The f-spin physics of rare-earth iron pnictides: influence of d-electron antiferromagnetic order on the heavy fermion phase diagram
aa r X i v : . [ c ond - m a t . s t r- e l ] J u l The f -electron physics of rare-earth iron pnictides: influence of d -electronantiferromagnetic order on the heavy fermion phase diagram Jianhui Dai, Jian-Xin Zhu, and Qimiao Si Zhejiang Institute of Modern Physics, Zhejiang University, Hangzhou 310027, China Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA Department of Physics and Astronomy, Rice University, Houston, TX 77005, USA
Some of the high T c iron pnictides contain magnetic rare-earth elements, raising the questionof how the existence and tunability of a d -electron antiferromagnetic order influences the heavyfermion behavior of the f -moments. With CeOFeP and CeOFeAs in mind as prototypes, we derivean extended Anderson lattice model appropriate for these quaternary systems. We show that theKondo screening of the f -moments are efficiently suppressed by the d -electron ordering. We alsoargue that, inside the d -electron ordered state (as in CeOFeAs), the f -moments provide a rarerealization of a quantum frustrated magnet with competing J - J - J interactions in an effectivesquare lattice. Implications for the heavy fermion physics in broader contexts are also discussed. PACS numbers: 71.10.Hf, 71.27.+a, 74.70.Tx, 75.10.-b
The homologous rare-earth iron arsenides exhibit anti-ferromagnetic (AF) ground states in addition to the hightemperature superconductivity [1, 2, 3, 4, 5]. The sys-tems of interest here are the arsenides R O x F − x FeAs,with R = Ce, Sm, Nd, Pr,... being magnetic rareearths, which have superconducting transition temper-atures higher [2, 3, 4, 5] than the maximal T c ≈
26 K ofLaO x F − x FeAs [1]. The parent compounds of these sys-tems, ROFeAs, have a layered structure, with FeAs and R O layers sandwiching each other. They typically showa collinear AF order and a structure distortion, whichare successively suppressed by carrier doping in favor ofsuperconductivity [6]. Also of interest are the iron phos-phides. LaOFeP was the iron pnictide reported to showsuperconductivity below T c ≈ f -electrons exhibitheavy fermion behavior with a Kondo temperature T K ≈
10 K [9]. CeOFeAs has the d -electron collinear AF or-dering below T ( d ) N ≈
130 K. Its f -electrons display a no-ticeable AF order below T ( f ) N ≈ d -4 f couplings between Ce-OFeP and CeOFeAs, as suggested by a first-principleLDA+DMFT study [11]. However, a more complete the-oretical estimate using a full density of states, whichis strongly peaked away from the Fermi energy, sug-gests that the effective Kondo couplings in CeOFeP andCeOFeAs may in fact be comparable [12]. Muon-spin-relaxation and neutron scattering experiments [13, 14]may also be interpretted in terms of a sizable Kondo coupling in CeOFeAs.In this Communication, we discuss the possibility thatthe distinction in the d -electron magnetism between Ce-OFeP and CeOFeAs plays an important role in influenc-ing their heavy fermion behavior. This mechanism is ex-pected to play an especially important role when we con-sider not only the end materials CeOFeP and CeOFeAs,but also the series CeOFeAs − x P x , which has been pro-posed to realize a continuously varying d -electron AF or-der and the associated quantum critical point [15].Studying the effect of the d -electron AF order on theheavy fermion phase diagram not only sheds new light onthe properties of the iron pnictides, but also represents anew twist to the heavy fermion physics in general. Typ-ically, AF order in heavy fermion metals is induced bythe RKKY interactions among the f -moments, and theheavy fermion phase diagram involves the competitionbetween RKKY and Kondo coupling [16, 17]. A tunable d -electron AF order adds a new dimension to the heavyfermion phase diagram.In the following, we will consider this effect within anextended Anderson lattice model (ALM) appropriate forthe stoichiometric R -1111 compounds R OFe X ( X =Asor P). The model incorporates the inter-layer hybridiza-tion between pnictogen X p -orbitals and rare earth Rf -orbitals. We note in passing that the derived modeltakes into account the microscopic crystal structure andsymmetry of the R -1111 compounds. Given that thereare many materials of the same ZrCuSiAs-type structure[18], with many of them containing magnetic rare-earthelements, we expect that our model will also be germaneto many such related compounds [19]. General considerations.
The lattice structure of the R -1111 compound series is schematically shown in Fig. 1.Let Fe-atoms be in the ( x, y )-plane with the coordinate( ~r, ~r = ( i x , i y ), i x and i y are both integers (thenearest Fe-Fe distance is set to unity). The coordinates of X - and R -atoms are ( ~r p , ηz p ) and ( ~r f , ηz f ) respectively, c b a R Fe O R X X
FIG. 1: (Color online) The lattice structure of R-1111 series.The small black and red (connected with solid line) circlesrepresent Fe and O ions, respectively, and the big blue/brownand dashed grey circles are the R and X ions, respectively.The small solid red, solid brown, and doted green squaresdescribe the Fe, R , and X plaquettes, respectively. Left panel: ab -plane; right panel: ac -plane. The dashed and doted linesdenote V pf and V pd , respectively. where ~r p = ( i x + 1 / , i y + 1 / ~r f = ( i x − / , i y +1 / η = e iπ ( i x + i y ) , z p and z f are the distances of X -and R -atoms to the Fe-plane. We denote the d -, p -, and f -electrons by d ( α ) σ ( ~r ), p ( µ ) σ ( ~r p , ηz p ), and f ( m ) σ ( ~r f , ηz f ),with orbital indices α = d xy , d xz , d yz , d x − y , d z − r , µ = p x , p y , p z , and m = 1 , · · · , l . The model Hamiltonian.
The hybridization part of theHamiltonian is given by H hybrid = H pd + H pf , where H pd = X ~r V ( µ,α ) pd [ p ( µ ) † σ ( ~r p , ηz p ) D ( α ) σ ( ~r ) + h.c.] , (1) H pf = X ~r V ( µ,m ) pf [ p ( µ ) † σ ( ~r p , ηz p ) F ( m ) σ ( ~r f , ηz f ) + h.c.] . (2)Here we introduce D ( α ) σ ( ~r ) = P (cid:3) d ( α ) σ ( ~r ) ≡ d ( α ) σ ( i x , i y ) + d ( α ) σ ( i x + 1 , i y ) + d ( α ) σ ( i x , i y + 1) + d ( α ) σ ( i x + 1 , i y + 1) and F ( m ) σ ( ~r f , ηz f ) = P (cid:3) f ( m ) σ ( ~r f , ηz f ) ≡ f ( m ) σ ( i x − / , i y +1 / , ηz f )+ f ( m ) σ ( i x +3 / , i y +1 / , ηz f )+ f ( m ) σ ( i x +1 / , i y − / , ηz f ) + f ( m ) σ ( i x + 1 / , i y + 3 / , ηz f ) as the plaquetteoperators of d - and f -electrons around X -atoms. (Sum-mations over the repeated spin and channel indices areimplied hereafter unless otherwise specified.)The interaction part of the Hamiltonian, H int = H int,d + H int,p + H int,f , contains the usual on-siteCoulomb interactions ( U p , U d , and U f ) and the Hund’scoupling ( J H,d ). The total Hamiltonian is then H = H + H hybrid + H int , with H containing the primitivesite energies of d -, p -, and f -electrons denoted by ε ( α ) d , ε ( µ ) p , and ε ( m ) f , respectively.It is expected that U p is small compared to the otherCoulomb interactions. We will therefore set U p = 0, inwhich case the p -orbitals can be readily integrated out.The obtained effective Hamiltonian ˜ H takes the form˜ H = H + H d + H f + H df + H int,d + H int,f . (3)Here H d = P ~r V ( αα ′ ) d [ D ( α ) † σ ( ~r ) D ( α ′ ) σ ( ~r ) + h.c. ], H f = P ~r V ( mm ′ ) f [ F ( m ) † σ ( ~r f , ηz f ) F ( m ′ ) σ ( ~r f , ηz f ) + h.c.], and H df = P ~r V ( αm ) df [ D ( α ) † σ ( ~r ) F ( m ) σ ( ~r f , ηz f ) +h.c.], with V ( αα ′ ) d = − P µ V ( µ,α ) pd V ( µ,α ′ ) pd /ε ( µ ) p , V ( mm ′ ) f = − P µ V ( µ,m ) pf V ( µ,m ′ ) pf /ε ( µ ) p , V ( αm ) df = − P µ V ( µ,α ) pd V ( µ,m ) pf /ε ( µ ) p . In the momentum K -space(in the reduced Brillouin zone corresponding to twoFe-atoms in the conventional cell with lattice constant a = √ H d = P K V ( αα ′ ) d g ( ηη ′ ) d ( K ) d ( α ) † η K σ d ( α ′ ) η ′ K σ , H f = P K V ( mm ′ ) f g f ( K ) f ( m ) † η K σ f ( m ′ ) η K σ , and H df = P K V ( αm ) df g ( ηη ′ ) df ( K )[ d ( α ) † η K σ f ( m ) η ′ K σ + h.c.], where d ( α ) η K σ and f ( m ) η K σ are the Fourier transform of d - and f - electronoperators in the sublattices η = A or B , respectively.The K -dependence of the dispersions and d - f hy-bridization is only encoded in the form factors, givenby g ( AA ) d ( K ) = g ( BB ) d ( K ) = 4 + 2 cos( K x a ) cos( K y a ), g ( AB ) d ( K ) = g ( BA ) d ( K ) = 8 cos( K x a/
2) cos( K y a/ g f ( K ) = 16 cos ( K x a/
2) cos ( K y a/ g ( AA ) df ( K ) = g ( BB ) df ( K ) = 8 cos ( K x a/
2) cos( K y a/ g ( BA ) df ( K ) = g ( BA ) df ( K ) = 8 cos( K x a/
2) cos ( K y a/ The d -electron correlations. For moderate large U d ,we may start from the strong coupling limit yielding thefrustrated J - J Heisenberg model for the d -electrons [20,21, 22]. The itinerancy of the d -electrons will furtherreduce the ordered moments and eventually lead to aparamagnetic phase [15]. In fact, both the weak- andstrong-coupling limits suggest that the staggered magne-tization M d = P α M ( α ) d = − (1 /N ) P K h{ d ( α ) † η K ↑ d ( α ) η K + Q ↑ − d ( α ) † η K ↓ d ( α ) η K + Q ↓ }i is a dominating order parameter with Q = ( π, π ) and N being the number of K points in thereduced Brillouin zone. For the purpose of demonstrat-ing the effect of d -electron order on the Kondo effect, wetreat M ( α ) d as the mean field parameters and approxi-mate H int,d by J d P K σM ( α ) d [ d ( α ) † η K σ d ( α ) η K + Q σ + h.c.], with J d being the effective coupling strength. The AF order-ing gap, ∆ ( α ) AF = J d M ( α ) d , is sizable for FeAs but vanishesfor FeP. Kondo effect vs. d-electron ordering.
In order to un-derstand the competition between the Kondo effect and d -electron AF order, we first neglect the f -electron or-dering. We are then led to consider H ALM = X K [ ε ( α ) d δ αα ′ δ ηη ′ + V ( αα ′ ) d g ( ηη ′ ) d ( K )] d ( α ) † η K σ d ( α ′ ) η ′ K σ + X K [ ε ( m ) f δ mm ′ + V ( mm ′ ) f g f ( K )] f ( m ) † η K σ f ( m ′ ) η K σ + X K [ V ( αm ) df g ( ηη ′ ) df ( K ) d ( α ) † η K σ f ( m ) η ′ K σ + h.c.]+ X K [ σ ∆ ( α ) AF d ( α ) † η K σ d ( α ) η K + Q σ + h.c.]+ U f X ~r f n ( m ) f, ↑ ( ~r f , ηz f ) n ( m ) f, ↓ ( ~r f , ηz f ) . (4)In the absence of d -electron ordering, Eq.(4) is theALM with weak f -electron dispersion and momentum-dependent hybridization. (The effect of momentum-dependent hybridization on the Kondo effect has recentlybeen studied in other contexts [23, 24].) For sufficientlylarge U f , and with the f -levels being well below Fermienergy, we are in the Kondo limit.To concretely demonstrate how the d -electron AF or-der influences the Kondo effect, we consider the resultingKondo lattice model with a single f -electron channel andtwo d -electron bands. In the slave-boson representation,this becomes H KLM = X k ǫ ( αα ′ ) d ( k ) d ( α ) k σ † d ( α ′ ) k σ + λ (cid:18) N L X k f † k σ f k σ − (cid:19) + X k [ σ ∆ AF d ( α ) k σ † d ( α ) k + Q ,σ + H.c.] − J K X k V df ( k ) b α [ f † k σ d ( α ) k σ + H.c.] . (5)Here, the Lagrange multiplier λ enforces the single oc-cupancy of f -electrons. The mean-field parameter b α = h f † k σ d ( α ) k σ i / T K ∝ b . The anisotropic hybridizationform factor V df ( k ) = 4 cos k x / k y /
2. The energy dis-persion for d -electrons are taken to be [25]: ǫ (1) ( k ) = − t cos k x − t cos k y − t cos k x cos k y ,ǫ (2) ( k ) = − t cos k x − t cos k y − t cos k x cos k y ,ǫ (12) ( k ) = ǫ (21) ( k ) = − t sin k x sin k y , with t = − t = 1 . t = t = − .
85. In ournumerical study, we choose J K = 0 .
04, temperature T = 10 − | t | , and the lattice size N L = 3200 × d -electrons are fixed atthe half-filling n d = 2 . d -electron AF order rapidly sup-presses the Kondo scale. This suppression is closely re-lated to the depression of the d -electron density of states(DOS) in the collinear AF state of undoped iron arsenides(see the inset of Fig. 2). The feature of low energy DOS issensitive to the degree of nesting and the DOS minimumis not necessarily located precisely at the Fermi energy(see, e.g. , the case of M d = 0 .
021 in Fig. 2); the latterexplains the effective Kondo scale first rising and thendropping with the AF order. Furthermore, the incom-plete nesting of the Fermi surface keeps the depressedDOS finite (unlike, say, in the superconducting state) atthe Fermi energy such that the T = 0 ground state hasthe f -moment always Kondo screened on the lattice.We should stress that, for the purpose of a semi-quantitative assessment of the proposed mechanism, wehave considered the upper limit for the Kondo scale inthe AF state: we have coupled the f -moments to only the -0.5 -0.25 0 0.25 0.5E/|t |00.511.52 ρ d ( E ) ( s t a t e s / | t | ) M d b × -3 FIG. 2: (Color online) Mean-field Kondo parameter b as afunction of the Q = ( π,
0) (in the notation of the one-Fe Bril-louin zone) staggered magnetization M d . The Kondo temper-ature T K ∝ b . M d is measured in µ B / Fe. The inset showsthe d -electron density of states for M d = 0 (dotted-black),0.021 (black-red), 0.165 (dashed-green), 0.296 (dash-dotted-blue). quasiparticles of the d -electron AF state and have alsoneglected the f -moment ordering; moreover, a genuine f -electron quantum phase transition will be induced bybreaking the Kondo screening upon the inclusion of thestandard RKKY-Kondo competition [26, 27, 28]. We cantherefore infer that the mechanism proposed here pro-vides a viable basis to understand the distinct f -electronheavy fermion behaviors in CeOFeP ( M d ≈
0) and Ce-OFeAs ( M d ≈ . − x P x series. In general, there willbe two magnetic quantum critical points x c and x c , as-sociated with the d - and f -electrons, respectively. TheRKKY interaction would then dominate in the interme-diate region of x , leading likely to a ferromagnetic orderbefore the heavy fermion state is approached. Magnetic frustration of the f -electrons. We now turnto the exchange interactions among the f -moments. Con-sider first the superexchange interaction, which can bederived by integrating out the virtual valence fluctua-tions of the f -electrons. From Eq. (4), we end up with˜ H f = P ~r J ( m,m ′ ) f ~S ( m ) F ( ~r f , ηz f ) · ~S ( m ′ ) F ( ~r f , ηz f ), where ~S ( m ) F ( ~r, ηz f ) = P (cid:3) ~S ( m ) f ( ~r f , ηz f ) are summations of f -electron spins in the corresponding plaquettes associatedwith ~r , and J ( m,m ′ ) f ≈ V ( mm ′ ) f ] ( U f + ε ( m ) f − ε ( m ′ ) f ). Thisis the superexchange interaction associated with the R - X - R path, which does not mix the odd and even sublat-tices of the f -sites in a single R O layer (see Fig. 3(a)).There will also be a superexchange interaction from the R -O- R path, due to the hybridization between the 4 f -orbitals of X -atoms and the 2 p -orbitals of O-atoms; thissuperexchange mixes the odd and even sublattices (seeFig. 3(b)). In the notations of an effective square lat-tice of the f -sites ( c.f. Fig. 3(c)) the R -O- R path gives (b) (c) (a) FIG. 3: (Color online) The would-be ordering patterns ofthe f -electrons due to the superexchange interactions via (a)the R - X - R process alone or (b) the R -O- R process alone.(c) illustrates the combined exchanges, viewed as J - J - J interactions of an effective square lattice within an R O layer,which are expected to turn the orders of (a) and (b) into ahelical one. The blue and brown circles label the rare-earthsites in the same way as in Fig. 1. rise to the nearest-neighbor (n.n.) interaction J ( O )1 andthe next-nearest-neighbor (n.n.n.) J ( O )2 , while the R - X - R path yields the n.n.n. J ( X )2 and the third-nearest-neighbor (n.n.n.n.) J ( X )3 . (Note that J ( X )2 and J ( X )3 correspond to the n.n. and n.n.n. interactions inthe odd/even sublattices separately.) The resulting f -electron spin Hamiltonian becomes a J - J - J Heisenbergmodel (Fig. 3(c)) H f = { X n.n. J + X n.n.n. J + X n.n.n.n. J } ~S i · ~S j , (6)where J = J ( O )1 , J = J ( O )2 + J ( X )2 , and J = J ( X )3 .In this way, the f -moments of CeOFeAs provides a re-alization of a geometrically frustrated quantum magneticsystem in two dimensions. Quantum frustrated magnetshave been the subject of theoretical studies for a longtime, and continue to attract extensive interest [29].However, suitable materials with spin-1 / f -moments in CeOFeAs and relatedarsenides.The ~S F · ~S F form given earlier corresponds to J ( X )3 /J ( X )2 being equal to 1 /
2, and further bond-angleconsiderations imply that J ( X )3 /J ( X )2 will be larger than1 / J ( O )2 /J ( O )1 ≈ /
2. We will thereforeexpect J > J / J /J . In this range,the N´eel and collinear orderings are excluded. Instead,an incommensurate helical phase with the ordering vec-tor ( q, π ) or ( q, q ) is the most likely ground state, wherecos q = J − J J [30]. Neutron scattering and muon spinrelaxation experiments in polycrystal CeOFeAs appearto have seen a helical f -electron ordering [10, 13].In the d -electron paramagnetic regime, there will alsobe an RKKY interaction. The latter is expected to beferromagnetic given the relatively small size of the Fermisurfaces, and this is consistent with the enhanced ferro- magnetic fluctuations of the heavy fermion state observedin CeOFeP [9]. Still, the frustrating J - J - J superex-change interactions will continue to operate, helping tosuppress the tendency for AF ordering. Discussion and summary.
A number of other conse-quences of the p - f hybridization are relevant to the iron-pnictides phase diagram. First, in the heavy fermionphase, the momentum-dependence of the induced d - f hybridization will generally smear the hybridizationgap (which has nodal lines along K x = ± π/a and K y = ± π/a ), and this could be visible in the optical-conductivity spectrum. Second, the induced d - f hy-bridization depends on the Fe- X and X - R distances. In-creasing pressure along the c -axis will decrease the dis-tances and increase the hybridizations, and eventuallyenhance T K [11]. Finally, in light of the fact that the f -electron ordering is further suppressed by the competing J - J - J interactions, the transition or crossover from thesuperconducting to the heavy fermion phases may takeplace at sufficiently high pressures in the carrier-dopedsuperconducting materials.In summary, we have considered a mechanism forweakening the Kondo screening effect through the an-tiferromagnetic order of the conduction electrons, andimplemented it in an extended Anderson lattice Hamil-tonian. For the iron pnictides, our mechanism issemi-quantitatively viable to explain the observed ex-istence/absence of heavy fermion behavior in CeOFePand CeOFeAs, respectively. More broadly, our mecha-nism goes beyond the standard picture of heavy fermionphysics, viz. the RKKY and Kondo competition, and cantherefore shed new light on the phase diagram of heavyfermion systems in general. Finally, we have proposedthat the f -electrons in the parent iron arsenides representa rare model system for quantum frustrated magnetismin two dimensions.We thank E. Abrahams, M. Aronson, G. H. Cao, X. H.Chen, X. Dai, C. Geibel, N. L. Wang, T. Xiang, Z. A. Xu,and H. Q. Yuan for useful discussions, and the U.S. DOECINT at LANL for computational support. This workwas supported by the NSF of China, the 973 Program,and the PCSIRT (IRT-0754) of Education Ministry ofChina (J.D.), the NSF Grant No. DMR-0706625 and theRobert A. Welch Foundation (Q.S.), and by U.S. DOEat LANL under Contract No. DE-AC52-06NA25396 (J.-X.Z.). [1] Y. Kamihara, T. Watanabe, M. Hirano, and H. Hosono,J. Am. Chem. Soc. , 3296 (2008).[2] G. F. Chen, Z. Li, D. Wu, G. Li, W. Z. Hu, J. Dong, P.Zeng, J. L. Luo, and N. L. Wang, Phys. Rev. Lett. ,247002 (2008).[3] Z.-A. Ren, J. Yang, W. Lu, W. Yi, X.-L. Shen, Z.-C. Li,G.-C. Che, X.-L. Dong, L.-L. Sun, F. Zhou, and Z.-X. Zhao, Europhys. Lett. , 57002 (2008).[4] X. H. Chen, T. Wu, R. H. Liu, H. Chen, and D. F. Chen,Nature , 761 (2008).[5] C. Wang, L. Li, S. Chi, Z. Zhu, Z. Ren, Y. Li, Y. Wang,X. Lin, Y. Luo, S. Jiang, X. Xu, G. Cao, and Z. Xu,Europhys. Lett. , 67006 (2008).[6] C. de la Cruz, Q. Huang, J. W. Lynn, J. Li, W. RatcliffII, J. L. Zarestky, H. A. Mook, G. F. Chen, J. L. Luo, N.L. Wang, and P. Dai, Nature , 899 (2008).[7] Y. Kamihara, H. Hiramatsu, M. Hirano, R. Kawamura,H. Yanagi, T. Kamiya, and H. Hosono, J. Am. Chem.Soc. , 10012 (2006); T.M. McQueen, M. Regulacio,A. J. Willams, Q. Huang, J. W. Lynn, Y. S. Hor, D.V. West, M. A. Green, and R. J. Cava, Phys. Rev. B , 024521 (2008); J. J. Hamlin, R. E. Baumbach, D. A.Zocco, T. A. Sayles, and M. B. Maple, J. Phys.: Condens.Matter , 365220 (2008).[8] Y. Kamihara, M. Hirano, H. Yanagi, T. Kamiya, Y.Saitoh, E. Ikenaga, K. Kobayashi, and H. Hosono, Phys.Rev. B , 214515 (2008).[9] E.M. Bruning, C. Krellner, M. Baenitz, A. Jesche, F.Steglich, and C. Geibel, Phy. Rev. Lett. , 117206(2008).[10] J. Zhao, Q. Huang, C. de la Cruz, S. Li, J. W. Lynn, Y.Chen, M. A. Green, G. F. Chen, G. Li, Z. Li, J. L. Luo,N. L. Wang, and P. Dai, Nature Mater. , 953 (2008).[11] L. Pourovskii, V. Vildosola, S. Biermann, and A.Georges, Europhys. Lett. , 37006 (2008).[12] X. Dai, private communications (2009).[13] H. Maeter, H. Luetkens, Yu. G. Pashkevich, A. Kwadrin,R. Khasanov, A. Amato, A. A. Gusev, K. V. Lamonova,D. A. Chervinskii, R. Klingeler, C. Hess, G. Behr, B.Buechner, and H.-H. Klauss, arXiv:0904.1563.[14] S. Chi, D. T. Adroja, T. Guidi, R. Bewley, S. Li, J.Zhao, J. W. Lynn, C. M. Brown, Y. Qiu, G. F. Chen, J. L. Lou, N. L. Wang, and P. Dai, Phy. Rev. Lett. ,217002 (2008).[15] J. Dai, Qimiao Si, Jian-Xin Zhu, and E. Abrahams, Proc.Natl. Acad. Sci. , 4118(2009).[16] P. Gegenwart, Q. Si, and F. Steglich, Nat. Phys. , 186(2008).[17] S. Doniach, Physica B , 231 (1977); C. M. Varma, Rev.Mod. Phys. , 219 (1976).[18] R. Pottgen and D. Johrendt, Z. Naturforsch. B , 1135(2008).[19] C. Krellner, T. F¨orster, H. Jeevan, C. Geibel, and J.Sichelschmidt, Phys. Rev. Lett. , 066401 (2008).[20] Q. Si and E. Abrahams, Phys. Rev. Lett. , 076401(2008).[21] T. Yildirim, Phy. Rev. Lett. , 057010 (2008).[22] F. Ma, Z.-Y. Lu, and T. Xiang, Phys. Rev. B , 224517(2008).[23] P. Ghaemi, T. Senthil, and P. Coleman, Phys. Rev. B , 245108 (2008).[24] H. Weber and M. Vojta, Phys. Rev. B , 125118 (2008).[25] S. Raghu, X.-L.Qi, C.-X. Liu, D. J. Scalapino, and S.-C.Zhang, Phys. Rev. B , 22053(R) (2008).[26] Q. Si, S. Rabello, K. Ingersent, and J. L. Smith, Nature(London) , 804 (2001).[27] T. Senthil, M. Vojta, and S. Sachdev, Phys. Rev. B ,035111 (2004).[28] I. Paul, C. P´epin, and M. Norman, Phys. Rev. Lett. ,026402 (2007).[29] H. T. Diep (ed.), Frustrated Spin Systems (World Scien-tific, 2005).[30] A. Moreo, E. Dagotto, T. Jolicoeur, and J. Riera, Phys.Rev. B , 6283 (1990); M. P. Gelfand, R. R. P. Singh,and D. A. Huse, ibid.40