Kondo resonance narrowing in d- and f-electron systems
aa r X i v : . [ c ond - m a t . s t r- e l ] J un Kondo resonance narrowing in d - and f -electron systems Andriy H. Nevidomskyy ∗ and P. Coleman Department of Physics and Astronomy, Rutgers University, Piscataway, N.J. 08854, USA (Dated: November 10, 2018)By developing a simple scaling theory for the effect of Hund’s interactions on the Kondo effect, we show howan exponential narrowing of the Kondo resonance develops in magnetic ions with large Hund’s interaction. Ourtheory predicts an exponential reduction of the Kondo temperature with spin S of the Hund’s coupled moment, alittle-known effect first observed in d-electron alloys in the 1960’s, and more recently encountered in numericalcalculations on multi-band Hubbard models with Hund’s interactions. We discuss the consequences of Kondoresonance narrowing for the Mott transition in d-band materials, particularly iron pnictides, and the narrow ESRlinewidth recently observed in ferromagnetically correlated f-electron materials.
PACS numbers: 75.20.Hr, 71.27.+a, 71.20.Be
The theory of the Kondo effect forms a cornerstone in cur-rent understanding of correlated electron systems [1]. Morethan forty years ago, experiments on d -electron materialsfound that the characteristic scale of spin fluctuations of mag-netic impurities, known as the Kondo temperature, narrowsexponentially with the size S of the impurity spin [2] (Fig1). An explanation of this effect was proposed [2] based onan early theory of the Kondo effect by Schrieffer [3], whofound that strong Hund’s coupling leads to a S -fold reduc-tion of the Kondo coupling constant. Surprisingly, interest inthis phenomenon waned after the 1960’s. Motivated by a re-cent resurgence of interest in f - and d -electron systems, espe-cially quantum critical heavy electron systems [4], and pnic-tide superconductors [5], this paper revisits this little-knownphenomenon, which we refer to as “Kondo resonance narrow-ing”, placing it in a modern context.The consequences of Kondo resonance narrowing haverecently been re-discovered in calculations on multi-orbitalHubbard and Anderson models [6, 7]. Numerical renormal-ization group studies found that the introduction of Hund’scoupling into the Anderson model causes an exponential re-duction in the Kondo temperature [6]. The importance ofHund’s effect has also arisen in the context of iron pnictidesuperconductors [8, 9], where it appears to play a key role inthe development of “bad metal” state in which the d -momentsremain unquenched down to low temperatures.In this paper, we show that Kondo resonance narrowing canbe simply understood within a scaling theory description ofthe multi-channel Kondo model with Hund’s interaction. Themain result is an exponential decrease of the Kondo temper-ature that develops when localized electrons lock together toform a large spin S , given by the formula ln T ∗ K ( S ) = ln Λ − (2 S ) ln (cid:18) Λ T K (cid:19) . (1)Here, T K is the “bare” spin / Kondo temperature and Λ = min ( J H S, U + E d , | E d | ) is the scale at which the locked spin S develops under the influence of a Hund’s coupling, while U and E d are the interaction strength and position of the bare d -level. Although this result is implicitly contained in the earlyworks of Schrieffer [3] and Hirst [10], a detailed treatment has T K FeFeCrFe MnMnMnMnHosts: S TiNiTi CoVCo FeCr
FIG. 1: Measured values [11] of the Kondo temperature T ∗ K in hostalloys Au, Cu, Zn, Ag, Mo, and Cd containing transition metal im-purities, plotted vs. the nominal size S of the spin. Solid line is thefit to Eq. (1) with Λ ≡ J H S . to our knowledge, not previously been given.To develop our theory, we consider K spin / impurityspins at a single site, ferromagnetically interacting via Hund’scoupling J H , each coupled to a conduction electron channelof band-width D via an antiferromagnetic interaction J : H = X k ,σ,µ ε k c † k σµ c k σµ − J H K X µ =1 s µ ! + J K X µ =1 s µ · σ µ , (2)where ε k is the conduction electron energy, µ = 1 , K is thechannel index and σ µ = P k c † k αµ σ αβ c k βµ is the conductionelectron spin density in channel µ at the origin. We implicitlyassume that Hund’s scale KJ H is smaller than D . When de-rived from an Anderson model of K spin- / impurities, then D = min ( E d + U, | E d | ) is the cross-over scale at which localmoments form while J = | V k F | (1 / ( E d + U ) + 1 / | E d | ) is theSchrieffer–Wolff form for the Kondo coupling constant [1],where V k F is the Anderson hybridization averaged over theFermi surface and E d < is negative.The behavior of this model is well understood in the two µ µ T χ( ) T IIIII I eff (b)(a)
Nozieres FL Locked large spins PM T K* T K* T K H
J S H J Sg /1 (Λ) T Λ FIG. 2: (a) Schematic showing the behavior of the running couplingconstant g eff (Λ) = J (Λ) ρK eff on a logarithmic scale, with K eff theeffective number of conduction electron channels per impurity spin( K eff = 1 in region I and K in regions II and III). (b) Schematicshowing effective moment µ eff ( T ) = T χ ( T ) in terms of the suscep-tibility χ ( T ) , showing the enhancement (15) in region II and the lossof localized moments due to Kondo screening in region III. extreme limits [12]: for J H = ∞ , the K spins lock to-gether, forming a K -channel spin S = K/ Kondo model.The opposite limit J H = 0 describes K replicas of the spin- / Kondo model. Paradoxically, the leading exponential de-pendence of the Kondo temperature on the coupling constant T K ∼ De − / Jρ in these two limits is independent of the sizeof the spin. However, as we shall see in the cross-over be-tween the two limits, the projection of the Hamiltonian intothe space of maximum spin leads to a (2 S ) -fold reduction inthe Kondo coupling constant.We now study the properties of this model as a function ofenergy scale or cut-off Λ . Qualitatively, we expect three dis-tinct regions depicted in Fig. 2(a):(I) Λ ≫ J H S : a spin- / disordered paramagnet character-ized by a high temperature Curie magnetic susceptibility χ I ( T ) = K (3 / gµ B ) k B T (3)(where g is the g-factor of the electron), with effective moment ( µ Ieff ) = 3 K/ ;(II) T ∗ K ≪ Λ ≪ J H S : an unscreened big spin S = K/ isformed above an emergent Kondo energy scale T ∗ K ;(III) Λ ≪ T ∗ K : the Nozi`eres Fermi liquid ground state of the K -channel S = K/ Kondo problem.We employ the “Poor Man’s scaling” approach [1, 13], inwhich the leading renormalization flows are followed as theelectrons in the conduction band are systematically decimatedfrom the Hilbert space. By computing the diagrams depictedin Fig. 3, we obtain the following renormalization group (RG)
FIG. 3: The diagrams appearing in (a) one-loop and (b) two-loop RGequations for Kondo coupling J (open circles), with solid lines denot-ing the conduction electron propagators and dashed line – the impu-rity spin. (c) The lowest order diagrams in the RG flow of Hund’scoupling (a square vertex denotes bare J H ). equations in region I:(I): d ( Jρ )d ln Λ = − Jρ ) + 2 ( Jρ ) (4) d ( J H ρ )d ln Λ = 4( Jρ ) J H ρ (5)where ρ is the density of states of the conduction electronsat the Fermi level. The first equation is the well-known betafunction for the Kondo model, which to this order is inde-pendent of Hund’s coupling. As we decimate the states ofthe conduction sea, reducing the band-width Λ down to theHund’s scale J H S , to leading logarithmic order we obtain ρJ (Λ) = 1ln (cid:16) Λ T K (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Λ= J H S ≡ ρJ I . (6)There is a weak downward-renormalization of the Hund’scoupling J H described by Eq. (5), which originates in the two-loop diagrams (Fig. 3c)). In the leading logarithmic approxi-mation, we may approximate J H by a constant.Once Λ is reduced below J H S , the individual local mo-ments become locked into a spin S = K/ , similar to the ef-fect discussed in Ref. 14 for the case of two impurities coupledby ferromagnetic RKKY interaction. The low-energy proper-ties of the system in region II are described by a Kondo modelof spin K/ with K conduction electron channels: H IIeff = J ∗ (Λ) K X µ =1 S · σ µ , (7)To obtain the value of J ∗ , we must project the original modelinto the subspace of maximum spin S . By the Wigner-Eckarttheorem, the matrix element of any vector operator acting inthe basis of states | S z i of big spin S = K/ is related by aconstant prefactor to S itself, i.e. h SS z | s µ | SS z i = g S h S z | S | S z i (8)Summing both sides of the equation over impurity index µ =1 , . . . K , one obtains h SS z | P µ s µ | SS z i = g S K S z . How-ever since P µ s µ = K s ≡ S , one arrives at the conclusionthat g S K = 1 , therefore determining the value of the constantcoefficient g S = 1 /K in Eq. (8). Comparing Eqs. (2) and (7),we arrive at the effective Kondo coupling: J ∗ = J/K. (9)This equation captures the key effect of crossover from regionI to region II in Fig. 2. This result was first derived in the earlywork on the multi-channel Kondo problem by Schrieffer [3],where the limit of J H → ∞ was implicitly assumed, and alsoappears for the particular case of K = 2 in the study of thetwo-impurity Kondo problem by Jayaprakash et al. [14].To one loop order, the scaling equation for J ∗ (Λ) in regionII is identical to that of region I (4), namely d ( J ∗ ρ ) / d ln Λ ≈− J ∗ ρ ) , though its size is K times smaller. To avoid thediscontinuous jump in coupling constant at the crossover, itis more convenient to consider g eff ≡ J (Λ) ρK eff , where theeffective number of channels K eff = 1 and K in regions Iand II respectively. This continuous variable satisfies(II): d g eff d ln Λ = − K g eff + 2 K g eff . (10)so the speed at which it scales to strong coupling becomes K times smaller in region II (see Fig. 2a). Solving this RGequation to leading order, and setting g eff (Λ = T ∗ K ) ∼ , weobtain T ∗ K ∼ ( J H S ) ( D/J H S ) K e − K Jρ for the renormalizedKondo scale. Comparing this with the bare Kondo scale T K ∼ De − / Jρ , we deduce T ∗ K ∼ J H S (cid:18) T K J H S (cid:19) K ≡ T K (cid:18) T K J H S (cid:19) K − , (11)from which formula (1) follows. This exponential suppressionof the spin tunneling rate can be understood as a result of a S -fold increase in the classical action associated with a spin-flip.These results are slightly modified when the two-loop termsin the scaling are taken into account. The expression for T K now acquires a pre-factor, T K = D √ Jρ e − / Jρ and J H isweakly renormalized so that T ∗ K = ( ˜ J H S ) (cid:18) T K √ K ˜ J H S (cid:19) K , (12)where ˜ J H is determined from the quadratic equation x − x (cid:18) x + 4ln( D/T K ) (cid:19) + 4 = 0 . (13)where x = ln( ˜ J H S/T K ) and x ≡ ln( J H S/T K ) .The magnetic impurity susceptibility in region II becomes χ ∗ imp = ( gµ B ) k B T S ( S + 1) − (cid:16) TT ∗ K (cid:17) + O (cid:16) TT ∗ K (cid:17) , (14)from which we see that the enhancement of the magnetic mo-ment at the crossover is given by (see Fig. 2b)) (cid:18) µ IIeff µ Ieff (cid:19) = K + 23 . (15) When the temperature is ultimately reduced below the ex-ponentially suppressed Kondo scale T ∗ K , the big spins S be-come screened to form a Nozi`eres Fermi liquid [15]. A phase-shift description of the Fermi liquid predicts that [12, 16] theWilson ratio W ≡ χ imp χ / γ imp γ is given by W K = 2( K + 2)3 ≡ (cid:18) µ IIeff µ Ieff (cid:19) , (16)which, compared with the classic result W = 2 for the one-channel model [15], contains a factor of the moment enhance-ment. This result holds in the extreme limit J H ≫ T K . Moregenerally, W depends on the ratio η = U ∗ /J ∗ H of a channel-conserving interaction U ∗ to an inter-channel Hund’s coupl-ing J ∗ H in the Fermi liquid phase-shift analysis, giving rise to W K ( η ) = 2 (cid:18) K − η ) + 1 (cid:19) . (17)On general grounds we expect η ∼ T K /J H .We end with a discussion of the broader implications ofKondo resonance narrowing for d - and f -electron materials.This phenomenon provides a simple explanation of the dras-tic reductions in spin fluctuation scale observed in the classicexperiments of the sixties [2]. Our treatment brings out theimportant role of Hund’s coupling in this process. One of theuntested predictions of this theory is a linear rise of the Wilsonratio W with spin S = K/ (16), from a value W [1] = W [5 /
2] = d -electron magnetic moments would be unobserv-able. This is, in essence, the situation for Ti impurities in gold,where the Kondo temperature of S = 1 moments becomesso high that magnetic behavior is absent below the meltingtemperature of gold. On the other hand, the Kondo resonancenarrowing effects of Hund’s interaction can become so severe,that the re-entry from region II into the quenched Fermi liquidbecomes too low to observe. This is the case for S = 5 / Mnin gold, where T ∗ K is so low that it has never been observed;the recent observation of a “spin frozen phase” in DMFT stud-ies [7] may be a numerical counterpart.What then, are the possible implications for dense d -electron systems? In those materials, the ratio of Kondo tem-perature to the Hund’s coupling will be strongly dependent onstructure, screening and chemistry. In cases where J H ≪ T K ,the physics of localized magnetic moments will be lost andthe d -electrons will be intinerant. On the other hand, the situ-ation where J H ≫ T K will almost certainly lead to long rangemagnetic order with localized d -electrons. Thus in multi-bandsystems, the criterion J H ∼ T K determines the boundary be-tween localized and itinerant behavior, playing the same roleas the condition U/D ∼ in one-band Mott insulators.These issues may be of particular importance to the ongoingdebate about the strength of electron correlations in the FeAsfamily of high-temperature superconductors [5, 17, 18, 19].Current wisdom argues that in a multi-band system, the criti-cal interaction U c necessary for the Mott metal-insulator tran-sition grows linearly with the number of bands N [20, 21],favoring a viewpoint that iron pnictide materials are itinerantmetals lying far from the Mott regime.In essence, Hund’s coupling converts a one channel Kondomodel to a K -channel model (7). Large- N treatments of thesemodels show that the relevant control parameter is the ratio K/N [22], rather than /N . By repeating the large- N argu-ment of Florens et al. [20], we conclude that the critical valueof the on-site interaction U for the Mott transition is U c ∝ ( N/K ) V k F ρ. (18)Thus Hund’s coupling compensates for multi-band behavior,restoring U c to a value comparable with one-band models. Re-cent DMFT calculations on the two-orbital Hubbard model [6]support this view, finding that U c is reduced from U c ≈ D to U ∗ c ≈ . D when J H /U = 1 / .LDA+DMFT studies of iron pnictide materials [9] concludethat in order to reproduce the incoherent bad metal features ofthe normal state, a value of J H ∼ . eV is required, resultingin T ∗ K of the order of 200 K. Fitting this with Eq. (11) re-sults in a nominal T K ∼ K and a ratio T K /J H S ∼ . .By contrast, for dilute Fe impurities in Cu [11] T ∗ K ≈ K,from which we extract a bare ratio T K /J H S ∼ . and T K ∼ K. The bare Kondo temperature is essentially thesame in both cases, but T K /J H S is significantly increaseddue to screening of J H in the iron pnictides, placing themmore or less at the crossover J H ∼ T K . A further sign ofstrong correlations in iron pnictides derives from the Wilsonratio, known to be ∼ W = 4 .Finally, we discuss heavy f -electron materials, which lie atthe crossover between localized and itinerant behavior [25]. Inthese materials, spin orbit and crystal field interactions dom-inate over Hund’s interaction. In fact, crystal fields are alsoknown to suppress the Kondo temperature in f -electron sys-tems [26], but the suppression mechanism differs, involvinga reduction in the spin symmetry rather a projective renor-malization of the coupling constant. But the main reasonthat Hund’s coupling is unimportant at the single-ion level inheavy f -electron materials, is because most of them involveone f -electron (e.g. Ce) or one f -hole in a filled f -shell (Yb,Pu), for which Hund’s interactions are absent.Perhaps the most interesting application of Kondo reso-nance narrowing to f -electron systems is in the context of in-tersite interactions. Indeed, (2) may serve as a useful modelfor a subset of ferromagnetically correlated f -electron mate-rials, such as CeRuPO [27], where J H would characterize thescale of ferromagnetic RKKY interactions between moments,as in Ref. 14. In these systems, our model predicts the forma-tion of microscopic clusters of spins which remain unscreenedin region II down to an exponentially small scale ∼ T ∗ K .This exponential narrowing of the Kondo scale may provide a clue to the observation [28, 29] of very narrow ESR ab-sorption lines in a number of Yb and Ce heavy fermion com-pounds with enhanced Wilson ratios. In particular, our the-ory would predict that the Knight shift of the electron g-factorin region II is proportional to the running coupling constant K ( T ) ∝ g eff ( T ) ∼ / ln( T /T ∗ K ) , where T ∗ K is the resonance-narrowed Kondo temperature. A detailed study of the ESRlineshape in this context will be a subject of future work.We would like to acknowledge discussions with ElihuAbrahams, Natan Andrei, Kristjan Haule, Gabriel Kotliar andAndrew Millis in connection with this work. This researchwas supported by NSF grant no. DMR 0605935. ∗ Electronic address: [email protected][1] A. C. Hewson,
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