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Phase diagram of the one dimensional anisotropic Kondo-necklace model S. Mahmoudian1 and A. Langari1,∗ 1Physics Department, Sharif University of Technology, Tehran 11155-9161, Iran (Dated: February 2, 2008) The one dimensional anisotropic Kondo-necklace model has been studied by several methods. It is shown that a mean field approach fails to gain the correct phase diagram for the Ising type anisotropy. Wethenappliedthespinwavetheorywhichisjustifiedfortheanisotropiccase. Wehave derived the phase diagram between the antiferromagnetic long range order and the Kondo singlet phases. We have found that the exchange interaction (J) between the itinerant spins and local 8 ones enhances the quantum fluctuations around the classical long range antiferromagnetic order 0 and finally destroy the ordered phase at the critical value, Jc. Moreover, our results show that the 0 onsetofanisotropyintheXYtermoftheitinerantinteractionsdevelopstheantiferromagneticorder 2 for J <Jc. This is in agreement with the qualitative feature which we expect from the symmetry of the anisotropic XY interaction. We have justified our results by the numerical Lanczos method n where the structurefactor at theantiferromagnetic wave vector diverges as thesize of system goes a J toinfinity. 1 PACSnumbers: 75.10.Jm,75.30.Mb,75.30.Kz,75.40.Mg 2 ] l I. INTRODUCTION inspatialdimensionD=1,2,3. Theonedimensionalcase e isalwaysintheKondosingletphasewhichisjustifiedby - r the numerical Monte-Carlo7 and Density Matrix Renor- t QuantumphasetransitionsbetweenKondosingletand s malization Group8 (DMRG) results. While a quantum antiferromagnetically ordered states such as found in . t phase transition between the Kondo singlet phase and a heavy fermion compounds have been attracted much re- m searchinterestrecently1,2,3. Itisthereforeofgreatimpor- theantiferromagnetic(AF)orderedphasehappensbyin- creasingtheratio(t/J)ofinter-sitetothelocalexchange tancetounderstandtheapproachtothequantumcritical - interaction for D=2, 3. The effect of both intersite and d region within suitable theoretical models. The most im- n portant among them is the Kondo lattice model consist- intrasite anisotropieson the quantum critical point have o also been investigated by the same approach10. Mean ing of a free conduction band and an on-site antiferro- c field theory has also been applied to study the effect of magnetic Kondo interaction which favors non-magnetic [ magneticfieldontheKondonecklacemodel11. However, singlet formation. In the second order it also leads to 2 the effective inter-site Ruderman-Kittel-Kasuya-Yosida for D=1 the mean field approach always shows a non- v magnetic Kondo singlet phase even in the presence of (RKKY) interactions which favor magnetic order. Their 4 anisotropy in the easy axis term10. competitionleadstotheappearanceofthequantumcrit- 1 9 icalpoint. Insuchapictureonlyspindegreesoffreedom In the present work we want to study the possible 3 are involved in the quantum phase transition. Therefore quantum phase transition in the D = 1 Kondo-necklace . the Kondo lattice model may be replaced by a simpler model under the assumption of the anisotropy in the 2 model where the itinerant hopping part is simulated by XY-interaction (η) between the spin of itinerant elec- 1 7 an inter-site interaction of the itinerant spins. Kondo trons. However, the real space renormalization group 0 necklace (KN) model has been originally proposed by (RG) study12 shows that if the anisotropy in the XY- : Doniach4fortheone-dimensionalcaseasasimplifiedver- interaction is greater than some nonzero value a phase v sionoftheitinerantKondolattice(KL)model5. Thereby transition from the Kondo singlet to the antiferromag- i X the kinetic energy of conduction electrons is replaced by netic long range order takes place by changing the local r an intersite exchange term. For a pure XY-type inter- exchange coupling (J/t). It means that for η < ηc the a site exchange this may be obtained by a Jordan-Wigner system is always in Kondo singlet state but for η > ηc transformation. The intuitive argument is that at low there is a critical exchange coupling (Jc/t) which is the temperatures the charge fluctuations in the Kondo lat- borderbetweenthe KondosingletandAFlongrangeor- ticemodelarefrozenoutandtheremainingspinfluctua- dered phases. However, we will show that the quantum tion spectrum can be simulated by an antiferromagnetic phase transition between the Kondo singlet and AF or- inter-site interaction term of immobile τ spins coupled der existsfor anynonzerovalue ofη. Our resultis based by a Kondo interaction to the local noninteracting spins on the spin wave approach which is justified in the ab- S. sence of local exchange for the anisotropic case. This is The quantum phase transition between the spin liq- in agreement with the general statement which tells us uid and antiferromagnetic phases in the one dimen- about the universality of this transition. Moreover, the sionalKondonecklacemodelhasbeenstudiedbyseveral Lanczos numerical computations justify that our state- methods6,7,8. The phase diagram of the genuine Kondo ment is correct. necklacemodelhasbeenstudiedbymeanfieldapproach9 In Sec.II we will explain why the mean field approach 2 fails to produce the correct results in one spatial dimen- and γ(k)=cos(k). The ground state energy is sion and easy axis anisotropy. We then consider a more general case of anisotropy in the Hamiltonian in Sec.III. J EAF =E +Nt¯2 µ+ ts¯2δ , (5) We then implement a spin wave theory to the Hamilto- 0 0 4 − (cid:20) (cid:21) nian with anisotropy in Sec.IV and show that the local exchange will increase the quantum fluctuations which where destroytheAForderedstate. InSec.Vwepresentournu- 3J 1 mericalLanczosresultswhichverifiesourproposedphase E =N (µ )s¯2 µ + ω (k) d (k) . 0 α α − 4 − 2 − diagram. Finally, the conclusion and discussion are pre- (cid:16) (cid:17) k,αX=x,y,z(cid:16) (cid:17) sented in Sec.VI. (6) Thechemicalpotential(µ)hasbeenaddedasaLagrange multiplier to the mean field Hamiltonian to preserve the II. MEAN FIELD THEORY dimension of Hilbert space on each bond. The ground state energy should be minimized with respect to s¯, t¯ In this section we shortly review the mean field ap- and µ. It gives the following equations proach to the Kondo-necklacemodel. We will show that J this approach fails to represent the antiferromagnetic µ=tδs¯2 , phase for Ising like anisotropy. Let first consider the fol- − 4 lowing Hamiltonian t¯2 = 5 J 1 1 + 2 , =t τxτx+τyτy +δτzτz +J τ S , (1) 4 − 2tδ − 4N Xk (cid:16) 1+γ(k) 1+ γ(δk)(cid:17) HAKN i j i j i j i· i p q Xhiji(cid:0) (cid:1) Xi s¯2 = 5 + J 1 1+γ(k)+2 1+ γ(k) , 4 2tδ − 4N δ wheret isthe strengthofexchangecouplingbetweenthe r Xk (cid:16)p (cid:17) spin (τ) of itinerant electrons and J is exchange cou- (7) plingbetweenthespinofitinerantelectronsandlocalized spins (S). The anisotropy parameter is represented by which should be solved self consistently. For δ > 1, we δ. The bond operator representation13 is used to trans- expect to havethe long rangeantiferromagneticorder in form the spin operators in terms of bosonic operators10 z direction. However, the summation 1 1 in (s,t ,t ,t ). We start from the strong coupling limit N k √1+γ(k) whexre ythezmean field theory works well and the model Eq.(7)divergeswhichshowsthatthemeanPfieldapproach fails to work correctly in this case. has a Kondo singlet ground state. For the mean field We have even considered the extreme case of only an approach10weassumeasingletcondensation s =s¯and h i Ising term in the interaction between itinerant spins, a condensation for one triplet component to induce the antiferromagneticordertk,z =√Nt¯δk,Q+ηk,z,whereηk,z namely t → 0 and tδ = constant. It is clear that for J =0 the ground state is long ranged antiferromagnetic isthequantumfluctuationabovethetripletcondensation ordered. However, the above mean field theory fails to and Q = π. After performing the Bogoliubov transfor- show the nonzero AF ordering. We conclude that the mation the Hamiltonian in momentum space becomes meanfieldtheoryisnotsuitabletoshowthelongranged antiferromagnetic phase in one dimension. It should be mf =EAF + [ ω (k)t˜† t˜ +ω (k)η˜† η˜ ] related to the strong quantum fluctuations which have HAKN 0 α k,α k,α z k,z k,z dominant effect in one dimension. Moreover, the above k α=x,y X X (2) mean field theory starts from the strong coupling limit where EAF is the ground state energy, ω (k) are the (t=0) where the singlet condensation is supposed to be 0 α excitation energies (α = x,y,z) and t˜† , η˜† (t˜ , the case while for the antiferromagnetic phase we need k,x(y) k,z k,x(y) to considerthe weak coupling limit (J =0). Having this η˜† )arethebosoniccreation(annihilation)operatorsin k,z in mind, we start from the weak coupling limit with an the diagonal representation. (More details can be found AF ordered ground state for the Ising anisotropy case. in Ref.10). The excitation energies are The effect of nonzero local exchange (J = 0) increases 6 the quantum fluctuations which finally destroy the AF ω (k)= d2(k) 4f2(k), α=x,y,z. (3) α α − α longrangeorder. Inthisrespect,weapplythe spinwave where p theory to the anisotropic Kondo necklace chain. ts¯2 f (k) = f (k)= γ(k), x y 4 III. THE MODEL HAMILTONIAN J ts¯2 d (k) = d (k)=µ+ + γ(k), x y 4 2 The Hamiltonian defined in Eq.(1) is always in the ts¯2 J ts¯2 Kondo singlet phase for XY anisotropy (δ 1) while it fz(k) = δγ(k), dz(k)=µ+ + δγ(k), (4) is expected to have a quantum phase tran≤sition to the 4 4 2 3 AF ordered state at a critical exchange coupling (J ) We then implement the Holstein-Primakov for Ising anisotropy (δ > 1). To consider a general cta|sce transformation14 to write the spin operators in terms we will investigate the phase diagram of the following of bosonic operators. Starting from the weak coupling Hamiltonian in one dimension regime (J = 0) the polarized ground state is the Neel state which impose the following transformations for N N H=t (τizτiz+1+(1−η)τiyτiy+1)+J (−→τi ·−→Si), (8) spins of the itinerant electrons Xi=1 Xi=1 τzA =τ a†a , τzB = τ +a†a , (10) where η is the anisotropy parameter which defines the i − i i i − i i deviation from the Ising limit (η = 1). The main rea- τ+A =(√2τ)(1 a†iai)1/2a , son to consider the anisotropy in this form is that the i − 2τ i anisotropyintheitinerantpartoftheinteractionenables a†a us to investigate the effect of symmetry on the results τi+B =(√2τ)a†i(1− 2iτi)1/2, of the Kondo necklace model compared with the Kondo a†a model. The present model, the case of η = 0 has U(1) τ−A =(√2τ)a†(1 i i)1/2, symmetry and for η =0 it has Z2 symmetry. In the case i i − 2τ of full anisotropy (η6= 1, Ising limit) we expect to have a†a τ−B =(√2τ)(1 i i)1/2a . a quantum phase transition from the antiferromagnetic i − 2τ i long range order to the Kondo singlet phase. Moreover, Intheaboveequationsτ isthemagnitudeofspinandthe in the absence of local exchange coupling (J = 0) the Boson creation (a†) and annihilation (a) operators obey universal behavior of the XY term is the same as for all the following commutation relations, η = 0. It is also another motivation to see what is the 6 effectofη anisotropyonthephasediagramofthemodel. [a†,a ]=δ , [a†,a†]=0 , [a ,a ]=0. (11) Specially, our final results show that there exist an anti- i j i,j i j i j ferromagnetic phase for any nonzero value of η when J Asimilartransformationsisalsodefinedforthelocalized is smallerthanthe criticalexchangecoupling Jc(η). The spin (S) by replacing τ S and a e in Eqs.(10) latter result is in contradiction with the renormalization wheree† (e)representsth−e→creation(an−n→ihilation)Boson group result presented in Ref.12. operator for the localized spins. Generally, one should s s s define two types of Boson operators (in each sublattice) for both itinerant and localized spins. However, due to A−sublattice (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) the translational invariance symmetry of the model the (cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1) Bosonoperatorsfor itinerantspins ineachsublattice de- (cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) B−sublattice (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) τ (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)τ(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) τ (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) fine the same operator as defined in Eqs.(10). The same story exists also for the localized ones. Thelinearspinwavetheoryisimplementedherewhere FIG. 1: (Color online) One dimensional Kondo-necklace lat- the spin operators are described in linear form of the tice labeled bytwo sublattices. Boson operators and higher order terms have been ne- glected, √2τ √2τ IV. SPIN WAVE APPROACH τyA (a a†) , τyB (a† a ). (12) j ≃ 2i j − j j ≃ 2i j − j The spinwavetheoryis constructedona groundstate Similar expressions to Eq.(12) are applied for the other which is maximally polarized. The low-lying excitations component of spin operators where they are replaced by are expected to be created by flipping a spin and letting Boson ones within the linear spin wave approximation this disturbance to propagate in the crystal which is a which finally leads to the following form for the whole spin wave. These excitations are called magnons which Hamiltonian are bosonic quasi particles. N/2 To apply the spin wave approach the one dimensional = N(tτ2+JτS)+tτ (2a†a +a† a ) Kondo-necklace lattice is labeled by two sublattices A H − i i i+σ i+σ i=1 σ andB(seeFig.1). TheHamiltonian(8)canbewrittenin XX N/2 the following form where represents the summation tτ(1 η) σ − (a a† +a†a a†a† a a ) over the nearest neighbors − 2 i i+σ i i+σ − i i+σ − i i+σ P i=1 σ XX N/2 N/2 N/2 H=t (τizAτiz+Bσ +(1−η)τiyAτiy+Bσ) + 2J (Sa†iai+τe†iei)+2J√τS (aiei+a†ie†i).(13) i=1 σ XX i=1 i=1 X X N/2 +J (−→τiA −→SiB +−→τiB −→SiA). (9) To proceed further and diagonalize the Hamiltonian we · · firsttransformtheoperatorstotheirFouriercounterparts i=1 X 4 by the following relations, where the ground state energy is 2 2 aj = N eikjck , ck = N e−ikjaj, (14) E0(J)=N[ 5(1+J)+√1+2J], (20) r k r j −4 X X and the quasiparticle excitations are 2 2 e = eikjh , h = e−ikje . j k k j N N r Xk r Xj (15) ωα(k)=−1+√1+2J , ωβ(k)=1+√1+2J. (21) In the momentum space representation and for τ = s = 1/2 the Hamiltonian (Eq.(13)) is finally written in the The sublattice magnetization mA = N2 ihτ −a†iaii following form definestheorderparameterfortheAFlongrangeordered P phase which can be calculated in the diagonal bases of N (1 η) = (1+J)+ − γ(k)[c†c† +c c ] theHamiltonian. Forsimplicityweputt=1asthescale H − 4 4 k −k k −k of energy. In the weak coupling limit J = 0 the Hamil- k X tonian (Eq.(8)) has long range AF order. The quantum J +2 ckh−k+c†−kh†k+c−khk fluctuationswhicharecreatedbynonzeroJ decreasethe Xk (cid:16) amount of AF ordering and finally destroy the ordered +c†h† +h†h +h† h phaseatacriticalvalue,Jc,wherethe sublatticemagne- k −k k k −k −k tization becomes zero (1 η) (cid:17) + [(J/2+1) − γ(k)][c†c +c† c ]. (16) Xk − 4 k k −k −k mA = 12 −ha†iaii=1− 2√J1++12J =0=⇒Jc ≃6.46. TheHamiltoniancanbediagonalizedbythepara-unitary (22) transformation15 for the generalcase of η =1. However, Itisthefirstestimateofthecriticalvalueoftheexchange 6 forthecaseofη =1,onecanusetheBogoliubovtransfor- coupling. However,theconditionwhichwehavestatedis mationwhichisaspecialcaseofthegeneralpara-unitary the extreme value where the quantum fluctuations sup- transformation. press the sublattice magnetization totally. At this extreme condition the accuracy of the linear spin wave results is notjustified, since the amplitude ofthe quantum A. η=1 fluctuationsisnotsmall. However,this approachisvalid for small amplitude of fluctuations. Moreover, it shows The Hamiltonian of the general case η = 1 can be that the amplitude of fluctuations become large by in- diagonalizedbythepara-unitarytransformat6ion15forthe creasing the value of exchange coupling (J). Thus, the bosonic operators. In this formalism the Hamiltonian is qualitativepictureofthespinwaveapproximationiscor- written as a hermitian 2m-square matrix D. The para- rect. unitarytransformation(Γ)canbeconstructedifandonly To justify that the spin wave theory gives correct re- if D is positive definite. This procedure is presented in sults, we have plotted the ground state energy per site appendix. versustheexchangecoupling(J)inFig.2. Wehavecom- However, for the special case η = 1 the para-unitary paredthe results obtainedin Eq.(20)with the numerical transformationistheusualBoguliubovtransformation16. Lanczosoneswhichwillbe discussedinthe nextsection. In this case, the bosonic operators is transformed to a ItshowsverygoodagreementofthespinwavewithLanc- new setofbosonic operators(α,β) by the followingrela- zosresultswhichjustifiesthatthespinwavegroundstate tions, presented here can be a good candidate to represent the many body ground state in this case. c =u α +v β† , h =u β +v α† (17) k k k k −k k k k k −k whereu =cosh(θ )andv =sinh(θ )andθ isthe free k k k k k parameter of transformation. The parameter θ should k satisfythe followingrelationtohaveadiagonalHamilto- B. η=1 6 nian J Wehaveimplementedthepara-unitarytransformation tanh(2θ )= − , (18) k J +1 (seeappendix)todiagonalizetheHamiltonian(Eq.(16)). The diagonalized Hamiltonian in terms of new sets of which is actually independent of k. The diagonalized bosonic operators (φ,ψ) is Hamiltonian in terms of the new sets of operators is =E (J)+ (ω (k)α†α +ω (k)β†β ) (19) =E (η,J)+ [Ω (k)φ†φ +Ω (k)ψ†ψ ]. (23) H 0 α k k β k k H 0 α k k β k k k k X X 5 -0.1 diverges for any nonzero η and J < J . However, the c η=1 LSpanincWzoasve numerical values for Jc which are obtained by the spin -0.15 wave approach is far from the numerical Lanczos ones. This will be discussed in the Sec.VI. -0.2 N E0(J)/-0.25 V. NUMERICAL LANCZOS RESULTS -0.3 The numerical Lanczos method has been applied to the one dimensional anisotropic Kondo-necklace model -0.35 defined in Eq.(8). The anisotropic Hamiltonian does not commute with the total z-component spin. Thus, one -0.4 should consider the full Hilbert space for doing numer- 0 0.2 0.4 0.6 0.8 1 J ics. In this respect, we have performed the numerical computations with N = 12,16,20,24 spins. The com- FIG. 2: (Color online) Ground state energy per site putations have been done for different values of the ex- (E0(J)/N)versusJ forη=1obtainedbynumericalLanczos changecoupling(J)andtheanisotropyparameter(η)to method (extrapolated totheN with 4 digits accuracy) →∞ trace the quantum phase transition between the Kondo andlinearspinwavetheorywhichshowverygoodagreement. singletphaseandthe antiferromagneticone. The Kondo singlet ground state is not ordered while the antiferro- where E (η,J) = (1/N) e (η,J;k) is the ground magnetic phase has long range order. The long range 0 k 0 state energy and e (η,J;k) has the following expression order imposes that the structure factor diverges in the 0 P thermodynamic limit (N ) for the antiferromag- 1 netic wave vector k = π.→Th∞e structure factor of the e (η,J;k)= √ +√ + , 0 2√2 P −Q P Q z-component spin at the wave vector k is defined by (cid:0) (cid:1) P =4(1+J)−(2+J)γ(k) , Q= S+T2, N2−1 S =(2+J)γ(k)η−8J2(2+γ(kp)(η−1)), G(k)= hτ0zτrzieikr. (27) =4+J(4+γ(k)(η 1))+2γ(k)(η 1). (24) r=0 X T − − The frequencies Ω (k) and Ω (k) are α β 3 1 0.0 Ω (k)= 4(1+J) (2+J)γ(k) 0.1 α √2 − −A 2.5 0.2 0.3 1 p η=0.6 J 0.4 Ωβ(k)= 4(1+J) (2+J)γ(k)+ , (25) 2 00..56 √2 − A 0.7 where p G(k)1.5 A A001...890 = + 2. (26) 1 A S T p A remark is in order here, the results in this subsection 0.5 A will not reproduce the results of η = 1 case simply by A A putting η equal to one. The reason is related to a diver- A A A A A A A A A gentfactorof(1 η)−1 whichappearsintheintermediate 00 2 π 4 6 k − partofcalculations. Suchtermswillnotappearifweput η = 1 from the beginning and the calculations will be- FIG. 3: (Color online) The z-component structure fac- come much simpler as presented in the last subsection. tor versus k for η = 0.6, N = 24 and different J = In the diagonal bases of the Hamiltonian we can calcu- 0.0,0.1,0.2,...,1.0. late the sublattice magnetization as defined in Eq.(10). As we have discussed in the previous subsection we can Wehaveplottedthestructurefactor(G(k)) versusthe determine the critical border between the AF order and wave vector (k) for η = 0.6, N = 24 and different ex- theKondosingletphasebylookingforthepositionwhere changecouplings(J =0.0,0.1,0.2,...,1.0)inFig.3. The the sublattice magnetization becomes zero. structure factor gets a strong peak at k = π for some Our results which are summerized in Table.I. show values of J. This peak shows the presence of antiferro- thattheonsetofnonzeroXY-anisotropy(η =0)develops magnetic order. We will show in the next plots how the 6 thelongrangeAForderforsmalllocalexchangecoupling height of this peak evolve as the size of system changes. (J <J ). This is also justified by the numerical Lanczos The z-component spin of structure factor at the an- c results where the static structure factor at momentum π tiferromagnetic wave vector, k = π, has been plotted 6 3 3 0.0 0.0 0.1 0.1 2.5 η=0 0.2 2.5 0.2 J 0.3 η=0.1 0.3 0.4 J 0.4 0.5 0.5 2 0.6 2 0.6 0.7 0.7 π= ) 00..89 π= ) 00..89 G(k1.5 A A1.0 G(k1.5 A A1.0 1 1 0.5 A A A A 0.5 A A A A 0 0 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 1/N 1/N FIG. 4: (Color online) The z-component structure factor for FIG.5: (Coloronline)Thez-componentstructurefactorver- k = π versus the inverse of of system size (1/N) for η = 0.0 sus 1/N (inverse of size of system) at k = π for η = 0.1 and and different J = 0.0,0.1,0.2,...,1.0. The structure factor differentJ =0.0,0.1,0.2,...,1.0. ThedivergenceofG(k=π) diverges for J =0 like √N which represent the massless be- as 1 0 for J 0.3 is faster than √N. It shows the anti- N → ≤ havior of XX spin 1/2 chain. The other sets of data show no ferromagnetic ordered phase for J 0.3. divergence or weaker than √N which justifies no long range ≤ antiferromagnetic order. 3 0.0 0.1 2.5 η=0.2 0.2 in Figs.4-8 for different anisotropy (η) versus the in- 0.3 J 0.4 verse of system size (N). The system sizes are N = 0.5 2 0.6 12,16,20,24. Each figure contains different plots for 0.7 J =0.0,0.1,0.2,...,1.0. πk= )1.5 A A001...890 We have plotted the case of genuine Kondo-necklace G( model, η = 0 in Fig.4. For J = 0, the properties of the 1 XX spin 1/2 chain is reproduced which shows that the structure factor diverges18 proportional to √N. How- 0.5 A A A A ever,itdoesnotshowtheantiferromagneticorderofthis model. It only represent the critical properties of the 0 XX spin 1/2 chain which has algebraic decay of correla- 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 1/N tionfunctions. Turningonthelocalexchangetononzero values J = 0 we observe weaker growth of the structure FIG. 6: (Color online) G(k = π) (the z-component struc- 6 factorasthesystemsizeinincreased. Itisconcludedthat ture factor) versus 1/N for η = 0.2 and different J = the one dimensionalgenuineKondo-necklacemodel does 0.0,0.1,0.2,...,1.0. It is clear that G(k = π) diverges for nothaveanantiferromagneticphaseandisalwaysinthe J <0.4 as N goes to infinity. Kondo singlet state. Moreover, we expect the structure factor diverges stronger than √N for the antiferromagnetic ordered phase. at antiferromagnetic wave length (G(k = π)) versus the The presence of anisotropy (η = 0) breaks the U(1) inverseof system size (1/N) for η =0.2 in Fig.6, η =0.6 symmetry and reduces it to Z2 sy6 mmetry. This is the in Fig.7 and η = 1.0 in Fig.8. We have observed that onset of antiferromagnetic ordering formation which is G(k =π)divergesinthethermodynamiclimit(N ) →∞ the resultofsymmetrybreaking. Thestructurefactorat forJ Jc(η)whichverifiestheexistenceofantiferromag- ≤ k =π versus the inverse of length (1/N) for η =0.1 has netic ordering. Ourresultsforthe criticalexchangehave beenplottedinFig.5. ItisclearthatG(π)divergesfaster been summerized in table.I. The accuracy of our results than √N for J = 0 which verifies the formation of an- for the critical point (Jc) is about 0.05. Because, the ± tiferromagnetic ordering in the ground state. A similar size of system is limited to N = 24. To distinguish be- situation exists also for J = 0.1,0.2,0.3,0.4. However, tween the orderedand disorderedphase more accurately for J 0.5 the divergence of G(π) is not apparent as it is necessary to consider larger system sizes. ≥ N . We then conclude that for J J (η = 0.1) c → ∞ ≤ ≃ 0.4 the model has antiferromagnetic ordering while for VI. CONCLUSION AND DISCUSSION largervalues of J it is in the Kondosingletphase. Thus, aquantumphasetransitionoccursatJ fromtheantifer- c romagnetic order to the disordered Kondo singlet state. We have studied the phase diagram of the one dimen- Wehavealsoplottedthe z-componentstructurefactor sional anisotropic Kondo necklace model. We have fo- 7 3.5 TABLEI:Thecriticalexchangevalue(Jc)fordifferentvalues ofanisotropy(η). TheLanczosresultsarereadfromthedata 3 η=1 of G(π) versus1/N which has an accuracy about 0.05. ± 2.5 η 0.1 0.2 0.6 1.0 0.0 Jc (Lanczos) 0.3 0.35 0.4 0.45 π= ) 2 00..12 Jc (SpinWave) 6.13 6.21 6.39 6.46 G(k1.5 J 00..34 0.5 0.6 0.7 1 0.8 3 0.9 A A1.0 η=0.6 0.5 A A A A 2.5 0 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 1/N 2 0.0 πG(k= )1.5 J 0000....1234 FveIrGsu.s8:1/N(Coalotrkon=lineπ) Tfohre ηz-c=omp1.o0neanntdstdriuffcetruernetfaJcto=r 0.5 0.0,0.1,0.2,...,1.0. The divergence of G(k =π) for N 1 00..67 is clear for J <0.5. →∞ 0.8 0.9 A A1.0 0.5 A A A A giveninEq.(8). Ourresultsshowthatthe quantumfluc- 0 tuations grow with increasing J such that the sublat- 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 1/N tice magnetization becomes zero at a critical exchange coupling (J ). Moreover, the antiferromagnetic phase c FIG. 7: (Color online) The z-component structure factor exists for any nonzero value of anisotropy (η = 0) and versus 1/N for k = π and η = 0.6 and different J = J < J . It is in contradiction with the results p6 resented c 0.0,0.1,0.2,...,1.0. ThestructurefactordivergesforJ 0.4 and 1 0. ≤ in Ref.12 which gives the antiferromagnetic phase only N → for η > ηc 0.58. In other words our results state that ≃ η =0. Wehaveplottedthephasediagramoftheonedi- c mensionalanisotropicKondonecklacemodelintheJ η − cused our attention on the case of Ising anisotropy(easy plane in Fig.9. It is obviousthat for any nonzeroη there axisofthespinofitinerantelectrons,δ >1). Letussup- exists anantiferromagneticphase. Moreover,the critical pose that the interaction between the spin of itinerant exchange coupling (J ) depends strongly on η for small c electrons is only composed of τzτz type. It is clear that anisotropy, 0<η <0.1 while its dependence is weak for i j in the absence of local exchange interaction (J = 0) the η >0.1. model has a classical antiferromagnetic ordered ground Our result is in agreement with the symmetry argu- state with nonzero staggered magnetization. It is ex- ments which is based on a qualitative picture. Let us pectedthatanonzerolocalexchange(J =0)addsquan- first suppose that J = 0. The remaining Hamiltonian is 6 tumfluctuationswhichdestroytheantiferromagneticor- an anisotropic XY spin 1/2 chain. The quantum renor- der at a critical exchange (Jc). However, the mean field malization group17 verifies that the universality class of approach10 fails to produce this picture correctly. In the nonzero anisotropy (η =0) is the same as Ising case the mean field approach which is based on the strong (η = 1). Thus, if the antif6erromagnetic phase exists for coupling limit (J/t ) we have assumed a nonzero theIsingcaseitshouldalsoappearforanynonzerovalue → ∞ condensation for local singlet formation plus a nonzero of 0<η <1. Moreover, the Hamiltonian has U(1) sym- triplet occupation (t¯) which induces the antiferromag- metryatη =0whilethesymmetryisbrokentothelower netic order. The solution of the self consistent mean Z2forη =0. Itisthusexpectedthatthequantumphase field equations always give no magnetic order, t¯ = 0. transitio6n happens at the symmetry breaking point. We A divergent integral appears if we assume t¯= 0. How- expect that the addition of local exchange (J = 0) be- ever,themeanfiledapproachworksfairlywell6 intwoand comes irrelevant at the fixed point J = 0, wh6ich is in three dimensional models in addition to the case of XY agreement with our numerical results. However, this anisotropy (0<δ <1) of the one dimensional system10. needs to be proved by more sophisticated methods like To overcome the problem which appears in the mean weak-couplingrenormalizationgroupwhich is out of the fieldapproachforIsinganisotropyoftheonedimensional scope of this article. model we decided to start from the weak coupling limit To verify that our result gives the correct phase (J 0). The linear spin wave theory has been imple- diagram we have implemented the numerical Lanczos → mented which is based on a classical antiferromagnetic method. Our results show that the z-component struc- ground state. To study the phase diagram we have con- turefactordivergesatthe antiferromagneticwavevector sidered a general case of Hamiltonian which has been for any nonzero η and J < J . However, the value of c 8 and D is a square matrix J Kondo singlet 2J+4−(1−η)γ(k) 0 (1−η)γ(k) J 4 4 2 0.45 0 J J 0 D = 2 2 . (A3) 0.4 (1−η)γ(k) J 2J+4−(1−η)γ(k) 0  4 2 4  0.3  J 0 0 J   2 2    The Hamiltonian willbe diagonalizedby the following AF ordered transformation ( ) to the new set of boson operators, J Λ= Γ, (A4) J 0 0.1 0.6 1 η where FIG. 9: Phase diagram of the one dimensional anisotropic Λ† =(φ†k ψk† φ−k ψ−k), (A5) Kondo necklace model where a quantum phase transition and the diagonalrepresentationis given in Eq.(23). The (dashed line) separates the ordered antiferromagnetic phase from the Kondosinglet one. new set of operators should satisfy the bosonic commutation relation, namely [Λ ,Λ†]=δˆ , (A6) the quantum critical point (J ) can not be determined i j i,j c with very high accuracy because of the finite size effect. where δîs the para Kronecker symbol defined by In this study we have been limited to N = 24 because the full Hilbert space should be considered in computa- δˆ =1 , 1 i 2, i,i tion and the Lanczos is basically limited to such values. ≤ ≤ δˆ = 1 , 3 i 4, However, it is proposed that a density matrix renormal- i,i − ≤ ≤ izationgroupcangetmoreaccuratevalues. Moreover,we δˆ =0 , i=j. (A7) i,j have understood that the critical exchange (J ) which is 6 c given by spin wave theory is far from the real one which In other words the transformation matrix ( ) should J is givenby numericalLanczosmethod. The reasonis re- obey the following equations, latedtothestrongquantumfluctuationswhichexistclose Iˆ † =Iˆ or †Iˆ =Iˆ, (A8) tocriticalpointandruintheaccuracyoflinearspinwave J J J J results. whereIîstheparaunitmatrix,Iˆ diag(1,1, 1, 1). A ≡ − − theoremwhichhas been provedin Ref.15 states that the matrixD canbe paraunitarydiagonalizedintoamatrix VII. ACKNOWLEDGMENTS with all diagonal elements positive if and only if D is positive definite. According to Eq.(A3), D is positive definite except at the point (η = 0, J = 0). To find the This work was supported in part by the center of para unitary transformation which satisfies Eq.(A8) the excellence in Complex Systems and Condensed Matter Hamiltonian and the transformation will be written in (www.cscm.ir). The authors acknowledgethe support of the following matrix form Sharif University of Technology. A B D = . (A9) B A ! APPENDIX A: THE PARA-UNITARY TRANSFORMATION and U† V† This section is based on the approach proposed in = − . (A10) J V† U† ! Ref.15todiagonalizeabosonicHamiltonian. TheHamil- − tonian in Eq.(13) can be written in the following matrix where A,B,U and V are 2 2 matrices. form The 2 2 matrix K is ×defined by K†K = A B. × − The eigenvalues of the Hermitian matrix K(A+B)K† = Γ†DΓ (A1) are identified by det[K(A+B)K† λ2I] = 0, i = 1,2 H − i k where I is the 2 2 unit matrix and the corresponding X eigenvectors (χ )×are normalized such that χ†χ = 1 . where Γ† is a row vector i i i λi We define f =K†χ and obtain p such that i i i Γ† =(c† h† c h ), (A2) (A+B)f =λ p . (A11) k k −k −k i i i 9 The transformationmatricesU andV aregivenby their REFERENCES column vectors with the following relations, 1 u = (f +p ), i i i 2 1 v = (f p ). (A12) i i i 2 − ∗ [email protected];http://spin.cscm.ir 10 A. Langari and P. Thalmeier, Phys. Rev. B 74, 24431 1 M. A. Continentino, Brazilian Journal of Physics 35, 197 (2006). (2005). 11 P. Thalmeier and A. Langari, Phys. Rev. B 75, 174426 2 M. Vojta, Rep.Prog. 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