epl draft Magnetic Order Beyond RKKY in the Classical Kondo Lattice Kalpataru Pradhan and Pinaki Majumdar 9 0 Harish-Chandra Research Institute, Chhatnag Road, Jhusi, Allahabad 211019, India 0 2 n a PACS 75.30.Kz–Magnetic phase boundaries J PACS 75.30.Et–Exchange and superexchangeinteractions 3 PACS 71.20.Eh–Rare earth metals and alloys 2 Abstract. - We study the Kondo lattice model of band electrons coupled to classical spins, ] in three dimensions, using a combination of variational calculation and Monte Carlo. We use l e the weak coupling ‘RKKY’ window and the strong coupling regime as benchmarks, but focus on - thephysically relevant intermediate coupling regime. Even for modest electron-spin coupling the r t phaseboundariesmoveawayfromtheRKKYresults,thenoninteractingFermisurfacenolonger s . dictates magnetic order, and weak coupling ‘spiral’ phases give way to collinear order. We use t a theseresultstorevisit theclassic problem of4f magnetism anddemonstratehowbothelectronic m structureand coupling effects beyond RKKYcontrol themagnetism in these materials. - d n o c [ The Kondo lattice model describes local moments on a Kasuya-Yosida(RKKY)model[8]. Theeffectivespin-spin lattice coupled to an electron band. Such local moments interactionin this limit is oscillatoryandlong range,con- 2 varisefromelectroncorrelationandHundscouplinginthed trolled by the free electron susceptibility, χ0(q), and the 0shellsoftransitionmetalsorthef shellsofrareearths. Al- magnetic ground state is generally a spiral. (b). When 4thoughhistoricallythe ‘Kondolattice’ aroseas the lattice the electron-spin coupling is very large compared to the 1 version [1] of the Kondo impurity problem, and refers to kinetic energy,the ‘double exchange’(DE) limit, the elec- 0 S = 1/2 moments coupled to conduction electrons, there tron spin is ‘slaved’ to the orientation of the core spin . 0 are also systems with local electron-spin coupling where andthe electronicenergyis minimised bya ferromagnetic 1 the moment is due to a spin with 2S 1. In that case (FM) background [9]. This leads intuitively to a spin po- 8 ≫ 0the quantum fluctuations of the local moment, and the larised ground state. :Kondo effect itself, are not relevant. Such a system can v Inmanymaterialstheratioofcouplingtohoppingscale be described by a classicalKondo lattice model (CKLM). i is 1,butnotquiteinthe doubleexchangelimit. Inthat XThis limit is relevant for a wide variety of materials, e.g, ≥ caseonehastosolvethecoupledspin-fermionmodelfrom rthe manganites [2], where S = 3/2 moments couple to a firstprinciples. Doingso,particularlyinthree dimensions itinerantelectronsviaHundscoupling,or4f metals[3–6], andatfinitetemperature,hasbeenachallenge. Westudy e.g, Gd with S = 7/2, or the Mn based dilute magnetic thisproblemusingacombinationofvariationalcalculation semiconductors[7]whereS =5/2. Insomeofthesemate- and full spin-fermion Monte Carlo. rials, notably the manganites and the magnetic semicon- ductors, the coupling scale is known to be large, while in Our principal results are the following: (i) We are able thef metalstheyhavebeentraditionallytreatedasbeing tomapoutthemagneticgroundstateallthewayfromthe weak. RKKY limit to double exchange, revealing the intricate TheCKLMinvolvestheorderingof‘classical’spins,but evolution with coupling strength. (ii) We demonstrate theeffectiveinteractionbetweenspinsismediatedbyelec- thatthephaseboundariesdependsensitivelyonelectronic tron delocalisation and cannot be described by a short hopping parameters. This is not surprising in the RKKY range model. In fact the major theoretical difficulty in regime, but the dependence at stronger coupling is un- analysing these systems is the absence of any simple clas- known. (iii) We use our results to revisit the classic 4f sicalspin model. Nevertheless, there are two limits where magnets,widely modelledasRKKYsystems,andsuggest theCKLMiswellunderstood. (a).Whentheelectron-spin that with increasing 4f moment, the effective coupling in coupling is small, one can perturbatively ‘integrate out’ thesesystemspushesthembeyondtheRKKYregime. We the electrons andobtain the celebratedRuderman-Kittel- work out the signatures of this ‘physics beyond RKKY’. p-1 Kalpataru Pradhan and Pinaki Majumdar Model: The Kondo lattice model is given by modeltounderstandthe weakcouplingphases. Atstrong coupling, J/t , there is no exact analytic H but eff H = t c† c µ n J ~σ .S (1) we can constru→ct∞approximate self consistent models [15] − X ij iσ jσ− X i− X i i hijiσ i i of the form HDefEf =−PhijiDijp1+Si.Sj, with the Dij related to the electronic kinetic energy. Unfortunately, Wewilluset=1asthenearestneighbourhoppingampli- when J (t) neither the RKKY model nor the DE ap- tude, and explore a range of t′, the next neighbour hop- ∼O proximationarevalid. Thisregimerequiresnewtoolsand ping, on a cubic lattice. Changing t′ will allow us to ex- we will use a combination of (i) variational calculation plorechangesinthe(bare)Fermisurface,andparticle-hole (VC) [16] for the magnetic ground state, and (ii) spin- asymmetry. µ is the chemical potential, and J > 0 is the fermion Monte Carlo using a ‘travelling cluster’ approxi- localelectron-spincoupling. We assumethe S tobe clas- i mation [17] (TCA-MC) at finite temperature. sical unit vectors, and absorb the magnitude of the core For the variational calculation we choose a simple spininto J wherevernecessary. ~σ is the electronspinop- i parametrisation [18] for the spin configuration: S = α, erator. We work with µ, rather than electron density (n), iz S = √1 α2 cosq.r and S = √1 α2 sinq.r . This asthecontrolvariablesothatregimesofphaseseparation ix − i iy − i encompasses the standard ferromagnet and antiferromag- (PS) can be detected, and study the magnetic properties net, as well as planar spiral phases, canted ferromagnets, for varying n, t′/t, J/t, and temperature T/t. and A and C type antiferromagnets. For a fixed µ and J Although there have been many studies in the ‘double we compute the electronic energy (α,q,µ) andminimise exchange’ (J/t ) limit [10], the attempts to explore E → ∞ it with respect to α and q. The electronic density at the thefulln J T phasediagramhavebeenlimited. (a)An − − chosen µ is computed on the minimised state. Since the effective actionobtainedfrom the CKLMvia gradientex- magnetic background only mixes electronic states k, pansion [11] has been analysed. This mapped out some | ↑i and k q, the electronic eigenvalues ǫ±(k,q) are sim- ofthecommensurateandspiralphasesintwodimensions, | − ↓i ple, and only an elementary numerical sum is required to where the phases are fewer. It did not explore the finite calculate (α,q,µ)= ǫ±(k,q)θ(µ ǫ±(k,q)). temperature physics, e.g, the Tc scales, and seems to be E Pk,± − While the VC providesafeeling forthe possibleground inaccurate when handling commensurability effects near states, it has the limitation that (i) it samples only one n=1. (b) The model has been studied within dynamical family of (periodic) functions in arriving at the ground mean field theory [12] (DMFT), and the broadregimes of state,and(ii) finite temperature properties,e.g, the mag- ferromagnetism, antiferromagnetism (AFM), and incom- netisationand the criticaltemperature are not accessible. mensurate order have been mapped out. Unfortunately For this we ‘anneal’ the system towards the equilibrium the effective ‘single site’ character of DMFT does not al- distribution P S Tr e−βH using the TCA based low a characterisation of the incommensurate phases and c,c† { } ∝ MonteCarlo. Inthis method the acceptanceofa spinup- misses out on the richness of the phase diagram. The loss dateis determinedby diagonalisingacluster Hamiltonian of information about spatial fluctuations also means that constructed around the update site, and avoids iterative critical properties, either in magnetism or transport, can- diagonalisation[14] ofthe full system. We can accesssys- not be correctly captured. (c) An ‘equation of motion’ 3 3 tem size 10 using a moving cluster of size 4 . approach [13] has been employed to study general finite ∼ S spins coupled to fermions, and results have been ob- TheTCAcapturesphaseswithcommensuratewavevec- tained in the classical limit as well. However, except the tor Q quite accurately, but access to the weak coupling ferro and antiferromagnetic phases other magnetic states incommensurate phases is poor. To get an impression do not seem to have been explored. (d) The full spin- of the ordering temperature for these phases we compute 1 fermionMonteCarlo,usingexactdiagnolisation,hasbeen the energy difference ∆E(n,J) = N(Edisord −Eord), be- employed [14] in one and two dimensions but severe size tween the ordered state and a fully spin disordered state limitations prevent access to non trivial ordered states. in a large system. ord is calculated from the variational E Method: The problem is technically difficult because it groundstate,and disordbydiagonalisingtheelectronsys- E involves coupled quantum and classical degrees of free- tem in a fully spin disordered background on a large lat- dom, and there is in general no equivalent classical spin tice. ∆ (n,J) is the ‘condensation energy’ of the ordered E Hamiltonian. The probability distribution for spin con- state, and provides a crude measure of the effective ex- figurations is given by P S Trc,c†e−βH so the ‘effec- changeandTc. WherewecouldcomparethetrendtoMC tive Hamiltonian’ is H { S} ∝= 1log Tr e−βH, the data, the agreement was reasonable. eff{ } −β c,c† fermion free energy in an arbitrary background S . It Results: Theresultsofthevariationalcalculationinthe i cannot be analytically calculated except when J/{t }1. ‘symmetric’(t′ =0)caseareshowninFig.1. Weemployed ≪ 3 When J/t 1, the (free) energy calculated perturba- a grid with upto 40 k points, and have checked stability tively to (J≪2) leads to the RKKY spin Hamiltonian [8], with respect to grid size. Let us analyse the weak and O Heff = J S .S , where J J2χ0 andχ0 is the strong coupling regimes first before getting to the more noRnKlKocYal suPscijeptiijbiilityj of the freije∼(J =i0j) electriojn sys- complex intermediate coupling regime. tem. χ0 islongrangeandoscillatory. Wewillanalysethis (i) RKKY limit: The key features for J/t 0 are: ij → p-2 Kondo Lattice Beyond RKKY Fig. 1: Colour online: Magnetic ground state for the particle- ′ holesymmetricmodel (t =0) forvaryingelectron density(n) and electron-spin coupling (J). The phases are characterised by their ordering wavevector Q, indicated by the colour code in the legend to the right, and their net magnetisation α (if any). Among the ‘commensurate’ phases, Q= {0,0,0} is the Fig.2: Colouronline: Thefinitetemperaturephasediagramin usualferromagnet,{0,π,π}and{π,π,π}areantiferromagnets the particle-hole symmetric case, for various J. Panels (a)-(d) withnonetmagnetisation, whilethe {0,0,π}antiferromagnet show the different ordered phases and their estimated transi- has α = 0 for J → 0 but picks up finite magnetisation with tion temperature as we move from the weak coupling to the increasing J. At n = 1 the system is always a Q = {π,π,π} double exchange limit. The legend for the phases is shown antiferromagnet. The incommensurate phases have ordering on the right. The transition temperatures are based either on wavevectors {Qx,Qy,Qz} of which at least one component is Monte Carlo results (shown as symbols), or the ∆E estimate neither0norπ. Forsuchphasestheexactwavevectordepends (firm lines) described in the text. Notice that the Tc for the on the value of n and J. For example, for J → 0 the (blue) ferromagnetic, Q = {0,0,0}, phase increases (and saturates) checkerboardregioninthelefthandcorner,totherightofQ= with increasing J. At n=1 the order is at Q={π,π,π} and {0,0,π},haswavevectorQ={0,Qy,π},whereQy variesfrom the corresponding Tc initially increases with increasing J and 0toπasonemoveslefttoright. The(green)shadedregionsin then decreases. Except for Q = {0,0,0} and {π,π,π} other thephasediagram,not indicatedinthelegend,arewindowsof phases vanish by the time J/t = 10. The Monte Carlo esti- phaseseparation. Nohomogeneousphasesareallowedinthese mate of ferromagnetic Tc are shown as circles, while that of regions. The results in this figure are based on a variational the antiferromagnet is marked on the n = 1 axis by a square calculation using a 203 k point grid, and cross-checked with symbol. As in Fig.1 the (green) shaded regions indicate phase dataon 403. separation. (i) the occurence of ‘commensurate’ planar spiral phases, the wavevector qmax(n) of the peak in χ0(q,n). The ab- with wavenumber Q which is 0,0,0 , or 0,0,π , etc, solute maximum in χ0(q,n) remains at q = 0,0,0 , as { } { } { } over finite density windows, (ii) the presence of planar the electrondensityisincreasedfromn=0,andatacrit- spirals with incommensurate Q over certain density in- icaldensityq shifts to 0,0,π . Withfurther increase max { } tervals, (iii) the absence of any phase separation, i.e, in density q evolves through 0,q,π to the C type max { } only second order phase boundaries, and (iv) the pres- 0,π,π ,then q,π,π ,andfinally the Gtype AFM with { } { } ence of a ‘G type’, Q = π,π,π , antiferromagnet at π,π,π , where the Fermi surface is nested. The absence { } { } n = 1. Although the magnetic state is obtained from of‘conical’phases,withfinite(α)andaspiralwavevector, the variationalcalculation,muchinsightcanbe gainedby is consistent with what is known in the RKKY problem. analysing the Heff . Since the spin-spin interaction is Thereisnophaseseparation,i.e,discontinuitiesinn(µ), RKKY longrangeitisusefultostudytheFouriertransformedver- forJ/t 0sincetheµ nrelationisthatoftheunderlying sion HRefKfKY ≡ PqJ˜q|Sq|2, where J˜q = Pi−jJijeiq.Rij tightbin→dingsysteman−dfreeofanysingularity. Thephase and Sq = PiSieiq.Ri. The coupling J˜q = J2χ0(q,n) transitions with changing n are all second order. With is controlled by the spin susceptibility, χ0(q,n), of the growingJ/t,however,somephaseboundariesbecomefirst J = 0 tight binding electron system. For our choice of order and regimes of PS will emerge. variationalstatetheminimumofHeff correspondstothe (ii) Strong coupling: For J/t , it makes sense to → ∞ wavevector at which χ0(q,n) has a maximum. We inde- quantise the fermion spin at site Ri in the direction of pendently computed χ0(q,n) and confirmed [19] that the the core spin Si, and project out the ‘high energy’ un- wavevector Q(n) obtained from the VC closely matches favourablestate. Thisleadstoaneffectivespinlessfermion p-3 Kalpataru Pradhan and Pinaki Majumdar theplanarspiralsbegintopickupanetmagnetisation,α, and now become ‘conical’ phases. Windows of phase sep- aration also appear, particularly prominent between the Q ,π,π andGtypeAFM(nearn=1),andsuggestthe x { } possibilityofinhomogeneousstates,etc,inthepresenceof disorder. Theprimesignatureof‘physicsbeyondRKKY’, however, is that the RKKY planar spirals now pick up a net magnetisation and much of the phase diagram starts to evolve towards the ferromagnetic state. (iv) Finite temperature: The TCA based MC readily captures the FM and π,π,π AFM phases at all cou- { } pling. However, it has difficulty in capturing the more complexspiral,A,andC typephaseswhenwe‘cool’from Fig.3: Colouronline: ThemagneticgroundstateintheRKKY the paramagnetic phase. In the intermediate J regime it limit, showing the dependence of the ordering wavevector Q usually yields a ‘glassy’ phase with the structure factor ′ on electron density and particle-hole asymmetry (via t). The having weight distributed over all q. In our undertstand- legend for thevariousstates is shown on theright. Thecalcu- ing this is a limitation of the small cluster based TCA, lationsweredoneatweakcoupling,J =0.5. Notethegrowing ′ and the energies yielded by VC are better than that of asymmetryofthephases(aboutn=1)ast increases. Itisalso clearthatifthehoppingparametert′changes(duetopressure, ‘unordered’ states obtained via MC. To get a feel for the etc) themagneticgroundstatecanchangeeveniftheelectron orderingtemperaturewehavecalculatedtheenergydiffer- densityremainsfixed,asdiscussedfor4f systemsin[20]. This ence∆ , definedearlier,asoftendone inelectronicstruc- E is particularly prominent in the top right hand corner of the ture calculations. This provides the trend in Tc across figure. In constructing this phase diagram we have ignored a the phases,Fig.2,andwhereverpossiblewe haveincluded narrow sliver of phaseseparation near n=1. dataaboutactualT (symbols)obtainedfromtheMCcal- c culation. Broadly,withincreasingJ the ∆ andT scales c E increasebutthenumberofphasesdecrease. TheT ofthe c problemwhosebandwidthiscontrolledbytheaveragespin G type AFM is expected to fall at large J but even at overlaphSi.Sjibetweenneighbouringsites. Theoverlapis J/t=10 it is larger than the peak FM Tc. largestforafullypolarisedstate,andtheFMturnsoutto (v) Interplay of FS and coupling effects: Till now we be the ground state at all n=1. At n=1 ‘real hopping’ have looked at the particle-hole symmetric case where 6 is forbidden so the fermions prefer a G type AFM back- t′ = 0. The tight-binding parametrisation of the ab ini- ground to gain kinetic energy (t2/J) via virtual hops. tio electronic structure of any material usually requires a O The FM and G type AFM have a first order transition finite t′, in addition, possibly, to multiple bands. We will between them with a window of phase separation, eas- use the t t′ parametrisationof band structure due to its − ily estimated at large J/t. The fully polarised FM phase simplicity. Itwillalsoallowustomimicthephysicsinthe has a density of states (DOS) which is simply two 3D 4f metals. tightbindingDOSwithsplittingJ betweenthebandcen- At weak coupling the magnetic order is controlled as ters. If we denote this DOS as NFM(ω,J) then the en- usualbythebandsusceptibility,χ0(q,n)which,now,also ergyoftheFMphaseis (µ,J)= µ N (ω,J)ωdω, depends on t′. At fixed n the magnetic order can change EFM R−∞ FM and the particle density is n(µ,J) = µ N (ω,J)dω. simply due to changes in the underlying electronic struc- There will be corresponding expressiRo−n∞s whFeMn we con- ture. Our Fig.3 illustrates this dependence, where we use sider electrons in the π,π,π AFM background, with J =0.5tostayinthe RKKYregimeandexplorethevari- DOS N (ω,J). On{ce we k}now µ = µFM that sat- ation of magnetic order with n and t′. The range of t′ AFM AFM isfies (µ,J) = (µ,J) we can determine the variation is modest, 0 0.3 , but can lead to phase PS wiEnFdMow from theEAdFenMsity equations. Since the FM changes (at fixed n) ∼in{som−e de}nsity windows. We have ph2ats(ecohskasaa+dcisopsekrsaio+n cǫoFksMk a=),ǫa0,nkd±thJe/2A,FwMheprheaǫs0e,kha=s croAssccohmecpkleicdattehde pahnadsemsowriethretahleisptiecakveirnsiχon0(qo)f.this has x y z −dispersion ǫAFM = ǫ2 +(J/2)2, it is elementary to been demonstrated recently [20] in the 4f family for the k ±q 0,k heavy rare earths from Gd to Tm. These elements all work out µFAFMM. The analysis can be extended to several havethesamehcpcrystalstructure,andthesameconduc- competing phases. It is significant that even at J/t=10, tion electron count, 5d16s2, so nominally the same band whichmightoccurforstrongHunds’couplinginsomema- filling. However, the electronic structure and Fermi sur- terials, the FM phase occurs only between n= 0,0.7 . face changes due to variation in the lattice parameters { } (iii) Intermediate coupling: The intermediate coupling and unit cell volume (lanthanide contraction) across the regimeiswhereoneisoutsidetheRKKYwindow,butnot series. It has been argued [20] that this changes the loca- solargeacouplingthatonlytheFMandGtypeAFMare tion qmax of the peak in χ0, and explains the change in possible. Towards the weak coupling end it implies that magnetic order from planar spiral (in Tm) to ferromag- p-4 Kondo Lattice Beyond RKKY tally measuredT is takenas ‘confirmation’of the RKKY c picture. Should’nt we also worry about the effect of the grow- ingJ (S)onthemagnetic order itself? Ifthemaximum eff J ,forGdwithS =7/2,weresmallerthantheeffective eff hopping scale t, then we need not - the RKKY scheme would be valid for the entire 4f family. However, mea- surements and electronic structure calculations [3] in Gd suggest that J 0.3eV and J (7/2) 1eV. The effec- eff ∼ ∼ tive t is more ambiguous, since there are multiple bands crossing the Fermi level, but the typical value is 0.3eV. ∼ This suggests J /t 3, clearly outside the RKKY win- eff ∼ dow! What is the consequence for magnetic order, and physical properties as a whole? Fig.4showsthet′ J magneticphasediagramatT =0 − forn=1.7. Att′ =0,theverticalscan,changingJ reveals howtheorderedstatechangeswithincreasingJ evenwith electronic parameters (and hence χ0 and FS) fixed. We have already seen this in Fig.1 The spirit of RKKY is to assume J 0, and move horizontally, changing t′ across → the series so that one evolves from a planar spiral to a ferromagnet. Wesuggestthatinthef metals,theparam- eter points are actually on a ‘diagonal’, with increasing t′ (our version of changing electronic structure) being ac- companied by increase in J . To capture the trend we eff set, t′ = 0 and J = 1.0 for S = 1, where the system eff is known to be a spiral, and t′ = 0.1 and J = 3.0 for eff S = 7/2 (the case of Gd), and explore the linear varia- tion shown in Fig.4. This parametrisation is only meant to highlight the qualitative effect of changing electronic Fig. 4: Colour online: Top: the magnetic ground state at n = 1.7 for varying t′ and J. The ordering wavevector is structureandJeff andsincerealt′ values,etc,wouldneed to be calculated from an ab initio solution. marked on the phases. The magnetic order has a pronounced dependence on both the ‘bandstructure’ (through t′) and the Withinthis framework,whilethesmallS resultissame electron-spin coupling. We highlight three kinds of parame- for both RKKY and explicit inclusion of Jeff, the order ter variation. (i) Varying J at fixed t′, the points on the y obtained at intermediate S depends on whether one ig- axis, shows how changing electron-spin coupling can change nores J (as in RKKY) or retains its effect. For a given eff ′ the ground state. (ii) Varying t at weak coupling, J = 1, il- t′ the phase on the diagonal is quite different from the lustrateshowbandstructureaffectstheRKKYmagneticorder. phase on the horizontal line. (iii) In the 4f elements we think what happens is a combina- In fact there is evidence from earlier ab initio calcula- tion of (i) and (ii) above, as shown by points on the diagonal. tions[21]thatinadditiontounitcellvolumeandc/aratio, Bottom: An impression of the real space spin configuration the strength of the 4f moment (and so J ) also affects forthethreeparametersets(i)-(iii)inthetoppanel. Each2×2 eff ′ the magnetic order. As an illustrative case, the optimal patternisforat, J combination. Thebottomleftspinineach patternisset on thereferencesiteR={0,0}, say. Theneigh- spiral wavevector in Ho evolves towards Q = 0,0,0 as { } bouring three spins are at {xˆ,0}, {0,yˆ}, and at {xˆ,yˆ}, where the effective moment is (artificially) varied from 2µB to xˆ and yˆ are unit vectors on the lattice. There is no variation 4µB (Fig.2 in Nordstrom and Mavromaras [21]). If mag- in the z direction so we only show the in-plane pattern. Top netism in this element, and the 4f family in general,were ′ row: scan (i) above, changing J at t = 0. Middle row: scan completely determined by RKKY there would be no de- ′ (ii),changingt atJ =1. Bottomrow: scan(iii),simultaneous pendence on J . In fact the authors suggested that one changein t′ and J. should re-examefinfe the basic assumptions of the ‘standard model’ of 4f magnetism [5], which gives primacy to the RKKY interaction (and magnetoelastic effects) since the netisminGd. AsimilareffectisvisibleinourFig.3where ab initioresultssuggestarolefortheeffectiveexchangein at n = 1.7, say, the ordering wavevector changes from a the magnetic order. Our aim here has been to clarify the spiral to FM as t′ changes from zero to 0.15. In this sce- physics underlying such an effect within a minimal model nario,J doesnotaffectthemagneticorderbutmerelysets Hamiltonian. This approach would be useful to handle thescaleforT . TheRKKYinteractionstrengthscalesas noncollinearphasesincomplexmanybandsystems,with- c J2 J2S(S+1),andasimilarscalingoftheexperimen- out any weak coupling assumption, once a tight binding eff ∼ p-5 Kalpataru Pradhan and Pinaki Majumdar parametrisationof the electronic structure is available GUINEA F., and MARTIN MAYOR V., Phys. Rev. B Let us conclude. We have examined the Kondo lattice 64, 054408 (2001). modelwithlargeS spinsandestablishedthegroundstate [17] KUMAR S. and MAJUMDAR P., Eur. Phys. J. B 50, 571 (2006). allthewayfromthe RKKYregimetothe strongcoupling [18] Our choice corresponds to a periodically varying az- limit. The intermediate coupling window reveals a com- petition between RKKY effects, which tend to generate a imuthal angle, φi. The most general periodic state in- volvesperiodicvariationinthepolarangleθi aswell[16]. planar spiral, and the tendency to gain exchange energy [19] PRADHANK.and MAJUMDARP., to be published. via ferromagnetic polarisation. This generally leads to a [20] HUGHES I. D., DANE M., ERNST A., HERGERT W., ‘conical’helix,givingwayatstrongcouplingtothedouble LUDERSM., POULTERJ., STAUNTONJ. B., SVANE exchange ferromagnet. Using these results we re-visited A., SZOTEK Z., TEMMERMAN W. M., Nature, 446, the classic 4f magnets to demonstrate how the magnetic 650 (2007). phases there are probably controlled non RKKY spin- [21] NORDSTROM L. and MAVROMARAS A., Europhys. fermion effects. One can add anisotropies and magneto- Lett. 49, 775 (2000). elastic couplings to our model to construct a more com- prehensive description of 4f magnetism. We acknowledgeuse ofthe Beowulfcluster atHRI, and thank Sanjeev Kumar and B. P. Sekhar for collaboration on an earlier version of this problem. REFERENCES [1] HEWSONA.C.,TheKondoProblemtoHeavyFermions, Cambridge University Press (1997). [2] TOKURA Y. (editor), Colossal Magnetoresistive Oxides, CRC Press (2000), CHATTERJI T. (editor), Colossal Magnetoresistive Manganites, Springer (2004). [3] SANTOS C., NOLTINGW. and EYERT V., Phys. Rev. B 69, 214412 (2004), HEINEMANN M. and TEMMER- MAN W. M., Phys.Rev. B 49, 4348 (1994). [4] LEGVOLDS.,Ferromagnetic Materials,Vol.1,Chap.3., (North Holland, Amsterdam, 1980). [5] JENSENJ.andMACKINTOSHA.K.,Rare Earth Mag- netism, Clarendon, Oxford (1991). [6] ELLIOTT R.J., Phys.Rev. 124, 346 (1961). [7] For an overview see, JUNGWIRTH T., SINOVA J., MASEKJ.,KUCERAJ.,andMACDONALDA.H.,Rev. Mod. Phys.78, 809 (2006). [8] RUDERMANM.A.,andKITTELC.,Phys.Rev.96,99 (1954), KASUYA T., Prog. Theor. Phys. 16, 45 (1956), YOSIDAK., Phys.Rev.106, 893 (1957). [9] ZENER C., Phys. Rev. 82, 403 (1951), ANDERSON P. W., and HASEGAWA H., ibid. 100, 675 (1955), de GENNES P-G., ibid. 118, 141 (1960). [10] CALDERONM.J.andBREYL.,Phys.Rev.B58,3286 (1998),MOTOMEY.andFURUKAWAN.,J.Phys.Soc. Jpn. 69 (2000) 3785. [11] PEKKER D.,MUKHOPADHYAYS.,TRIVEDIN.,and GOLDBARTP. M, Phys. Rev. B 72, 075118 (2005). [12] CHATTOPADHYAYA.,MILLISA.J.andDASSARMA S., Phys. Rev.B 64, 012416 (2001). [13] KIENERTJ., et al., Phys. Rev. B 73, 224405 (2006). [14] DAGOTTOE.,YUNOKIS.,MALVEZZIA.L.,MOREO A.,HuJ.,CAPPONIS.,POILBLANCD.,FURUKAWA N., PhysRev B 58 6414 (1998). [15] KUMAR S., and MAJUMDAR P., Eur. Phys.J. B 46. [16] For examples of variational calculations, see HAMADA M.,andSHIMAHARAH.,Phys.Rev.B51,3027(1995), ALONSO J. L., CAPITAN J. A., FERNANDEZ L. A., p-6