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Random Kondo alloys S. Burdin1 and P. Fulde1 1 Max-Planck-Institut fu¨r Physik komplexer Systeme, Nöthnitzer Strasse 38, 01187 Dresden, Germany 7 Abstract 0 0 The interplaybetweenthe Kondoeffect anddisorderis studied. This is done by applyinga matrixcoherentpotential 2 approximation(CPA)andtreatingthe Kondointeractiononamean-fieldlevel. The resulting equationsareshownto n agree with those derived by the dynamical mean-field method (DMFT). By applying the formalism to a Bethe tree a structure with infinite coordination the effect of diagonal and off-diagonal disorder are studied. Special attention is J paidto the behaviorofthe Kondo-andthe Fermiliquid temperature asfunction ofdisorderandconcentrationofthe 4 Kondo ions. The non monotonous dependence of these quantities is discussed. 2 ] l I. INTRODUCTION e - r The Kondoeffect is oneofthe mostinvestigatedphenomena insolid-statephysics. Partofthe reasonis thatit can t s not be treated perturbationally since it is a strong coupling effect. Therefore it requires special theoretical tools to . t deal with it. Kondo physics occurs when strongly correlated electrons like 4f electrons in Ce3+ or holes in Yb3+ are a m weakly hybridizing with the conduction electrons of their surrondings. This results in low-energy excitations which in the case of concentrated systems may result in heavy quasiparticles. For recent reviews of the field we refer to - d Refs. [1-2-3]. A realistic starting point for Kondo systems is the Anderson impurity or Anderson lattice model. Due n to the hybridization mentioned above it involves spin as well as charge degrees of freedom. Often the charge degrees o of freedom are less interesting and are therefore eliminated by a Schrieffer-Wolf transformation [4]. The result is an c antiferromagneticinteractionbetweenthe spinsofthe conductionelectronsandthestronglycorrelatedlocalised,e.g., [ 4f electrons. This leads to the Kondo Hamiltonian. 1 Competing with the Kondo effect is the Ruderman-Kittel-Kasuya-Yoshida(RKKY) interaction. While the Kondo v effect leads to the formationofa singletbetween the spins ofthe 4f andconduction electrons,the RKKYinteraction 8 lowers the energy of a system of local spins interacting with each other via conduction electrons. Therefore if the 9 latter is more important than the former, the local spins will remain uncompensated and eventually order and not 5 participate in the singlet formation. 1 0 TheaimofthepresentinvestigationistostudytheeffectofdisorderonKondophysics[5-6-8-9-10-7-11]. Ithasbeen 7 suggestedinseveralworksthatdisorderleadsto non-Fermi-liquidbehavior(NFL)atlowtemperatures. Forexample, 0 it has been shown in Refs. [5-6-7] that a distribution of Kondo temperatures T can result from local disorder, the K / NFLfeaturesbeeingrelatedtothepresenceofvery-low-T spinswhichremainunquenchedatanyfinitetemperature. t K a Another possible scenario attributes the NFL behavior to the proximity to a quantum critical point resulting from m disorderedRKKY interactions [8-9-10]. More recently,it has been suggestedthat a NFL behavior canoccur between - the local Fermi-liquid (FL) and coherent heavy FL phases characterizing respectively a diluted and a dense Kondo d alloy [11]. n We assume that we are in a regime where the Kondo effect is more important than the RKKY interaction so that o c the latter may be neglected. Instead we concentrate on the singlet formation energy and on how it can be expressed : intermsoftheKondotemperatureT andofthetemperatureT atwhichthedifferentlowenergyexcitationsform v K FL coherent quasiparticles. In particular we study how T and T behave as functions of conduction electron band i K FL X filling n , local spin concentration x and disorder. c r a II. MODEL HAMILTONIAN AND METHODS OF SOLUTIONS We consider the Kondo alloy model (KAM) with the Hamiltonian H = t c† c + JK Sσσ′c† c , (1) ij iσ jσ 2 i iσ′ iσ ijσ i∈Aσσ′ X XX where the first term describes nearest-neighbor hopping of conduction electrons on a lattice with sites occupied randomly by atoms of kind A and B. The corresponding concentrations are c = x and c = 1 x. The hopping A B − 2 matrix elements have three different values, i.e., t if i,j A A ∈ t =γ t if i,j B . (2) ij ij B  ∈ t otherwise AB  Hereγij isthestructurefactoroftheunderlyingperiodiclatticewithFouriertransformγk ≡ ijγijexpik.(Ri−Rj). The second term in Eq. (1) describes the Kondo interaction between the conduction electron spin and local spin operators S , the latter being attached to atoms of type A only. P i We shalltreatthe Kondo alloymodeldefined by Eq.(1) by applying a number ofapproximations. A rathersimple oneisthatweassumearandomdistributionofsitesAandB. AsregardstheKondointeractionweshallconsidertwo different ways of treating the randomness. They are similar to each other but based on different physical pictures. One approach is a generalisation of a CPA matrix approach originally introduced in Refs [12-13]. The Kondo interaction is treated here within a mean-field approximation. The second approachis a matrix generalisationof the dynamical mean field theory (DMFT) which is exact in the limit of infinite dimensions [14-15]. Averaging over the randomness is done here without simplifying the Kondo interaction. A mean-field approximation can be introduced before or after the DMFT approximation and leads to the same set of self-consistent equations as obtained in the first, i.e., the generalised CPA approach. Theanalyticalexpressionsobtainedfromthesetwoapproachesareapplicabletoanylatticestructure. Thenumerical resultspresentedbelowapplytheDMFT toaBethelatticeinsteadofaregularone. TheKondointeractionistreated in this case within the mean-field approximation. III. THE MATRIX-CPA METHOD A. Mean-field treatment of the Kondo interaction We begin with a mean-field approximation for the Kondo interaction. Following the standard theory [16-17-18- 19-20], the spin operators are written in the fermionic representation Sσiσ′ = fi†σfiσ′ −δσσ′/2, with the constraint f† f =1. The Hamiltonian Eq. (1) becomes therefore σ iσ iσ P J H = tijc†iσcjσ + 2K fi†σfiσ′c†iσ′ciσ . (3) ijσ i∈Aσσ′ X XX The systems we want to describe here involve physical spins 1/2 with a SU(2) symmetry. The mean-field approach as introduced in Refs [16-17] is in this case an approximation which becomes exact in the limit of SU(N ) → ∞ symmetry [18-19-20]. The Hamiltonian is n 1 H = t c† c +r c† f +f† c µ c† c c λ f† f . (4) ij iσ jσ iσ iσ iσ iσ − iσ iσ− 2 − iσ iσ − 2 Xijσ Xσ Xi∈A(cid:16) (cid:17) Xσ Xi (cid:16) (cid:17) Xσ Xi∈A(cid:18) (cid:19) where n is the average number of conduction electrons per site i while µ denotes the chemical potential. In the c following, we discard the spin index σ since in mean-field approximation the contributions to H of the different spin components decouple [16-17-18-19-20]. The Kondo interaction is approximated by an effective hybridization r = J [ f c† ] between the conduction electrons and the fermionic operators, where the denotes the thermal K h i ii h···i averagewithrespectto the Hamiltonian(4)for randomconfigurationsofsitesAandB,and[ ]denotesthe average ··· with respect to these configurations. Note that the same form of the Hamiltonian is obtained by starting from an Anderson lattice instead of a Kondo lattice, and treating it within the mean-field slave boson approximation [18-19- 20]. WehavestartedherefromtheKondoHamiltonianbecauseweareinterestedinthecaseofnearintegervalencyof the impurity, i.e.,thef electroncountis supposedtobe verycloseto one. The abovemean-fieldapproximationleads to an f like band. It models the low-energy excitations which result from the Kondo interaction or alternatively Anderso−n Hamiltonian. The conditions f† f =1 are taken into account by Lagrange parameters λ . We set all σ iσ iσ i ofthemequaltoλwhichimpliesthattheaboveconditionsaresatisfiedonaverageonly. Thussmalllocalfluctuations P in the f electron count are possible here like in the Anderson model. The quantities µ, λ and r are determined by self-consistency conditions. For that purpose local Green’s functions areintroduced. TheyaredifferentformagneticsitesA,nonmagneticsitesB andforf aswellasconductionelectrons. The Gff(τ τ′) T f (τ)f†(τ′) , Gfc(τ τ′) T f (τ)c†(τ′) , and Gcc(τ τ′) T c (τ)c†(τ′) are finite ij − ≡ −h τ i j i ij − ≡ −h τ i j i ij − ≡ −h τ i j i 3 temperature Green’s functions defined for imaginary time τ, where T denotes the imaginary-time chronological τ ordering. We also define the averagedlocal Green’s functions 1 1 Gff Gff , Gfc Gfc , (5) A ≡ x ii A ≡ x ii i∈A i∈A X X 1 1 Gcc Gcc , Gcc Gcc . (6) A ≡ x ii B ≡ 1 x ii i∈A − i∈B X X The chemical potential µ, the Lagrange multiplier λ, and the effective hybridization r are determined by the self-consistent saddle point equations: r/J = 1 f c† =Gfc(τ =0−) , (7) − K x i∈Ah i ii A 1/2= x1 Pi∈Ahfi†fii =GfAf(τ =0−) , (8) nc/2= Pihc†icii =Gcc(τ =0−) , (9) with Gcc Gcc =xGcc+(1 x)Gcc. P ≡ i ii A − B P B. Configuration averages ForthedeterminationoftheGreen’sfunctionoftheconductionelectronswechooseageneralizationoftheCoherent PotentialApproximation(CPA)toamatrixformasintroducedinRefs[12-13]. Withinthatapproximation,thesystem can be viewed as a medium with three interacting fermionic bands: two bands correspondingto conduction electrons on sites A or B, and a third one representing the excitations of the strongly correlated f electrons. Therefore the − dynamics related to the spins of the A sites is described in a simplified form, i.e., in the form of f electrons with a − dispersive band. The 3 3 Green’s function matrix is of the following form × xˆ xˆ Gff xˆ xˆ Gfc xˆ yˆ Gfc i j ij i j ij i j ij G = xˆ xˆ Gcf xˆ xˆ Gcc xˆ yˆ Gcc , (10) ij  i j ij i j ij i j ij  yˆxˆ Gcf yˆxˆ Gcc yˆyˆ Gcc e  i j ij i j ij i j ij    where xˆ =1 yˆ are projection operators, which are unity (zero) if site i is occupied by an A (B) atom. Averaging i i − the local Green’s function matrix with respect to different configurations of randomly distributed types of atoms we find xGff xGfc 0 A A Gii = xGcAf xGcAc 0  . (11) 0 0 (1 x)Gcc h i − B e   In this expression, the vanishing of the mixed A B matrix elements follows directly from xˆ yˆ = 0, which ensures i i − that a givensite is either of kind A or B. Averagingoverthe different configurationsof A and B sites restoreslattice translation symmetry. Therefore we define Gk e−ik.(Ri−Rj) Gij (12) ≡ Xij h i e e Within the single component CPA, the systemis approximatedby an effective medium, characterisedby a local, i.e., k independent, but frequency dependent self-energy [21-22-23-24-25]. The latter is determined self-consistently by − requiring that the scattering matrix of the atoms A and B within this effective medium vanishes on average. The matrix form of the CPA introduced in Refs [12-13] generalises the scalar procedure to an effective medium with two bands of conduction electrons. Here we generalise the 2 2 matrix form of the CPA to a 3 3 one. The averaged × × Green’s function matrix characterising the effective medium is given by the relation −1 G(iω ) =iω Iδ Σ(iω )δ Wγ , (13) n n ij n ij ij − − (cid:18)h i (cid:19)ij where iω iπT(2n+1) denotes tehe fermionic Matsubeara freequencies. Infthe following we leave out the n index. n ≡ Invoking the reciprocal space Green’s function matrix defined by Eq. (12), the relation (13) becomes G−k1(iω)=iωI Σ(iω) Wγk . (14) − − e e e f 4 Here I is a 3 3 unit matrix, W is the transfer matrix, × 0 0 0 e f W = 0 t t , (15) A AB   0 t t AB B f   and Σ is a local self-energy matrix, σ (iω) e Σ (iω) 1 Σ(iω) A σ (iω) . (16) 2 ≡  σ (iω) σ (iω) Σ (iω) 1 2 B e   which is determined by the set of self-consistent equations (see appendix A): xGff(iω) xGfc(iω) 0 A A Gii(iω) = xGcAf(iω) xGcAc(iω) 0 = Gk(iω) , (17) h i 0 0 (1−x)GcBc(iω) Xk e   e λ r (1 x) Gff(iω) Gfc(iω) −1 Σ (iω) = − A A , (18) A − r µ − x Gcf(iω) Gcc(iω) (cid:18) (cid:19) (cid:18) A A (cid:19) x Σ (iω) = µ . (19) B − − (1 x)Gcc(iω) − B The self-energies σ and σ are determined by requiring that the mixed A B elements of the local Green’s function 1 2 − matrix in Eq. (17) vanish for the same reason as in Eq. (10). We find that σ (iω)=0 , (20) 1 which reflects the fact that there is no direct interaction between f fermions and the B electronic band, describing − − the electrons on nonmagnetic sites. We find also an explicit expression for σ , i.e., 2 1 2x+x(iω+µ r2/(iω+λ))Gcc(iω) (1 x)(iω+µ)Gcc(iω) σ (iω)= t − − A − − B , (21) 2 − AB xt Gcc(iω) (1 x)t Gcc(iω) A A − − B B which results from the direct hopping of conduction electrons between A and B sites. A complete solutionof the initial Kondo alloy systemis obtainedby solving simultaneously the mean-fieldEqs.(7, 8, 9) together with the matrix Eq.(17). Thereby the k dependent averageGreen’s function matrix is determinedby − the relationEq.(14), withthe lattice structurefactorγk andthe localself-energymatrixΣ. Thelatter isdetermined by the self-consistent CPA Eqs. (16, 18, 19). e C. Kondo temperature We define the Kondo temperature T as the temperature at which the effective hybridization r (obtained from K Eqs. (7, 8, 9)) vanishes. We find 2/J = dǫρ0(ǫ+µ )tanh[(ǫ/2T ]/ǫ , (22) K A 0 K Z where ρ0 and µ are respectively the local electronic density of states (DOS) on a magnetic site, and the chemical A 0 potential of a random A B alloy without Kondo interaction. An explicit expression for T has been derived in K − Ref. [26] in the weak-coupling regime J <<t ,t ,t : K A B AB TK =De−1/(JKρ0A(EF)) 1 (EF/D)2FK(nc) , (23) − p D−EF dωρ0(E +ω) ρ0(E ) F (n )=exp A F − A F , (24) K c ω 2ρ0(E ) Z−(D+EF) | | A F ! where D is the half-bandwidth of the non-interacting local DOS ρ0(ω). A 5 IV. DMFT EQUATIONS The matrix form of the CPA introduced in Refs. [12-13] was generalized to the KAM (1) after an appropriate mean-field approximation was made for the Kondo interaction. It allowed for keeping the dynamical aspects of the A sites with their attached spins by means of introducing an additional f like band of excitations. It supplemented − the two bandsresulting fromthe conductionelectronsoftheA andB sites. Inthe followingwe developfor the KAM a matrix DMFT computational scheme which can be formulated without a mean-field approximation for the Kondo interaction. Note that our approach is different from the dynamical cluster approximation introduced in Ref. [27] since the latter concerns diagonal disorder only. A. DMFT matrix formalism for a binary alloy The Kondo-alloy Hamiltonian Eq. (1) can be written as H = γ P†WP c† c + JK xˆ Sσσ′c† c , (25) ij i j iσ jσ 2 i i iσ′ iσ ijσ i σσ′ X X X where we introduced the transfer matrix t t W= A AB , (26) t t AB B (cid:18) (cid:19) and the projection operators xˆ P = i , (27) i yˆ i (cid:18) (cid:19) with their conjugates P† = xˆ , yˆ , (28) i i i where xˆ 1 yˆ is unity if i is an A site and zero ot(cid:0)herwise. H(cid:1)ere, we have implicitly mapped the initial KAM (1), i i ≡ − characterised by a single disordered conduction band, into a two-band effective model. Thereby each site of the underlying periodic lattice acts like being occupied simultaneously by atoms of A and B type. As before, the initial physicalHilbertspacecorrespondingtoasinglekindofatompersiteisrecoveredbyintroducingprojectionoperators. TheyguaranteethatasiteactseitherasanAorB atom. Here,wefollowtheDMFTformalism[14-15],whichisexact inthe limitofalargecoordinationnumberz. Consideringthatthe energyofthe systemis anextensivequantity,this limitrequiresarescalingofthehoppingenergiest =γ P†WP =t˜ /√z,wheret˜ remainsfinite(i.e.,independent ij ij i j ij ij of z) when z . From the lattice Hamiltonian Eq. (25) we obtain a local effective action for site 0 →∞ S(xˆ0) = − βdτ βdτ′c†0σ(τ)P†0K(τ −τ′)P0c0σ(τ′)−xˆ0J2K βdτSσσ′(τ)c†0σ(τ)c0σ′(τ) . (29) σ Z0 Z0 σσ′ Z0 X X Here,thekernelKisa2 2matrix,whichisadynamicalgeneralizationoftheWeissfieldusualyintroducedforastatic mean-field approximatio×n. The projection operators P and P† select the diagonal matrix element (respectively ) of K depending on wether site 0 is occupied by0an A or0B atom. The resulting local effectKivAe action (xˆ ) B 0 K S remains a scalar quantity, which can have two values, i.e., (xˆ = 1) = and (xˆ = 0) = . This is a key 0 A 0 B S S S S quantityintheDMFTprocedure. ItprovidesarelevantsimplificationsincethelocalelectronicandmagneticGreen’s functions characterizing the lattice Hamiltonian (25) can now be computed from (xˆ ), which invokes local degrees 0 offreedomonly. Next, wedetermine the self-consistentrelationsallowingto expresSsthe kernelK asa function ofthe local Green’s functions. Following the standard DMFT formalism, we find K(iω)=(iω+µ)I γ γ W P P†G(0)(iω) W . (30) − 0i j0 i j ij Xij h i Here G(0) is the cavity Green’s function, corresponding to the lattice Hamiltonian (25), but with site 0 excluded. ij In order to establish a self-consistent relation for the kernel, we perform an infinite order perturbation expansion of 6 Green’s functions in terms of the hopping elements γ P†WP . Following the DMFT scheme [14-15], the Green’s ij i j functionG foragivendistributionofsitesAandB is expressedasasumofallpossiblepathsi i i i j ij 1 2 p → → ··· → connecting site i to site j through the sequence of structure factors γ . In the limit of large z we may exclude i1i2 returning paths since their contribution is of order 1/zn+1 when n is the number of returns. Thus, each path is factorised in terms of local dressed irreducible scalar propagators Π which contain information about the local ii interactions: G = Π γ P†WP Π γ P† WP Π Π γ P† WP Π . (31) ij ii ii1 i i1 i1i1 i1i2 i1 i2 i2i2··· ipip ipj ip j jj paths X For the sake of simplicity we have dropped the explicit time dependencies. We will show below that, after averaging over the randomness, these local propagators can be related to a local self-energy. The large z expansion Eq. (31) is a scalar relation,similarto the one obtainedin the usualDMFT approach. The only difference arisesfromthe scalar hopping elements t =P†WP γ , which here are random. In the following we cast this relation into a 2 2 matrix form, with a periodijic effecitive hjoipjping matrix Wγ between nearest neighbors. We define × ij xˆ xˆ G xˆ yˆ G G P G P† = i j ij i j ij , (32) ij ≡ i ij j yîxˆjGij yîyˆjGij (cid:18) (cid:19) and xˆ Π 0 Π P Π P† = i ii . (33) ii ≡ i ii i 0 yîΠii (cid:18) (cid:19) The large z expansion for the Green’s function matrix G is obtained by multiplying Eq. (31) with the projection ij − operators P† (from the left) and P (from the right). After averaging with respect to the different configurations of i j A and B sites, we find [G ]= Π γ W Π γ W Π Π γ W Π . (34) ij ii ii1 i1i1 i1i2 i2i2··· ipip ipj jj paths X (cid:2) (cid:3) In the large z limit, we consider only direct paths connecting sites i and j. We assume now that the occupation of − a givensite by an A or B atom is purely randomand does not depend on the configurationsof the neighboring sites. Therefore, in Eq. (34), each irreducible propagator matrix Π can thus be averagedseparately, and we find ii [G ]= Π γ W Π γ W Π Π γ W Π , (35) ij 0 ii1 0 i1i2 0··· 0 ipj 0 paths X where Π [Π ]. The matrix relation Eq. (35) between the average Green’s functions, the averaged local dressed 0 ii ≡ propagator,andthehoppingelementsisformallyidenticaltoascalarexpansionobtainedforaregularperiodicsystem within the standard DMFT formalism [14-15]. We introduce the averagedlocal Green’s function xG 0 G [G ]= A , (36) loc ≡ ii 0 (1 x)GB (cid:18) − (cid:19) whereG andG arethe localGreen’sfunctions correspondingto anAsite (respectivelyB site), onceaveragesover A B all the other sites configurations have been taken. Using Eq. (35), the relation between the cavity and full Green’s function reads P P†G(0) =[G ] [G ]G−1[G ] . (37) i j ij ij − i0 loc 0j h i Averaging over a random distribution of sites A and B restores the translation symmetry of the underlying lattice. The averagedGreen’s function matrices are thus periodic in space, and we can define their Fourier transforms Gk e−ik.(Ri−Rj)[Gij] . (38) ≡ ij X From Eq. (35), we find that the Green’s functions are characterised by a local 2 2 self-energy matrix Σ × G−k1(iω)=(iω+µ)I Σ(iω) Wγk , (39) − − 7 where Σ is relatedto the averagedlocalpropagatorbythe matrixidentity Π−1(iω)=(iω+µ)I Σ(iω). The matrix elements of Σ can be expressed in terms of the local average Green’s function0 matrix G by ta−king the inverse of loc Gloc(iω)= Gk(iω) . (40) k X Finally, using the relation (37) for the cavity Green’s function, together with the expression (30) for the kernel, we find K(iω)=Σ(iω)+G−1(iω) . (41) loc Equations (39, 40, 41) provide a self-consistent relation between the matrix kernel K and the averagedlocal Green’s function matrix G : loc Gloc(iω)= (iω+µ)I−K(iω)+G−lo1c(iω)−Wγk −1 . (42) k X(cid:0) (cid:1) In turn, the local Green’s functions G and G invoked in the definition (36) of G can be computed for a given A B loc kernel K, by considering the cases xˆ =0 and xˆ =1 in the local effective action Eq. (29). Since the effective action 0 0 on sites B is quadratic in terms of electronic opperators, we obtain an explicit expression for G : B B S G (iω)= −1(iω) . (43) B KB The local Green’s functions G is obtained from the local effective action on an A atom: A SA = − βdτ βdτ′c†0σ(τ)KA(τ −τ′)c0σ(τ′)− J2K βdτSσσ′(τ)c†0σ(τ)c0σ′(τ) . (44) σ Z0 Z0 σσ′ Z0 X X Appart from the self-consistent relation (42), which can be treated using analytic (and eventually numerical) simple calculations, the main difficulty consists in computing G from the local effective action (44). Even if the initial A difficulty of studying a lattice model has been consequently reduced into a single site effective model, this issue remains a many body problem. The Kondo interaction part has to be considered using a numerical scheme or appropriate analytical approximations. Onceaself-consistentsolutionisobtainedforKandG ,thek dependentcorrelationfunctionsfortheconduction loc − electronscanbeobtainedusingEqs.(39,41). Here,wedescribethe systemwithtwobandsofconductionelectronsA andB,whosecorrelationsarecharacterisedbythe2 2matrixGk. Invokingtheidentityxîxˆj+xîyˆj+yîxˆj+yîyˆj =1, the physicalsingle bandaverageGreen’s functions [G× ] canbe obtainedby adding the four matrix elements of[G ]. ij ij As a consequence, the k dependent averagecorrelationfunction for the physical single band of conduction electrons is also obtained by addin−g the four matrix elements of Gk. B. Equivalence of the CPA and the DMFT TheequivalenceofthedynamicalCPAandtheDMFTwaspreviouslyprovenbyKakehashiongeneralgrounds[28]. As discussed before by applying a 3 3 matrix CPA approach we were able to describe the important dynamical × aspects of the spins of the A sites. Therefore it is reassuring that we can demonstrate the equivalence of the 3 3 × matrixCPAapproachwithcorrespondingDMFTequations,whenweintegrateoverthef electrondegreesoffreedom − in the CPA approach and make a mean-field approximationwithin the DMFT approach. 1. Expression of the CPA equations using a 2 2 matrix formalism × For a demonstration of the equivalence of the two methods we start from a modified version of Eqs. (16- 19). It is easy to show that after some algebraic modifications the following relations can be derived from these equations: 1 r2 Gff(iω) = + Gcc(iω) , (45) A iω+λ (iω+λ)2 A r Gfc(iω) = − Gcc(iω) . (46) A iω+λ A 8 The self-consistent CPA Eqs. (16, 17, 18, 19) can be cast into the form 1 xGcAc(iω) = ∆k(iω) iω+µ−ΣCBPA(iω)−tBγk , (47) k X (cid:0) (cid:1) 1 (1−x)GcBc(iω) = ∆k(iω) iω+µ−ΣK(iω)−ΣCAPA(iω)−tAγk , (48) k X (cid:0) (cid:1) where ∆k(iω) = iω+µ−ΣK(iω)−ΣCAPA(iω)−tAγk iω+µ−ΣCBPA(iω)−tBγk −(−σ2(iω)−tABγk)2 , (49) (cid:0) (cid:1)(cid:0) (cid:1) 1 2x+x(iω+µ Σ (iω))G (iω) (1 x)(iω+µ)G (iω) K A B σ (iω) = t − − − − . (50) 2 AB − xt G (iω) (1 x)t G (iω) A A B B − − Here we have set 1 x 1 ΣCPA(iω) = − , (51) A − x Gcc(iω) A x 1 ΣCPA(iω) = . (52) B −1 xGcc(iω) − B We have also introduced the definition r2 Σ (iω) . (53) K ≡ iω+λ A complete resolution of the model is obtained by the following self-consistent scheme: (i) Calculate Gcc and Gcc from Eqs. (47, 48, 49, 51, 52, 53), as function of the mean-field parameters r, λ and µ. A B (ii) Calculate Gfc and Gff by using Eqs. (45, 46). A A (iii) Optimise the parameters r, λ and µ so as to satisfy the mean-field Eqs. (7, 8, 9). 2. DMFT and mean-field approximation for the Kondo term AcompleteresolutionofthematrixDMFTself-consistentrelationsrequiresanimpuritysolver,inordertocompute the local electronic Green’s functions G related to the local effective action given by Eq. (44). In order to A A S demonstrate the formal equivalence between the matrix-DMFT and the matrix-CPA approaches, we use the mean- field approximation for the impurity solver. Before, we define the local self-energy due to the Kondo interaction on sites A: Σ (iω) (iω) G−1(iω) (54) K ≡KA − A Since there is no local interaction on sites B, we have G (iω)= −1(iω). The relation (41) can be expressed as B KB Σ (iω)+ΣCPA(iω) (iω) Σ(iω)= K A KAB (55) (iω) ΣCPA(iω) (cid:18)KAB B (cid:19) with ΣCPA(iω) = ((1 x)/x)G−1(iω) (56) A − A ΣCPA(iω) = (x/(1 x))G−1(iω) (57) B − B These expressions are identical to Eqs. (51, 52) obtained within the matrix-CPA approach. Here, as within the CPA approach, the off-diagonal self-energy (denoted before σ ) is determined by requiring the vanishing of the AB 2 off-diagonal elements of G in Eq. (36). InKanalogy to Eq. (21), we find loc 1 2x+x(iω+µ Σ (iω))G (iω) (1 x)(iω+µ)G (iω) K A B (iω)=σ (iω)= t − − − − . (58) AB 2 AB K − xt G (iω) (1 x)t G (iω) A A B B − − 9 Combining Eq. (55) with the self-consistent relations Eqs. (36, 39, 40) we find 1 xGA(iω) = ∆k(iω) iω+µ−ΣCBPA(iω)−tBγk , (59) k X (cid:0) (cid:1) 1 (1−x)GB(iω) = ∆k(iω) iω+µ−ΣK(iω)−ΣCAPA(iω)−tAγk , (60) k X (cid:0) (cid:1) with ∆k(iω) = iω−µ−ΣK(iω)−ΣCAPA(iω)−tAγk iω−µ−ΣCBPA(iω)−tBγk −(−σ2(iω)−tABγk)2 , (61) which are formally(cid:0) equivalent to the relations Eqs (47, 4(cid:1)8(cid:0), 49) obtained from the mat(cid:1)rix form of the CPA approach. The matrix-DMFT approach developed here is performed without any approximation concerning the local Kondo interaction. An impurity solver is required in order to calculate the local Green’s functions from the local effective action defined in Eq. (44), and then to compute the Kondo self-energyΣ . For example, the mean-field approxi- A K S mationcanbe performedasdescribedinthe previoussection(CPA),leadingtothe samesetofsaddle pointrelations as Eqs. (7, 8, 9). This method, developped in the framework of a Kondo Alloy Model can be generalised to other alloy models with strong local correlations. C. Bethe lattice with infinite coordination The DMFT formalism described in the previous section is exact in the limit of an infinite coordination number z. It can be applied to any underlying periodic lattice, which has to be defined by its structure factor γk. In order to study numerically the Kondo Alloy Model defined by the Hamiltonian Eq. (1), it appears as very convenient to consider a Bethe lattice. For a similar approachapplied to ferromagneticsemiconductors see Ref [29]. In this specific case, the self-consistent equations are much simpler, and the general physical properties of the system are preserved. Applying the DMFT formalism described in the previous section to a Bethe lattice, we obtain a local effective action for the two kind of sites SA = − βdτ βdτ′c†Aσ(τ)KA(τ −τ′)cAσ(τ′)− J2K βdτSσσ′(τ)c†Aσ(τ)cAσ′(τ) , (62) σ Z0 Z0 σσ′ Z0 X X β β = dτ dτ′c† (τ) (τ τ′)c (τ′) . (63) SB − Bσ KB − Bσ σ Z0 Z0 X They are formaly equivalent to the compact expression Eq. (29). The main simplification obtained by considering a Bethe lattice rests on the fact that the cavity Green’s functions involved in Eq. (30) can now be replaced by local full Green’s functions. This procedure is exact in the limit of a large coordination number z. The Bethe lattice self-consistent relations for the Kernels and are thus A B K K (iω) = iω+µ xt˜2Gcc(iω) (1 x)t˜2 Gcc(iω) (64) KA − A A − − AB B (iω) = iω+µ xt˜2 Gcc(iω) (1 x)t˜2Gcc(iω) , (65) KB − AB A − − B B where, in the large z limit, the nearest-neightbor hoppings have been rescaled: t t˜ /√z, with similar definitions A A for t˜ and t˜ . We then apply the mean-field approximation,described in the first≡section, as an impurity solver for B AB the local effective action of an A site. Within mean-field approximation, the effective actions Eqs. (62, 63) are A S quadratic and the local Green’s functions can in turn be expressed explicitly as functions of the kernels and A B K K Gff(iω) Gfc(iω) iω+λ r −1 A A = , (66) (cid:18) GcAf(iω) GcAc(iω) (cid:19) (cid:18) r KA(iω)(cid:19) Gcc(iω) = −1(iω) . (67) B KB Together with Eqs.(64, 65) and with the mean-field equations Eqs.(7, 8, 9) we have a complete set of self-consistent relations for the local Green’s functions and the effective parameters r, λ and µ. 10 V. APPLICATIONS OF THE FORMALISM A. Off-diagonal randomness: non-magnetic random alloy 1. Formalism In this section we consider off-diagonal randomness, i.e., hopping matrix elements. The model is defined by the Hamiltonian Eq. (1) without the Kondo interaction. This is a standard situation for the CPA and we discuss this case here only because we want to combine it later with the Kondo problem. We know that the CPA misses certain localization effects. Their importance in connexion with Kondo effect has been discussed in Ref. [11]. Since the spin componentsaredecoupled,thesystemcorrespondstoarandomtight-bindingmodelofconductionelectrons,identical to the one considered in Refs. [12-13]. Thus G (τ τ′) T c (τ)c†(τ′) is the electron Green’s function defined ij − ≡ −h τ i j i forimaginarytime. Sinceherewedonotconsiderthe Kondointeraction,theself-consistentequationfortheaveraged Green’sfunction canequivalently be obtainedeither fromthe matrixformofthe CPAapproach(Eqs.(47, 48, 49,51, 52)) or from the matrix DMFT approach (Eqs. 56, 57, 59, 60, 61). In both cases the Kondo self-energy Σ =0. We K find 1 xGA(iω) = ∆k(iω) iω−ΣCBPA(iω)−tBγk , (68) k X (cid:0) (cid:1) 1 (1−x)GB(iω) = ∆k(iω) iω−ΣCAPA(iω)−tAγk , (69) k X (cid:0) (cid:1) where ∆k(iω) = iω−ΣCAPA(iω)−tAγk iω−ΣCBPA(iω)−tBγk −(−σ2(iω)−tABγk)2 , (70) with (cid:0) (cid:1)(cid:0) (cid:1) ΣCPA(iω) = ((1 x)/x)G−1(iω) , (71) A − A ΣCPA(iω) = (x/(1 x))G−1(iω) , (72) B − B and 1 2x+iω(xG (iω) (1 x)G (iω)) A B σ (iω)= t − − − . (73) 2 AB − xt G (iω) (1 x)t G (iω) A A B B − − Here G and G are the local Green’s functions G on a site i of kind A (or B respectively), obtained by averaging A B ii over all the other site configurations A or B. For the sake of simplicity, we drop the chemical potential µ. This convention implies that the Fermi level energy is zero when the electron band is half-filled. We define the local density of electronic states (DOS) associated with G and G A B ρ (ω) (1/π) mG (ω+i0+) , (74) A/B A/B ≡− I and the averagedlocal DOS ρ(ω)=xρ (ω)+(1 x)ρ (ω) . (75) A B − Here, within CPA or DMFT, all the local DOS’s of A sites are the same while a more accurate treatment would also show a spread there. It can be obtained by randomizing the distribution of the hopping matrix elements t , t and A B t . This could be done within the present DMFT formalism. By construction, ρ , ρ , and ρ have each a total AB A B spectral weight of unity, and A (B) atoms contribute with a weight x (1 x) to the averagedDOS ρ. − We didn’t find a general analytic solution for this set of equations, so that a numerical evaluation is required. Nevertheless, a dimensionless ratio emerges from these expressions α=t /√t t , (76) AB A B which compares the energy characterising the hopping of electrons between sites A and B with the hopping energies within an A and a B sublattice. Intuitively, if α > 1, the electronic levels of lowest energy will be dominated by hoppingbetweenA B neighboringsites. Inthe oppositecaseofα<1,hopping withinpureAorpure B sublattices − dominates.