RKKY Interaction and Intrinsic Frustration in Non-Fermi Liquid Metals
RRKKY Interaction and Intrinsic Frustration in Non-Fermi Liquid Metals
Jian-Huang She, and A. R. Bishop
Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA. (Dated: November 12, 2018 [file: RKKY-JHS3arxiv])We study RKKY interaction in non-Fermi liquid metals. We find that the RKKY interactionmediated by non-Fermi liquid metals can be of much longer range than for a Fermi liquid. Theoscillatory nature of RKKY interaction thus becomes more important in non-Fermi liquids, andgives rise to enhanced frustration when the spins form a lattice. Frustration suppresses the magneticordering temperature of the lattice spin system. Furthermore, we find that the spin system withlonger range RKKY interaction can be described by the Brazovskii model, where the orderingwavevector lies on a shell with constant radius. Strong fluctuations in such model lead to a first-order phase transition and/or glassy phase. This may explain some recent experiments where glassybehavior was observed in stoichiometric heavy fermion material close to a ferromagnetic quantumcritical point.
Introduction:
When magnetic moments are placedin a metal, the conduction electrons mediate an indirectinteraction between these moments. Such a long rang in-teraction is called the Ruderman-Kittel-Kasuya-Yosida(RKKY) interaction. RKKY interaction plays crucialroles in, e.g., heavy fermions, diluted magnetic semicon-ductors, graphene. The usual derivation of the RKKY in-teraction is based on the assumption that the conductionelectrons form a Landau Fermi liquid. However manystrongly correlated electron systems show non-Fermi liq-uid behavior, e.g. cuprates, heavy fermions, pnictides.The question we ask here is what is the form of RKKYinteraction in a non-Fermi liquid metal, and what are theconsequencies.Of particular interest are heavy fermion systems, wherelocal moements couple to the conduction electrons. TheDoniach phase diagram with competing Kondo couplingand RKKY interaction has been the paradigm for heavyfermions for decades [1]. In the last few years, as exper-imental results accumulate, there is a growing necessityto go beyond the Doniach phase diagram. Frustrationor the quantum zero point energy has been proposed asa new dimension in the global phase diagram of heavyfermions [2–6]. One obvious origin of frustration is frus-tration of lattice structure itself. However such geometricfrustration is not universally observed in heavy fermionmaterials. Here we propose that the non-Fermi liquidnature of conduction electrons in the Kondo liquid phaseleads to intrinsic frustration for the localized spin de-grees of freedom. This provides a more universal sourceof frustration.Our approach is based on the idea of quantum critical-ity and non-Fermi liquid (NFL) behavior. The standardpicture is that the critical fluctuations near a quantumcritical point (QCP) leads to NFL behavior. Here wedepart from this picture by starting with the assump-tion that in a certain range of the parameter space, theitinerant electrons form a NFL state. We then proceedto study its consequences on other degrees of freedom,e.g. the localized spins. Focusing on the regime withsmall Kondo coupling, i.e. a small Fermi surface, we findthat the magnetic transition temperature will be reduced T x NFLFL T frustration MagneticallyMagnetically disorderedordered
FIG. 1: Schematic electronic and magnetic phase diagrams.Distance dependence of the RKKY interaction is shown in theinsets. In the non-Fermi liquid region, RKKY interaction is oflonger range, leading to frustration. The magnetic transitiontemperature decreases with increasing frustration, and newphases can emerge near the QCP. by the frustration resulting from longer-range RKKY in-teraction produced by NFL itinerant electrons. Further-more, we find that the putative ferromagnetic (FM) QCPmay be replaced by a first-order phase transition or aglassy phase [7,8] (see Fig. 1).
Formalism:
We start with the Kondo lattice model, H = H C + H K . Here H C is the conduction elec-tron Hamiltonian, and ususally only the hopping termis included. The Kondo coupling between conductionelectrons and localized spins is of the form, H K = − J K (cid:80) iαβ S i · c † iα σ αβ c iβ . We depart from the usualapproach by considering the conduction electrons to bestrongly interacting themselves, i.e. H C = H (0) C + H (int) C .One way to motivate this is to consider the phenomeno-logical two fluid model [9–12]. In many heavy fermionsystems, below the coherence temperature T ∗ , the ex-perimental results can be understood in terms of thetwo fluid model, with one component the itinerant heavyelectrons, and the other component local moments. Theheavy electron Kondo liquid is not a simple Fermi liquid,e.g. the specific heat is logarithmically enhanced at lowtemperature. One has a model of interacting itinerantelectrons coupled with localized spins.The itinerant electrons induce a RKKY type interac- a r X i v : . [ c ond - m a t . s t r- e l ] A p r tion among the localized moments: H RKKY = (cid:88) ij J abij S ai S bj . (1)Here the coupling J abij = − J K χ abij [13–15], is determinedby the static spin susceptibility of the conduction elec-trons χ abij = − i (cid:126) (cid:90) ∞ (cid:104) [ s a ( r i , t ) , s b ( r j , (cid:105) e − ηt dt, (2)with the electron spin s a ( r i ) = (cid:80) αβ c † iα σ aαβ c iβ and η = 0 + . If the conduction electrons are in the param-agnetic state, the spin susceptibility is isotropic and di-agonal, i.e. χ abij = χ ( r ij ) δ ab . For Fermi liquids, the spinsusceptibility behaves as χ ( r ) ∼ (1 /r d ) cos(2 k F r + θ )at long distances, with d the spatial dimension. Thisleads directly to the standard form of the RKKY inter-action. The exponent d results from the sharp jump inthe momentum distribution n ( k ), characteristic of Fermiliquids. For non-Fermi liquid metals, the RKKY inter-action can have qualitatively different behavior. We stillassume the existence of a Fermi surface, i.e. singularityin n ( k ), thus the spin susceptibility still has 2 k F oscilla-tion. The exponent can take a different value. Thus wehave χ ( r ) ∼ (1 /r α ) cos(2 k F r + θ ). More detailed studiesof the NFL spin susceptibility will be presented below.Consider placing a lattice of spins in the non-Fermiliquid metal. We focus on the effect of the RKKY in-teraction to the spin system, and will not consider thecompetition between Kondo coupling and RKKY in-teraction [1]. This can be achieved by assuming thespins to be classical, or considering only the part ofthe phase diagram with a small Fermi surface. With S ( q ) = (1 /N ) (cid:80) i S i e i q · r i , one has in momentum space, H = (cid:80) q F ( q ) S ( q ) · S ( − q ), where F ( q ) = 1 N (cid:88) r i (cid:54) =0 J ( r i ) e i q · r i , (3)with r i defined on the lattice. The ordering wavevec-tor in the ground state is determined by minimizing thefunction F ( q ).For the conventional three dimensional RKKY interac-tion mediated by a Fermi liquid, this problem has beenstudied in [16], where different phases have been iden-tified as the conduction electron density changes. Atsmall k F a , where a is the lattice constant, the groundstate is ferromagnetic. As k F a increases, antiferromag-netic phases with different ordering wavevectors appear.In the case k F a →
0, the above summation can be re-placed by an integral, and F ( q ) ∼ − χ ( q ). The orderingwavevector is thus determined by maximizing the staticspin susceptibility.Now we consider in more detail what is the form ofthe static spin susceptibility in a NFL metal. Whenvertex corrections can be ignored, the spin susceptibil-ity can be calculated from the fermion bubble, with χ ab ( q ) ∼ (cid:82) σ a G ( k + q ) σ b G ( k ). When the momentumdistribution n ( k ) has a weaker singularity than a jumpat k F , e.g. a kink, the Friedel oscillation decays fasterthan that of Fermi liquids (see Appendix). Then one ex-pects χ ( r ) and J ( r ) to decay faster than that of Fermiliquids. An interesting question is whether it is possibleto have longer range RKKY interactions, which wouldgenerate the desired frustration among the spins [2–6].We will present two models of NFL metals that can giverise to such behavior. Longer range RKKY interaction in 1-d:
First,as a proof of principle that RKKY interaction in astrongly interacting electron system can be of longerrange than in a free system, let us first consider one di-mension. In 1-d, RKKY interaction mediated by freeelectrons is of the form J ( r ) ∼ Si(2 k F r ) − π , withthe sine integral function Si( x ). At large distance onehas J ( r ) ∼ cos(2 k F r ) /r . In momentum space, one has χ ( q ) ∼ (1 /q ) ln | q +2 k F q − k F | , with a maximum at q = 2 k F .The low energy dynamics of interacting electrons in 1-dis described by the Luttinger liquid theory. Due to spin-charge separation, the conduction electron Hamiltonaincan be written as a summation of the two channels [17], H C = (cid:88) α = c,s v α (cid:90) dx [ g α Π α + g − α ( ∂ x θ α ) ] , (4)with v c and v s the velocity of charge and spin densitywave respectively. The charge interaction constant g c = 1for noninteracting fermions, g c < g c > g s = 1 in the presence of SU(2) spinsymmetry. The oscillating part of the spin correlationfunction is [17] (cid:104) s ( x, τ ) · s (0 , (cid:105) ∼ cos(2 k F x ) | τ + ix/v c | g c | τ + ix/v s | g s . (5)The RKKY interaction, determined from the static spinsusceptibility, is of the form J ( x ) ∼ (cid:90) dτ cos(2 k F x ) | τ + ix/v c | g c | τ + ix/v s | g s ∼ cos(2 k F x ) x g c + g s − . (6)For g c < g s = 1, the exponent α = g c + g s − < d = 1.The RKKY interaction mediated by a Luttinger liquidis thus of longer range than that mediated by a non-interacting Fermi gas . Spin susceptibility in 2-d:
Now we consider twodimenional metals. For free electrons, the static spinsusceptibility reads χ ( q ) = (cid:40) χ for q < k F χ (cid:104) − (cid:112) − (2 k F /q ) (cid:105) for q > k F , (7)with χ = 1 /π , which has a one-sided square-root sin-gularity. The RKKY interaction is thus of the form q k f Χ (cid:72) a (cid:76) q k f (cid:45) (cid:45) (cid:45) F (cid:72) q (cid:76) (cid:72) b (cid:76) FIG. 2: (a) The static spin susceptibility χ ( q ) as function ofmomentum for Fermi liquid (dashed, black) and the gauge-fermion model with σ < / σ > / F ( q ) as function of momentum for an-gles θ = 0 , π/ , π/
4. The curves for different angles are al-most identical. Here the spins form a square lattice, and σ = 1 / , k F a = 0 . J ( r ) ∼ sin(2 k F r ) /r . For 2-d Fermi liquid, includinghigher order diagrams, there is also a square-root singu-larity for q < k F , with χ ( q ) = χ (2 k F ) + χ sing ( q ) [18],and χ sing ( q ) = A (cid:112) − ( q/ k F ) . (8)The new singularity gives contribution δχ ( r ) ∼ [(2 k F r ) cos(2 k F r ) − sin(2 k F r )] /r , and will not changethe long distance behavior of the RKKY interaction.A prototype of non-Fermi liquid metal in higher di-mensions is the system of 2-d degenerate fermions inter-acting via a singular gauge interaction [19–26], where thepresence of the gauge interaction leads to singular 2 k F response [23]. The fermion 2 k F vertex Γ k F has a powerlaw dependence on frequency, with Γ k F ∼ (cid:0) E F ω (cid:1) σ Γ k F .The exponent is of the form σ = N + π N ln N + O (cid:0) N (cid:1) for large N , and σ = √ π √ N + O (1) for small N .Here the spin index is generalized to take values from 1to N . Taking N = 2, one obtains σ = 0 .
35 from thelarge-N expansion, and σ = 0 .
56 in the small-N limit.The spin susceptibility is calculated from the polariza-tion bubble with vortex corrections, χ ( q , ω ) (cid:39) Π( q , ω ) = (cid:82) d p d(cid:15)G ( p + q / , (cid:15) + ω/ G ( p − q / , (cid:15) − ω/
2) [Γ (cid:15) p ( q , ω )] .For σ < /
3, the static spin susceptibility is of the form[23] χ ( q ) ∼ χ − C| q − k F | − σ , (9)and for σ > / χ ( q ) ∼ | q − k F | σ − , (10)with a singularity at q = 2 k F (see Fig. 2(a)). Fouriertransforming to real space, we find χ ( r ) ∼ (cid:90) | q − k F | σ − J ( qr ) qdq ∼ cos(2 k F r − θ ) r / − σ . (11)The exponent α = 5 / − σ can be much smaller thanthe space dimension d = 2.More generally, for non-Fermi liquid metals, one canemploy a scaling theory for the susceptibility (see e.g. [27,28]). Assuming the existence of a Fermi surface, thestatic spin susceptibility generally has a power law behav-ior near q = 2 k F , with χ ( q ) ∼ | q − k F | ν . For ν < / Longer range RKKY interaction in 2-d:
Let usnow consider the ground state of the spins embedded inthe 2-d metals with small k F a . For the Fermi liquid case(Eqs.(7,8)), χ ( q ) increases monotonically with decreasing q (see Fig. 2). The ground state is ferromagnetic. Fornon-Fermi liquid (Eqs.(9),(10)), the maximum of χ ( q ) isat q = 2 k F , indicating an instability of the ferromagneticstate. More precisely, one can calculate the interaction F ( q ) by first Fourier transforming χ ( q ) to real space toget χ ( r ), and then performing the lattice summation inEq. (3). The result for σ = 1 / F ( q ) has a minimum at q = 2 k F . Thesingularity in χ ( q ) is smeared out by the lattice effect.Another observation is that F ( q ) has a very weak de-pendence on the direction of momentum. In Fig. 2(b), F ( q ) for the three different angles are almost indistin-guishable. With the minimum of F ( q ) at q = 2 k F , theordering wavevector of the lattice spin system lies on ashell of radius 2 k F . Expanding F ( q ) around q , one ob-tains the Brazovskii model [29], H = (cid:88) q (cid:104) b + D ( | q | − q ) (cid:105) S ( q ) · S ( − q ) . (12)Brazovskii found that the large phase space available forfluctuations around a shell of minima leads to a first-order phase transition [29]. It has been found experi-mentally that putative FM-QCPs are replaced by firstorder transitions at low temperatures in several transi-tion metal compounds, e.g. MnSi, ZrZn , and heavyfermion systems, e.g. UGe , UCoAl, UCoGe (see [30]and references therein). It was realized earlier that com-peting orders [31] as well as fluctuations [32–35] can leadto first order quantum phase transitions. Here we finda new mechanism where the frustration resulting fromNFL behavior generates first order transitions.A further observation is that the extensive configura-tional entropy in the Brazovskii model should lead toslow dynamics and glassiness [36–39]. Glassy correla-tions emerge when the correlation length ξ = ( D/b ) / becomes of order the modulation length l = 2 π/q [37].The parameter b needs to be determined self-consistently.Within the large-N approximation, and including a smallquartic term with coupling u , we have b = b + uT (cid:90) d q (2 π ) G ( q ) , (13)with the Green’s function G ( q ) = 1 / [ b + D ( q − q ) ].The condition ξ/l ∼ T g (cid:39) πD u q − b /Dc − log( q a ) ,with the coefficient c of order unity and momentum cut-off Λ ∼ a − . We notice that here T g depends logarithmi-cally on cutoff instead of the 1 / Λ dependence for the 3-dmodel considered in [37].Glassy spin dynamics was recently observed in theheavy fermion system CeFePO [40]. CeFePO is a layeredKondo lattice system, in close proximity to a FM QCP.Spin-glass-like freezing was detected in the ac suscepti-bility, specific heat and muon-spin relaxation [40]. Theglass behavior in such a stoichiometric system points tonew mechanisms that do not reply on external random-ness. Our model provides such a possibility (see [41–46]and references therein for earlier attempts to obtain glassbehavior from frustrated deterministic models).
Away from QCP:
Having identified a glass transi-tion near the QCP, we proceed to study the behaviorof the lattice spin system away from QCP using a ran-dom exchange model that is widely used to describe spinglasses. Due to the cosin function, the RKKY interac-tion changes sign and magnitude with distance. It canbe well approximated by a random interaction [47–50], J ij ∼ J K (cid:15) ij r α , where (cid:15) ij is a random variable with cosinedistribution P ( (cid:15) ij ) = (1 /π )(1 − (cid:15) ij ) − / .When the itinerant electrons are away from the QCP,there is a crossover to the Fermi liquid behavior at lowenergy, or equivalently long distance, where the RKKYinteraction is substantially reduced. We will assume forsimplicity that the RKKY interaction can be neglectedbeyond a crossover scale r FL . Then the exchange inter-action is of the form J ij = (cid:26) A (cid:15) ij / | r i − r j | α for | r i − r j | < r FL | r i − r j | > r FL . (14)When r FL becomes of order the lattice constant, onlythe nearest neighbor interactions survive, i.e. H = J (cid:80)
0, andeach spin interacts with z neighouring spins. The spinsorder ferromagnetically below the transition temperature T (0) c = ˜ J S ( S + 1) /
6, with ˜ J = zJ . This correpondesto the case far away from the QCP.Then we add to the above mean field ferromagneticmodel random exchange interactions to model the frus-tration effect when approaching a QCP. The new Hamil-tonian can be written as H = − (cid:80) ( ij ) J ij S i · S j , wherethe interaction J ij is distributed according to P ( J ij ) = √ πJ exp (cid:104) − ( J ij − J ) J (cid:105) [51,52]. This model is readilysolved by the replica technique [52,53], and the transi-tion temperature to ferromagnetism is reduced by therandom interactions, with the result [52,54] T c = T (0) c (cid:34)
12 + 12 (cid:115) − S ( S + 1) ˜ J ˜ J (cid:35) , (15) zz c T c T c (cid:72) (cid:76) FIG. 3: Ferromagmetic transition temperature as function ofrange of random exchange interaction. where we have defined ˜ J = z / J .We fix the mean field ordering temperature in the ab-sence of random exchange interaction T (0) c and the vari-ance of the random distribution J , so that z is a measureof the range of random exchange interaction, i.e. r FL inEq.(14). We can define z c = ( S ( S + 1) /
3) ˜ J /J , andwrite T c in the form T c = T (0) c (cid:20)
12 + 12 (cid:114) − zz c (cid:21) , (16)which is plotted in Fig. 3. One can see that with in-creasing range of random exchange interaction, the FMordering temperature decreases. This then translates tothe picture that when approaching the QCP, as RKKYinteraction becomes of longer range, magnetic orderingis suppressed (see Fig. 1). Conclusions:
We have studied the RKKY interac-tion in non-Fermi liquid metals. The basic picture we findis summarized in Fig. 1. In the non-Fermi liquid phase,when including vertex corrections, the RKKY interactioncan be of longer range than in a Fermi liquid. Longerrange RKKY interaction leads to frustration for the lat-tice spin system placed in such a NFL metal. Magneticordering will be suppressed by frustration, and novel be-havior may emerge near the putative QCP. In particu-lar, the continuous second-order phase transitions maybe replaced by first-order transtions. Glassy dynamicsmay occur near the QCP without invoking disorder. Onecandidate material for such glass behavior is the heavyfermion system CeFePO. We focused here on FM QCP.One can also generalize the whole procedure to AFMQCP by increasing k F a . Another interesting questionis the competition between the Kondo coupling and thelonger range RKKY interaction.We acknowledge useful discussions with Sasha Bal-atsky, Cristian D. Batista, Andrey Chubukov, MatthiasGraf, Jason T. Haraldsen, John Hertz, John Mydosh,Stephen Powell, Jan Zaanen, and Jian-Xin Zhu. Thiswork was supported by the U.S. Department of Energyunder contract DE-AC52-06NA25396 at Los Alamos Na-tional Laboratory through the Basic Energy Sciences pro-gram, Materials Sciences and Engineering Division. Appendix: Friedel oscillation
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