales that are lower thanthe BCS gap ∆. This integration results in a boundaryterm, which encodes normal scattering ( ∝ V B ≪ ∆) aswell as Andreev scattering ( ∝ ∆ B ). Both V B and ∆ B (which can be taken to be purely real in our case) de-pend on microscopic details of the superconductor suchas its Fermi velocity, ∆ and δ . Also note that quasiparti-cle transmission into the superconductor is forbidden atenergies below the gap.We bosonize Eq. (33) in the standard manner: Ψ rσ ( x ) = η σ √ πa e − i √ [ r Φ c ( x ) − Θ c ( x )+ rσ Φ s ( x ) − σ Θ s ( x )] , (37)where η σ is a Klein factor ( η ↑ η ↓ = − η ↓ η ↑ , η σ η σ = 1), r =+ ( − ) for right- (left-) moving fermions and σ = +( − )for spin-up (-down) fermions. Φ c (Φ s ) and Θ c (Θ s ) arethe usual charge- (spin-) bosons.It follows that H lead = v F π X α ∈{ c,s } Z −∞ dx (cid:20) K α ( ∂ x Θ α ) + 1 K α ( ∂ x Φ α ) (cid:21) , (38)where K c ( K s ) is the Luttinger parameter in the charge(spin) sector ( K s = 1 due to spin rotational symmetry).2Similarly, H C = E c [ √ c (0) /π + N g ] , (39)where √ c (0) /π equals the number of electrons con-tained in x ∈ [ −∞ ,
0] up to a constant (we takeΦ c ( −∞ ) = 0).In order to bosonize H B , we use boundary condi-tions that are associated with perfect Andreev reflec-tion at the NS interface: Ψ R ↑ (0) = − i Ψ † L ↓ (0) andΨ R ↓ (0) = i Ψ † L ↑ (0). These boundary conditions providea reasonable approximation for highly transparent NSinterfaces with small Fermi surface mismatches. Fur-thermore, perfect Andreev reflection constitutes a renor-malization group fixed point when E c = V B = 0. Inbosonic language, the Andreev boundary conditions canbe translated as √ c (0) = π/ √ s (0) = π . Con-sequently the boundary interaction at the Andreev fixedpoint reads H B = − (2 /π ) V B cos[ √ c (0)] , (40)which describes single-particle backscattering (normal re-flection) at the interface. Terms proportional to ∆ B donot appear in Eq. (40), which is consistent with the factthat boundary Andreev reflection is never a relevant per-turbation at the Andreev fixed point. Next, we determine how E c and V B evolve as the tem-perature is lowered. Assuming that their bare values aresmall compared to ∆, the leading order RG flow equa-tions read dE c dl = E c ; dV B dl = (1 − K c ) V B , (41)where l = ln(∆ /T ). Thus E c is a relevant perturbation,and V B is relevant, marginal or irrelevant depending onwhether electron-electron interactions in the lead are re-pulsive ( K c < K c = 1) or attractive( K c > dV B /dl = (1 − K c / V B and backscattering is arelevant perturbation for non-interacting electrons. Theunderlying reason for this qualitative difference is thata perfectly transmitted channel at a NN interface has aconductance of 2 e /h , whereas a perfectly transmittedchannel at a NS interface has a conductance of 4 e /h (due to perfect Andreev reflection).Since either E c or V B is certain to grow under RG,the Andreev fixed point is unstable. The nature of thenew fixed point depends crucially on whether E c or V B reaches strong coupling first. If E c wins, the charge bosonat the boundary gets pinned to a value dictated by thegate voltage, √ c (0) /π → − N g , and the ground state isnon-degenerate. Since the gate charge N g is a continuousvariable, the discreteness of charge is unimportant in thiscase and the system is nowhere equivalent to a two-levelsystem.If V B prevails instead, then the charge boson at theinterface becomes pinned through √ c (0) /π → n , for any integer n . The discreteness of charge becomes impor-tant through the discreteness of n , although there are aninfinite number of degenerate ground states. This energydegeneracy between different values of n is lifted by H C (Eq. (39)). Choosing N g appropriately (e.g. N g = 1) thelowest energy state becomes two-fold degenerate ( n = 0and n = − two electrons. The remaining minima of thecosine potential are at energies & E c higher, where E c is the renormalized charging energy. Therefore in thestrong coupling limit of V B the ground state of Eq. (33)behaves as a two-level quantum system. Rare tunnelingevents between the n = 0 and n = − V B dominate over E c underrenormalization group. In terms of the parameters of theoriginal Hamiltonian, this condition reads∆ E c (0) ≫ (cid:18) ∆ V B (0) (cid:19) − Kc ; K c < , (42)where V B (0) and E c (0) are the values of the backscatter-ing potential and the charging energy at energy scales oforder ∆. In mesoscopic NSN devices Eq. (42) is difficultto satisfy unless electron-electron interactions in the leadare strongly repulsive and/or the bare charging energyis very small. The temperature at which V B reaches thestrong coupling regime is T ≡ ∆ e − l c , where l c is definedthrough V B ( l c ) ≃ ∆. Thus T ≃ ∆( V B (0) / ∆) − Kc ≪ ∆ . (43)If T ≫ E c ( l c ) = E c (0) e l c , inelastic charge excitations(i.e. transitions to n = 0 , −
1) are present at tem-peratures T > T > E c ( l c ). In this case, by anal-ogy with Ref. [50], the dimensionless Kondo coupling is νλ ∼ E c ( l c ) /T and the Kondo temperature scales like T K ∼ E c ( l c ) exp( − /νλ ) ≪ E c ( l c ) ≪ T . If T < E c ( l c ),inelastic excitations are forbidden as soon as T < T and T K could be a sizeable fraction of T . Yet because T istypically very small in weakly interacting leads ( K c ≃ T K is unlikely to be detectable. Prospects might bebetter in leads made from carbon nanotubes, where K c ≃ . ≪ /δ →
0, though itincreases rapidly as Γ /δ grows. In this section we haveseen that T K is very small when Γ /δ ≫ VIII. SUMMARY AND CONCLUSIONS
Partly motivated by the recent experimental searchfor charge-Kondo correlations in Tl-doped PbTe, wehave developed a theory for the charge- Kondo effectin artificially fabricated, chaotic superconducting islandsthat are weakly connected to non-superconducting leads.We have focused on tunable superconducting charge-qubits where the energy gap in the excitation spectrumis larger than the electrostatic charging energy. Thelow-energy transport and thermodynamic properties ofthese systems showcase Kondo-like behavior when theelectrostatic energies of two charge states differing bya Cooper pair become degenerate. A single-channelanisotropic Kondo model is found to describe supercon-ducting grains of size smaller than the superconductingcoherence length, which are coupled to normal conduc-tors either through atomic point contacts or semicon-ducting nanowires. The same model could also applyfor wider metal-superconducting contacts as long as thetunneling amplitude is independent of energy in a nar-row strip around the Fermi energy. For this model theKondo temperature can reach ≃ ≃ Acknowledgments
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In Section IV we introduced a low energy effective Kondo model, Eq. (7), from which we inferred the Kondoeffect by simply invoking standard renormalization group equations. While we provided physical arguments to justifythe emergence of such a model at temperatures lower than T ∗ , we did not present a rigorous derivation. The mainobjective of this Appendix is to validate Eq. (7) by deriving hallmark Kondo-like divergences from the full microscopic(or “first-principles”) Hamiltonian, Eq. (1).The procedure we follow has been introduced elsewhere. The task undertaken here consists of evaluating theamplitude for Andreev reflection to fourth order in tunneling. Andreev reflection transforms an itinerant electron intoan itinerant hole, while adding a Cooper pair to the superconducting island. This can be interpreted as a processthat flips the pseudospins of the superconducting island as well as of the incident fermion. The initial and final statesfor this process are | i i = c † k ↑ | ø i| M i ; | f i = c k ↓ | ø i| M + 2 i , (A1)which are eigenstates of Eq. (1) in absence of tunneling. Tunneling induces | i i → | f i transitions, and the transitionamplitude A i → f may be computed from certain matrix elements of an effective Hamiltonian derived perturbativelyfrom Eq. (1). One arrives at A i → f = A (2) i → f + A (4) i → f + ... , where A (2) i → f = X n h f |H T | n ih n |H T | i i E i − E n A (4) i → f = X n ,n ,n h f |H T | n ih n |H T | n ih n |H T | n ih n |H T | i i ( E i − E n )( E i − E n )( E i − E n ) − ǫ X n h f |H T | n ih n |H T | i i ( E i − E n ) − A (2) i → f X n |h f |H T | n i| ( E i − E n ) and | n i , | n i , ... denote intermediate states of energies E n , E n ...in absence of tunneling. In addition, ǫ = X n |h i |H T | n i| E n − E i = t X n X q Θ( ξ q ) E c − ξ q − p ǫ n + ∆ − E c + ξ q + p ǫ n + ∆ ! , (A2)5where in the second equality we have assumed particle-hole symmetry and energy-independent tunneling amplitudes.Θ( x ) is the step function. νA (2) i → f is identical to (Γ /δ ) f ⊥ of Section IV, and was first computed in Ref. [34]. It is finite and non-singular for E c < ∆. Yet A (4) i → f , which has not been previously computed, hosts infrared (IR) logarithmic divergences for any E c / ∆. These IR divergences signal the onset of Kondo correlations.Evaluating A (4) i → f is straightforward in principle but cumbersome in practice. For convenience we attach TablesIII-IV, which collect all possible intermediate state configurations for the first term in the expression of A (4) i → f . Manyof the individual amplitudes entering Tables III-IV are ultraviolet (UV) divergent in the infinite-bandwidth limit.Remarkably, all UV divergences in A (4) i → f end up cancelling one another after summing over all the amplitudes. Thisindicates that details of the energy spectrum at high energies are not important for the Kondo effect, and endowsuniversality to results derived in this paper. In fact, the bulk contribution to A i → f originates from states with | ǫ n | . ∆.Following a numerical sum of all the amplitudes, we arrive at νA i → f = Γ δ f ⊥ + 2 Γ δ f ⊥ f || ln ∆ ω + Γ δ g ⊥ , (A3)where g ⊥ is a dimensionless function of E c / ∆. To the present order approximation, Eq. (A3) can be recasted as νA i → f ≃ νλ ⊥ + 2( νλ ⊥ )( νλ || ) ln T ∗ ω , (A4)where νλ ⊥ = Γ δ f ⊥ (cid:20) δ g ⊥ f ⊥ + 2 Γ δ f || ln ∆ T ∗ (cid:21) and νλ || = Γ δ f || (cid:20) δ g || f || + 2 Γ δ f ⊥ f || ln ∆ T ∗ (cid:21) (A5)are the values of the Kondo couplings at the energy scale T ∗ (see Eq. (20)) and we have neglected O (Γ ) terms.Eq. (A4) provides a first-principles proof for the second line of Eq. (19) and establishes the anisotropic Kondo modelas an appropriate low energy theory for Eq. (1). Admittedly, the energy scale T ∗ does not appear naturally in ourzero-temperature calculation. Rather we introduce it in hindsight (i.e. with the knowledge that only at energies below T ∗ can the superconducting island behave as a two-level system) and arrange the rest of the terms accordingly. Thelogarithmic terms in Eq. (A5) indicate that the Kondo couplings begin to renormalize starting at energy scales of order∆. g ⊥ is plotted in Fig. (1), where it is shown that 4th order tunneling events can either enhance (if g ⊥ >
0) or deplete(if g ⊥ <
0) the Andreev reflection amplitude, depending on E c / ∆. We have not evaluated g || , which would requireadditional Tables of intermediate state configurations. We expect g || ≃ g ⊥ for E c ≃ . /δ ) | g ⊥ | ∼
1, which occurs at E c ≃ .
8∆ for Γ /δ ≃ .
1. Of course, the breakdown of theperturbative regime does not preclude a Kondo effect; it just means that we cannot compute the Kondo temperaturereliably using the weak-tunneling approach.6
TABLE III: Virtual elastic processes to fourth order in single-particle tunneling. n , m and q are dummy variables to be summedover. The amplitudes labelled with a number are either UV and/or IR divergent. The amplitudes labelled with a letter arenon-divergent. The amplitudes grouped with the same number or letter are identical to one another in presence of particle-holesymmetry. For each of these groups, half of the configurations involve electrons ( c † q ) and half involve holes ( c q ). For simplicitywe have assumed the tunneling amplitude to be simply a constant; however, our results can be readily generalized to includemore complicated situations. In the derivation of Fig. (1) we use P q → ν R dξ and P n → (1 /δ ) R dn . N.B.: in these Tables(and in these Tables only) E n stands for √ ǫ n + ∆ .Label | n i | n i | n i Contribution to A (4) i → f (in units of t )1 | ø i γ † n ↑ | M + 2 i c † qσ | ø i γ † n ↑ γ † m, − σ | M + 2 i | ø i γ † n ↑ | M + 2 i u n v n v m Θ( ξ q )( E c − E n ) ( ξ q + E n + E m ) | ø i γ † n ↑ | M + 2 i c qσ | ø i γ † n ↑ γ † mσ | M + 2 i | ø i γ † n ↑ | M + 2 i u n v n u m Θ( − ξ q )( E c − E n ) ( − ξ q + E n + E m ) | ø i γ † n ↑ | M + 2 i c † k ↑ | ø i γ † n ↑ γ † m ↓ | M + 2 i c † k ↑ c q ↑ | ø i γ † n ↑ | M + 2 i u n v n v m Θ( − ξ q )( E c − E n )( E n + E m )( E c + ξ q − E n ) | ø i γ † n ↑ | M + 2 i c † k ↑ | ø i γ † n ↑ γ † m ↓ | M + 2 i c † k ↑ c q ↓ | ø i γ † m ↓ | M + 2 i u m v m v n Θ( − ξ q )( E c − E n )( E n + E m )( E c + ξ q − E m ) | ø i γ † n ↑ | M + 2 i c † q ↓ | ø i γ † n ↑ γ † m ↑ | M + 2 i | ø i γ † m ↑ | M + 2 i − u m v m v n Θ( ξ q )( E c − E n )( ξ q + E n + E m )( E c − E m ) | ø i γ † n ↑ | M + 2 i c q ↓ | ø i| M + 2 i | ø i γ † m ↑ | M + 2 i − u m v m v n Θ( − ξ q )( E c − E n ) ξ q ( E c − E m ) | ø i γ † n ↑ | M + 2 i c q ↓ | ø i| M + 2 i c † k ↑ c q ↓ | ø i γ † m ↓ | M + 2 i − u m v m v n Θ( − ξ q )( E c − E n ) ξ q ( E c + ξ q − E m ) | ø i γ † n ↑ | M + 2 i c † q ↑ | ø i| M i c † q ↑ c † k ↑ | ø i γ † m ↓ | M i u n v n v m Θ( ξ q )( E c − E n ) ξ q ( ξ q +3 E c + E m ) | ø i γ † n ↑ | M + 2 i c † q ↑ | ø i| M i | ø i γ † m ↑ | M i u n v n u m Θ( ξ q )( E c − E n ) ξ q ( E c − E m ) A | ø i γ † n ↑ | M + 2 i c † qσ | ø i γ † n ↑ γ † m, − σ | M + 2 i c † qσ c † k ↑ | ø i γ † m, − σ | M i − u n v n v m Θ( ξ q )( E c − E n )( ξ q + E n + E m )( ξ q +3 E c + E m ) B | ø i γ † n ↑ | M + 2 i c qσ | ø i γ † n ↑ γ † mσ | M + 2 i c † k ↑ c qσ | ø i γ † mσ | M i − u n v n u m Θ( − ξ q )( E c − E n )( ξ q − E n − E m )( ξ q + E c − E m ) | ø i γ † n ↑ | M + 2 i c † k ↑ | ø i γ † n ↑ γ † m ↓ | M + 2 i c † k ↑ c † q ↓ | ø i γ † n ↑ | M i u m v m v n Θ( ξ q )( E c − E n )( E n + E m )( ξ q +3 E c + E n ) | ø i γ † n ↑ | M + 2 i c † k ↑ | ø i γ † n ↑ γ † m ↓ | M + 2 i c † k ↑ c † q ↑ | ø i γ † m ↓ | M i u n v n v m Θ( ξ q )( E c − E n )( E n + E m )( ξ q +3 E c + E m ) | ø i γ † n ↑ | M + 2 i c † q ↓ | ø i γ † n ↑ γ † m ↑ | M + 2 i c † q ↓ c † k ↑ | ø i γ † n ↑ | M i u m v m v n Θ( ξ q )( E c − E n )( ξ q + E n + E m )( ξ q +3 E c + E n ) | ø i γ † n ↑ | M + 2 i c q ↑ | ø i γ † n ↑ γ † m ↑ | M + 2 i c q ↑ c † k ↑ | ø i γ † n ↑ | M + 2 i u n v n u m Θ( − ξ q )( E c − E n )( ξ q − E n − E m )( E c + ξ q − E n ) | ø i γ † n ↑ | M + 2 i c q ↑ | ø i γ † n ↑ γ † m ↑ | M + 2 i | ø i γ † m ↑ | M i u n v n u m Θ( − ξ q )( E c − E n )( ξ q − E n − E m )( E c − E m ) E c k ′ ↓ c † qσ | ø i γ † n, − σ | M + 2 i c † qσ | ø i γ † m ↑ γ † n, − σ | M + 2 i | ø i γ † m ↑ | M + 2 i − u m v m v n Θ( ξ q )( E c − ξ q − E n )( − ξ q − E n − E m )( E c − E m ) c k ′ ↓ c † q ↓ | ø i γ † n ↑ | M + 2 i c k ′ ↓ | ø i γ † m ↓ γ † n ↑ | M + 2 i | ø i γ † n ↑ | M + 2 i − u n v n u m Θ( ξ q )( E c − ξ q − E n )( − E m − E n )( E c − E n ) B c k ′ ↓ c † qσ | ø i γ † n, − σ | M + 2 i c † k ↑ c k ′ ↓ c † qσ | ø i γ † n, − σ γ † m ↓ | M + 2 i c † k ↑ c k ′ ↓ | ø i γ † m ↓ | M + 2 i − u m v m v n Θ( ξ q )( E c − ξ q − E n )( − ξ q − E n − E m )( E c − E m ) c k ′ ↓ c † q ↑ | ø i γ † n ↓ | M + 2 i c k ′ ↓ | ø i γ † m ↑ γ † n ↓ | M + 2 i c k ′ ↓ c † k ↑ | ø i γ † n ↓ | M + 2 i − u n v n u m Θ( ξ q )( E c − ξ q − E n )( − E n − E m )( E c − E n ) c k ′ ↓ c † q ↓ | ø i γ † n ↑ | M + 2 i c † q ↓ | ø i γ † m ↑ γ † n ↑ | M + 2 i | ø i γ † n ↑ | M + 2 i u n v n v m Θ( ξ q )( E c − ξ q − E n )( − ξ q − E n − E m )( E c − E n ) c k ′ ↓ c † q ↑ | ø i γ † n ↓ | M + 2 i c † q ↑ | ø i| M i | ø i γ † m ↑ | M + 2 i − u n v n u m Θ( ξ q )( E c − ξ q − E n )( − ξ q )( E c − E m ) c k ′ ↓ c † q ↓ | ø i γ † n ↑ | M + 2 i c † k ↑ c k ′ ↓ c † q ↓ | ø i| M i c † k ↑ c k ′ ↓ | ø i γ † m ↓ | M + 2 i − u n v n u m Θ( ξ q )( E c − ξ q − E n )( − ξ q )( E c − E m ) c k ′ ↓ c † q ↑ | ø i γ † n ↓ | M + 2 i c † q ↑ | ø i| M i c † q ↑ c † k ↑ | ø i γ † m ↓ | M i u n v n v m Θ( ξ q )( E c − ξ q − E n )( − ξ q )( − ξ q − E c − E m ) c k ′ ↓ c † qσ | ø i γ † n, − σ | M + 2 i c † qσ | ø i γ † m ↑ γ † n, − σ | M + 2 i c † qσ c † k ↑ | ø i γ † n, − σ | M i − u m v m v n Θ( ξ q )( E c − ξ q − E n )( − ξ q − E n − E m )( − ξ q − E c − E n ) C c k ′ ↓ c † q ↓ | ø i γ † n ↑ | M + 2 i c † q ↓ | ø i γ † m ↑ γ † n ↑ | M + 2 i c † q ↓ c † k ↑ | ø i γ † m ↑ | M i u n v n v m Θ( ξ q )( E c − ξ q − E n )( − ξ q − E n − E m )( − ξ q − E c − E m ) c k ′ ↓ c † q ↓ | ø i γ † n ↑ | M + 2 i c † k ↑ c k ′ ↓ c † q ↓ | ø i| M i c † k ↑ c † q ↓ | ø i γ † m ↑ | M i u n v n v m Θ( ξ q )( E c − ξ q − E n )( − ξ q )( − ξ q − E c − E m ) c k ′ ↓ c † q ↑ | ø i γ † n ↓ | M + 2 i c † k ↑ c k ′ ↓ c † q ↑ | ø i γ † n ↓ γ † m ↓ | M + 2 i c † k ↑ c k ′ ↓ | ø i γ † n ↓ | M + 2 i − u n v n v m Θ( ξ q )( E c − ξ q − E n )( ξ q + E n + E m )( E c − E n ) c k ′ ↓ c † q ↓ | ø i γ † n ↑ | M + 2 i c † k ↑ c k ′ ↓ c † q ↓ | ø i γ † n ↑ γ † m ↓ | M + 2 i c † k ↑ c † q ↓ | ø i γ † n ↑ | M i − u m v m v n Θ( ξ q )( E c − ξ q − E n )( − ξ q − E n − E m )( − ξ q − E c − E n ) c k ′ ↓ c † q ↑ | ø i γ † n ↓ | M + 2 i c † k ↑ c k ′ ↓ c † q ↑ | ø i γ † n ↓ γ † m ↓ | M + 2 i c † k ↑ c † q ↑ | ø i γ † n ↓ | M i − u m v m v n Θ( ξ q )( E c − ξ q − E n )( − ξ q − E n − E m )( − ξ q − E c − E n ) C c k ′ ↓ c † q ↑ | ø i γ † n ↓ | M + 2 i c † k ↑ c k ′ ↓ c † q ↑ | ø i γ † n ↓ γ † m ↓ | M + 2 i c † k ↑ c † q ↑ | ø i γ † m ↓ | M i u n v n v m Θ( ξ q )( E c − ξ q − E n )( − ξ q − E n − E m )( − ξ q − E c − E m ) c k ′ ↓ c † q ↓ | ø i γ † n ↑ | M + 2 i c k ′ ↓ | ø i γ † n ↑ γ † m ↓ | M + 2 i c k ′ ↓ c † k ↑ | ø i γ † m ↓ | M i − u n v n u m Θ( ξ q )( E c − ξ q − E n )( − E n − E m )( E c − E m ) c k ′ ↓ c † q ↑ | ø i γ † n ↓ | M + 2 i c k ′ ↓ | ø i γ † n ↓ γ † m ↑ | M + 2 i | ø i γ † m ↑ | M i − u n v n u m Θ( ξ q )( E c − ξ q − E n )( − E n − E m )( E c − E m ) TABLE IV: Continuation of Table IIILabel | n i | n i | n i Contribution to A (4) i → f (in units of t )16 c k ′ ↓ c † k ↑ | ø i γ † n ↓ | M + 2 i c k ′ ↓ c † k ↑ c † qσ | ø i γ † m, − σ γ † n ↓ | M + 2 i c k ′ ↓ c † k ↑ | ø i γ † n ↓ | M + 2 i u n v n v m Θ( ξ q )( E c − E n ) ( ξ q + E n + E m ) c k ′ ↓ c † k ↑ | ø i γ † n ↓ | M + 2 i c k ′ ↓ c † k ↑ c qσ | ø i γ † mσ γ † n ↓ | M + 2 i c k ′ ↓ c † k ↑ | ø i γ † n ↓ | M + 2 i − u n v n u m Θ( − ξ q )( E c − E n ) ( ξ q − E n − E m ) c k ′ ↓ c † k ↑ | ø i γ † n ↓ | M + 2 i c † k ↑ | ø i γ † n ↓ γ † m ↑ | M + 2 i c † k ↑ c q ↑ | ø i γ † m ↑ | M + 2 i u m v m v n Θ( − ξ q )( E c − E n )( E n + E m )( ξ q + E c − E m ) c k ′ ↓ c † k ↑ | ø i γ † n ↓ | M + 2 i c k ′ ↓ c † k ↑ c † q ↑ | ø i γ † m ↓ γ † n ↓ | M + 2 i c k ′ ↓ c † k ↑ | ø i γ † m ↓ | M + 2 i − u m v m v n Θ( ξ q )( E c − E n )( ξ q + E n + E m )( E c − E m ) c k ′ ↓ c † k ↑ | ø i γ † n ↓ | M + 2 i c k ′ ↓ c † k ↑ c q ↑ | ø i| M + 2 i c k ′ ↓ c † k ↑ | ø i γ † m ↓ | M + 2 i − u m v m v n Θ( − ξ q )( E c − E n ) ξ q ( E c − E m ) c k ′ ↓ c † k ↑ | ø i γ † n ↓ | M + 2 i c k ′ ↓ c † k ↑ c q ↑ | ø i| M + 2 i c † k ↑ c q ↑ | ø i γ † m ↑ | M + 2 i − u m v m v n Θ( − ξ q )( E c − E n ) ξ q ( E c + ξ q − E m ) c k ′ ↓ c † k ↑ | ø i γ † n ↓ | M + 2 i c † k ↑ | ø i γ † n ↓ γ † m ↑ | M + 2 i c † k ↑ c q ↓ | ø i γ † n ↓ | M + 2 i u n v n v m Θ( − ξ q )( E c − E n )( E n + E m )( E c + ξ q − E n ) c k ′ ↓ c † k ↑ | ø i γ † n ↓ | M + 2 i c k ′ ↓ c † k ↑ c † q ↓ | ø i| M i c k ′ ↓ c † k ↑ | ø i γ † m ↓ | M + 2 i u n v n u m Θ( ξ q )( E c − E n ) ξ q ( E c − E m ) c k ′ ↓ c † k ↑ | ø i γ † n ↓ | M + 2 i c k ′ ↓ c † k ↑ c † q ↓ | ø i| M i c † k ↑ c † q ↓ | ø i γ † m ↑ | M i u n v n v m Θ( ξ q )( E c − E n ) ξ q ( ξ q +3 E c + E m ) c k ′ ↓ c † k ↑ | ø i γ † n ↓ | M + 2 i c k ′ ↓ c † k ↑ c † q ↑ | ø i γ † m ↓ γ † n ↓ | M + 2 i c † k ↑ c † q ↑ | ø i γ † n ↓ | M i u m v m v n Θ( ξ q )( E c − E n )( ξ q + E n + E m )( ξ q +3 E c + E n ) E c k ′ ↓ c † k ↑ | ø i γ † n ↓ | M + 2 i c k ′ ↓ c † k ↑ c qσ | ø i γ † mσ γ † n ↓ | M + 2 i c † k ↑ c qσ | ø i γ † mσ | M i − u n v n u m Θ( − ξ q )( E c − E n )( ξ q − E m − E n )( E c + ξ q − E m ) c k ′ ↓ c † k ↑ | ø i γ † n ↓ | M + 2 i c † k ↑ | ø i γ † n ↓ γ † m ↑ | M + 2 i c † k ↑ c † q ↑ | ø i γ † n ↓ | M i u m v m v n Θ( ξ q )( E c − E n )( E n + E m )( ξ q +3 E c + E n ) D c k ′ ↓ c † k ↑ | ø i γ † n ↓ | M + 2 i c k ′ ↓ c † k ↑ c † qσ | ø i γ † m, − σ γ † n ↓ | M + 2 i c † k ↑ c † qσ | ø i γ † m, − σ | M i − u n v n v m Θ( ξ q )( E c − E n )( ξ q + E n + E m )( ξ q +3 E c + E m ) c k ′ ↓ c † k ↑ | ø i γ † n ↓ | M + 2 i c k ′ ↓ c † k ↑ c q ↓ | ø i γ † m ↓ γ † n ↓ | M + 2 i c k ′ ↓ c † k ↑ | ø i γ † m ↓ | M i u n v n u m Θ( − ξ q )( E c − E n )( ξ q − E n − E m )( E c − E m ) c k ′ ↓ c † k ↑ | ø i γ † n ↓ | M + 2 i c k ′ ↓ c † k ↑ c q ↓ | ø i γ † m ↓ γ † n ↓ | M + 2 i c † k ↑ c q ↓ | ø i γ † n ↓ | M + 2 i u n v n u m Θ( − ξ q )( E c − E n )( ξ q − E n − E m )( E c + ξ q − E n ) c k ′ ↓ c † k ↑ | ø i γ † n ↓ | M + 2 i c † k ↑ | ø i γ † n ↓ γ † m ↑ | M + 2 i c † k ↑ c † q ↓ | ø i γ † m ↑ | M i u n v n v m Θ( ξ q )( E c − E n )( − E n − E m )( − E c − ξ q − E m ) D c k ′ ↓ c qσ | ø i γ † nσ | M + 2 i c qσ | ø i γ † nσ γ † m ↑ | M + 2 i | ø i γ † m ↑ | M + 2 i − u m v m u n Θ( − ξ q )( ξ q − E c − E n )( ξ q − E n − E m )( E c − E m ) A c k ′ ↓ c qσ | ø i γ † nσ | M + 2 i c k ′ ↓ c qσ c † k ↑ | ø i γ † nσ γ † m ↓ | M + 2 i c k ′ ↓ c † k ↑ | ø i γ † m ↓ | M + 2 i − u m v m u n Θ( − ξ q )( ξ q − E c − E n )( ξ q − E n − E m )( E c − E m ) c k ′ ↓ c q ↓ | ø i γ † n ↓ | M + 2 i c q ↓ | ø i| M + 2 i | ø i γ † m ↑ | M + 2 i u m v m u n Θ( − ξ q )( ξ q − E c − E n ) ξ q ( E c − E m ) c k ′ ↓ c q ↓ | ø i γ † n ↓ | M + 2 i c q ↓ | ø i| M + 2 i c † k ↑ c q ↓ | ø i γ † m ↓ | M + 2 i u m v m u n Θ( − ξ q )( ξ q − E c − E n ) ξ q ( ξ q + E c − E m ) c k ′ ↓ c q ↑ | ø i γ † n ↑ | M + 2 i c k ′ ↓ c q ↑ c † k ↑ | ø i| M + 2 i c k ′ ↓ c † k ↑ | ø i γ † m ↓ | M + 2 i u m v m u n Θ( − ξ q )( ξ q − E c − E n ) ξ q ( E c − E m ) c k ′ ↓ c q ↑ | ø i γ † n ↑ | M + 2 i c k ′ ↓ c q ↑ c † k ↑ | ø i| M + 2 i c q ↑ c † k ↑ | ø i γ † m ↑ | M + 2 i u m v m u n Θ( − ξ q )( ξ q − E c − E n ) ξ q ( ξ q + E c − E m ) c k ′ ↓ c qσ | ø i γ † nσ | M + 2 i c qσ | ø i γ † nσ γ † m ↑ | M + 2 i c † k ↑ c qσ | ø i γ † nσ | M i − u m v m u n Θ( − ξ q )( ξ q − E c − E n )( ξ q − E n − E m )( ξ q + E c − E n ) c k ′ ↓ c qσ | ø i γ † nσ | M + 2 i c k ′ ↓ c qσ c † k ↑ | ø i γ † nσ γ † m ↓ | M + 2 i c qσ c † k ↑ | ø i γ † nσ | M i − u m v m u n Θ( − ξ q )( ξ q − E c − E n )( ξ q − E n − E m )( ξ q + E c − E n ) c k ′ ↓ c q ↓ | ø i γ † n ↓ | M + 2 i c k ′ ↓ | ø i γ † m ↑ γ † n ↓ | M + 2 i c k ′ ↓ c † k ↑ | ø i γ † n ↓ | M i u m v m u n Θ( − ξ q )( ξ q − E c − E n )( − E n − E m )( E c − E n ) c k ′ ↓ c q ↑ | ø i γ † n ↑ | M + 2 i c q ↑ | ø i γ † n ↑ γ † m ↑ | M + 2 i | ø i γ † n ↑ | M i u m v m u n Θ( − ξ q )( ξ q − E c − E n )( ξ q − E n − E m )( E c − E n ) c k ′ ↓ c q ↓ | ø i γ † n ↓ | M + 2 i c k ′ ↓ c q ↓ c † k ↑ | ø i γ † n ↓ γ † m ↓ | M + 2 i c k ′ ↓ c † k ↑ | ø i γ † n ↓ | M i u m v m u n Θ( − ξ q )( ξ q − E c − E n )( ξ q − E n − E m )( E c − E n ) c k ′ ↓ c q ↑ | ø i γ † n ↑ | M + 2 i c k ′ ↓ | ø i γ † m ↓ γ † n ↑ | M + 2 i | ø i γ † n ↑ | M i u m v m u n Θ( − ξ q )( ξ q − E c − E n )( − E n − E m )( E c − E n ) c k ′ ↓ c q ↑ | ø i γ † n ↑ | M + 2 i c k ′ ↓ | ø i γ † m ↓ γ † n ↑ | M + 2 i c k ′ ↓ c † k ↑ | ø i γ † m ↓ | M + 2 i u m v m u n Θ( − ξ q )( ξ q − E c − E n )( − E n − E m )( E c − E m ) c k ′ ↓ c q ↓ | ø i γ † n ↓ | M + 2 i c k ′ ↓ | ø i γ † m ↑ γ † n ↓ | M + 2 i | ø i γ † m ↑ | M + 2 i u m v m u n Θ( − ξ q )( ξ q − E c − E n )( − E n − E m )( E c − E m ) C c k ′ ↓ c q ↑ | ø i γ † n ↑ | M + 2 i c q ↑ | ø i γ † n ↑ γ † m ↑ | M + 2 i c † k ↑ c q ↑ | ø i γ † m ↑ | M + 2 i u m v m u n Θ( − ξ q )( ξ q − E c − E n )( ξ q − E n − E m )( ξ q + E c − E m ) C c k ′ ↓ c q ↓ | ø i γ † n ↓ | M + 2 i c k ′ ↓ c q ↓ c † k ↑ | ø i γ † n ↓ γ † m ↓ | M + 2 i c † k ↑ c q ↓ | ø i γ † m ↓ | M + 2 i u m v m u n Θ( − ξ q )( ξ q − E c − E n )( ξ q − E n − E m )( ξ q + E c − E mm