Monte Carlo Simulations for the Magnetic Phase Diagram of the Double Exchange Hamiltonian M.J. Caldero´n and L. Brey 8 Instituto de Ciencia de Materiales (CSIC). Campus de la Universidad Aut´onoma, 28049, Madrid, Spain. 9 (February 1, 2008) 9 1 n one d orbital. a WehaveusedMonteCarlosimulationtechniquestoobtain The double exchange (DE) mechanism developed by J the magnetic phase diagram of the double exchange Hamil- Zener[3–5],explainstheexistenceofferromagnetismand 9 tonian. We have found that the Berry phase of the hopping metallicbehavioratlowtemperatures. Inthismodel,the 2 amplitudehasanegligible effect inthevalueof themagnetic electronsget mobility betweenthe manganese ions using criticaltemperature. Toavoidfinitesizeproblemsinoursim- oxygen, which is magnetically inert, as an intermediate. ] ulations we have also developed an approximated expression l Asaconsequence,the tunnelingtakesplacebetweentwo e for the double exchange energy. This allows us to obtain the configurations in which the Mn ions of different charge - critical temperature for the ferromagnetic to paramagnetic r (Mn3+andMn4+)interchangetheirvalencestates. This t transition more accurately. In our calculations we do not s conduction process is proportional to the electron trans- observe any strange behavior in the kinetic energy, chemical . at potentialorelectron densityof statesnearthemagneticcrit- fer integral t if the core spins of Mn3+ and Mn4+ are m ical temperature. Therefore, we conclude that other effects, aligned. Otherwise,the transferintegralisinverselypro- notincludedinthedoubleexchangeHamiltonian,areneeded portional to the Hund’s rule coupling energy, which for - d in order to understand the metal-insulator transition which the Mn ions is muchlargerthan t. In the DE model the n occurs in themanganites. ferromagnetismistheninducedviathis electronconduc- o PACS number 71.10.-w,75.10.-b tion process. c In the limit of large Hund’s coupling the spin of the [ electronintheactivedorbitalisparalleltothelocalcore 1 I. INTRODUCTION spin Si, which is treated as a classical rotator with the v normalization S = 1 and characterized by the angles 1 The recent discovery of colossal magnetoresistance θi and φi. Th|en,|semiclassically, the effective hopping 1 (CMR) [1] in mixed-valence compounds of the form Hamiltonian is 3 1 La31+−xA2x+Mn31+−xMn4x+O32−, (where A can be Ca, Sr or H = t C+C +h.c. . (1) 0 Ba) has revived the interest in these materials with per- − i,j i j 8 ovskite structure. The interest is focused on the phase hXi,ji(cid:0) (cid:1) 9 diagram and the magneto-transport properties. Both / Here C+ creates an electron at site i with spin parallel t features have been determined experimentally [2]. For i a toS , i,j denotesthenearest-neighborpairs,whichare 0.1 x 0.5and lowtemperatures the system is metal- i m ≤ ≤ only cohuntied once, and the hopping amplitude acquires lic and presents ferromagnetic order. As the tempera- - ture increases the system becomes insulator and param- a Berry’s phase and it becomes a complex number given d by [6], agnetic. The ferro-paramagnetic transition occurs at an n co cxoldoespsaelndmeangtncertiotriceasilsttaenmcpeeerffateuctrseaTrce(xo)bs∼erv3e0d0Kfo.r tTemhe- ti,j =t cosθi cosθj sinθi sinθjei(φi−φj) . (2) : peratures close to T (x). For x 0.1 and low tempera- (cid:18) 2 2 − 2 2 (cid:19) v c ≤ i tures the system is a ’layer’ antiferromagnetic with fer- X Thiscomplexhoppingappearsafterrotatingtheconduc- romagnetic coupling inside planes. tion electron spins so that the spin quantization axis at r In these compounds the electronically active orbitals a siteiisparalleltoS ,andthenprojectontothespinpar- aretheMndorbitals. ThenumberofdelectronsperMn i allel to S . In this paper we will call complex DE (cDE) is x+3, namely, four electrons per Mn3+ and three per j model the system governedby the Hamiltonian (1) with Mn4+. The cubic symmetry and the strong Hund’s rule the hopping given by (2). couplingmakethatthreeelectronsgettrappedinthet 2g For topologies where electrons close paths do not oc- states and therefore these electrons become electrically cur, it is possible to choose wave functions phase factors inert, forming a core spin S of magnitude 3/2. The rest such that oftheelectronsgotothee orbitalsandtheygetstrongly g coupledto S bythe Hund’s rulecoupling. Consequently, θ i,j t t =tcos , (3) all the spins on each Mn prefer to be parallel. For small i,j →| i,j | 2 valuesofx,theperovskitesshowalongrangeJahn-Teller orderwhichselectsapreferreddorbitalandthereforeitis being θi,j the angle between the semiclassical spins Si possible to assume that the electrons move only through and Sj. 1 InthisworkwewillusethenameDEmodelforthesys- The paper is organized as follows: in section II we de- temcontrolledbyequation(1)withthehopping(3). This scribe briefly the Monte Carlo algorithm we use and in is the HamiltonianproposedbyAndersonandHasegawa section III we present the results obtained for the cDE [4] as a generalization of the Hamiltonian describing the and DE Hamiltonians. In section IV we introduce an tunneling between two fixed sites i and j. approximationto the DE energy which allows us to per- De Gennes [5] and Kubo and Ohata [7] calculated, in formMonteCarlosimulationsinbiggersystems. Section a mean field theory approximation, the magnetic phase Vis devotedtoshowthe resultsobtainedintheframeof diagram of the DE model. Their results show a ferro- this approximation. Finally, we present the conclusions paramagnetic phase transition at a critical temperature in section VI. TMF c =1.6 C+C (4) t h i ji0 II. DESCRIPTION OF THE MONTE CARLO This transition is accompanied by a change in the tem- ALGORITHM perature dependence of the resistivity. In equation (4), Oˆ means the expectation value of the operator Oˆ at For calculating the magnetic phase diagrams, we have 0 h i zero temperature. performedclassicalMonteCarlo(MC)simulationsonthe However, recently it has been pointed out [8] that the classical core spin angles. The simulations are done in critical temperatures predicted by the mean field theory N N N cubic lattices with periodic boundary condi- × × of the DE model is much bigger than the observed ex- tions. Although the localized spins are considered clas- perimentally. Furthermore,theresistivityimpliedbythe sical, the kinetic energy of the conduction electrons is hopping(3)isincompatiblewithmanyaspectsoftheex- calculatedby diagonalizingthe DE Hamiltonianbecause perimental information. Millis et al [8] proposed that in we consider it as a quantum quantity. addition to the DE mechanism a strong electron-phonon ThestandardMetropolisalgorithm[13]wasusedinthe interaction plays a crucial role in these perovskites. The MC simulations. The sites to be considered for a change interplay between these two effects could reproduce the in the spin orientationare randomly chosen. Once a site experimental critical temperature and the observed be- is selected for a spin reorientation, the angle associated havior of the resistivity at the magnetic transition tem- with an attempted change of the spin is chosen at ran- perature [9,10]. dom fromwithin a specified range[14]. Then the energy On the other hand, Yunoki et al [11] claim that the change, ∆E, associated with the attempted update, is inclusion of the phase in the hopping term can lower calculated. If the quantity exp( ∆E/T) is smaller than − the critical temperature down to the experimental val- arandomnumberbetween0and1,thechangeisallowed, ues. Also, for large but finite Hund’s coupling, they ob- otherwise, it is rejected. Typically, 5000-7000 MC steps tainthatatlowconcentrationsphaseseparationbetween per spin are used for equilibration and 3000-5000 steps hole-poor antiferromagnetic and hole-rich ferromagnetic for spin are used for calculating averages. regions occurs. The existence of phase separationis also In the simulations we calculate the average of the in- obtained by solving the DE model plus an antiferromag- ternal energy, E, and the average of the absolute value netic coupling between next neighbors core spins [12]. of the magnetization, M, In this paper we are interested in two points; first, in 1 comparing the magnetic phase diagram of the two DE E = H , (5) N3h i models, equations (2) and (3), and second, in obtaining an approximatedexpressionfor the DE energy which al- 1 luonwitscuesllstoanpdertfoorombtMainonatcecuCraartleovcaalulceuslaotfiotnhse imnabgingegteicr M = N3h(cid:12)(cid:12)(cid:12)Xi Si(cid:12)(cid:12)(cid:12)i , (6) criticaltemperature. Themainresultsofourcalculations (cid:12) (cid:12) where denotes statistic(cid:12)al aver(cid:12)age. Since the MC up- are shown in Fig.1. Here we show the critical tempera- h i dating proceduregeneratesuniform rotationsofthe spin tureversusconcentration,forthecomplexDEmodeland system, a calculation of the MC average of the direction for the real DE model, in the case of a unit cell of size ofthemagnetizationisnotmeaningful. Alsowecalculate 4 4 4. We obtain that there is not big difference be- × × the average value of the width of the density of states, tween the critical temperatures of both models. This re- W, and the value of the chemical potential, µ, measured sultindicatesthatthecontributiontothepartitionfunc- with respect to the bottom of the density of states, tion of electronic configurations in which fermions move on closed loops in real space can be neglected. In Fig.1 W = ε ε (7) max min we also show Monte Carlo results obtained by using an h − i µ= ε ε , (8) approximated expression for the electron kinetic energy. h occ− mini We have checked in small systems that this approxima- where ε and ε are the minimum and the maxi- min max tion is realistic. From our results we obtain that the mum energies obtained from diagonalizing the electron critical temperature of the DE model is in the range of Hamiltonian, and ε is the higher energy of the occu- occ the experimental one [2]. pied states. 2 III. MONTE CARLO RESULTS FOR THE H t¯ C+C +h.c. (10) DOUBLE EXCHANGE MODELS 0 ≡− hXi,ji(cid:0) i j (cid:1) V δt C+C +h.c. , (11) In order to obtain the internal energy in each of the ≡− i,j i j hXi,ji(cid:0) (cid:1) MonteCarlosteps,itisnecessarytodiagonalizetheelec- tronic Hamiltonian. The diagonalization is very expen- being t¯the average of the absolute value of the hopping sive in terms of CPU time, and therefore we can only amplitude, study small size systems. Here we present the results obtained for a cubic unit cell of size N = 4 and for the 1 t¯= t , (12) electron concentrations x=0.1, 0.25 and 0.3. 3N | i,j | In figure 2, it is presented, for the DE and the cDE 0 hXi,ji Hamiltonians, the magnetization, M, as a function of and the temperature, T. Due to the finite size of the unit cells used in the simulations, M is different from zero at δt =t t¯ . (13) i,j i,j anytemperature,andwedefinethe criticaltemperature, − Tc, as the point where the second derivative of M with Here N0 is the total number of Mn ions in the system. respect T changes sign. This way of obtaining the crit- Note that since the Hamiltonian is hermitic, t¯is a real ical temperatures, implies uncertainties of around 10% quantity. of the value of Tc. In any case we obtain that the crit- Given a disordered system, characterized by a set of ical temperatures of the DE and the cDE Hamiltonians t , we want to obtain the expectation value of H. i,j are practically the same. In figure 3 we plot the internal T{he}Hamiltonian H can be diagonalized by using the 0 energy E as a function of T. Note that the difference Bloch states, between the internalenergies obtainedby using the cDE and using the DE Hamiltonians is rather small. That is k = 1 eik·Ri i , (14) the reason why both models give very similar magnetic | i √N | i 0 Xi critical temperatures. These results imply that in order to calculate total where i represents the atomic orbital at site i, Ri are kinetic energies the close paths are almost irrelevant. the lat|ticie vectors and k is a wave vector in the first In figure 4 we plot, as a function of the temperature, Brillouin zone. The energy of the state k is | i the width of the density of states W and the chemical potential µ, for the DE case. Note that at temperatures ε(k)= 2t¯(coskxa+coskya+coskza) , (15) − close to T , the bandwidth and the chemical potential c have a value bigger than the obtained in the fully disor- beingathe lattice parameter. Infunctionoftheseeigen- dered case, T . This is because in the DE models values, the expectation value of H0 is, → ∞ the ferromagneticto paramagnetictransitionis a second occ order phase transition, and therefore the internal energy E = ε(k) , (16) 0 is continuous at the critical temperature. Moreover, the Xk bandwidth and the chemical potential present a contin- uous behavior near T . In the DE models we do not wherethesumisovertheoccupiedstates. Fromthevalue c observe any change in the electronic states at the ferro- of E we can obtain the expectation value of C+C , 0 i j paramagnetic critical temperature. E C+C = 0 , (17) h i ji0 −6t¯N 0 IV. APPROXIMATION FOR THE DOUBLE EXCHANGE ENERGY which is independent of the value of t¯. To obtainthe expectation value of V , we first notice that since the set δt are randomhly idistributed, k In the MC calculations of the DE models, the size of V k =0,anditi{snie,cje}ssarysecondorderperturbathion| the matrix to diagonalize impose a restriction on the di- | i theory in order to get a correction to the expectation mension of the unit cell used in the simulation. In order value of H , to be able to perform simulations in bigger systems, we 0 have developed a second order perturbation theory for occ obtaininganexpressionfortheelectronkineticenergyin E2 k VG0(εk)V k , (18) ≃− h | | i a background of randomly oriented core spins. Xk We start writing the Hamiltonian (1) in the form, where H H +V (9) 0 k k ≡ G (h¯ω)= | ih | (19) 0 with Xk ¯hω−εk 3 is the Green function of the perfect crystal. Because the M as a function of T by using only E for the internal 0 values δt are not correlated, it is easy to obtain the energy, see figure 6. Comparing these results with the i,j { } expression, onesobtainedwiththeapproximatedDEenergy,E +E , 0 2 weestimatethattheinclusionofE intheinternalenergy 2 E = a (t¯ t )2 , (20) lowers the critical temperature more than a 20%. 2 2 i,j − − hXi,ji In Fig.7, we plot for different values of the concentra- tion x, the statistical average of the absolute value of with the hopping amplitude, t , as a function of the tem- i,j | | perature. This quantity is proportional to the electron 1 occ ε(k+k′) 1 bandwidth. We find that t is a continuous func- a = 6+ . i,j 2 −2N2 (cid:18) t¯ (cid:19)ε(k) ε(k′) tion of T, and there are noth|sign|ails of any change in the 0 Xk Xk′ − electronic structure near T . c (21) The bandwidth near T is around 1.15 times bigger c than the bandwidth in the T limit. Li et al [15] Adding E andE , we obtainthe followingexpression → ∞ 0 2 haveobtainedthatintheT limitofthecDEmodel, for the energy of the system, →∞ only 0.5%ofthe electronstates arelocalized. In order ∼ to know the difference between the density of states at θ E ≃−2hCi+Cji0 cos 2ij −a2 (t¯−ti,j)2 . (22) T ∼Tc andatT →∞, wehavecalculatedthe statistical hXi,ji hXi,ji averageoftheelectrondensityofstates. Wefindthatthe numberofelectronstateswithenergiesbetween W(T ) c Infigure5weplothCi+Cji0 anda2 asafunctionofthe and −W(T →∞) is less than 1% of the total nu−mber of electrondensityx. Thefirsttermintheaboveexpression states. Therefore, even if all these states were localized, is a ferromagnetic coupling for the core spins and the the percentage of localized states near T is not larger c second term is an antiferromagnetic coupling. Note that than 2%. We conclude that in the DE models it is not hCi+Cji0isonlylinearonxatsmallvaluesoftheelectron possible to relate the metal-insulator transition with the concentration. ferro-paramagneticone. The critical temperatures we obtain are around 1.5 times smaller than the obtained in mean field theory V. MONTE CARLO RESULTS WITH THE by Kubo and Ohata [7]. However, it is around 8 times APPROXIMATED DOUBLE EXCHANGE smaller than the obtained by Millis et al. [8] using also ENERGY a mean field approximation. We think that the differ- ence occurs because they use a lineal dependence, see In figure 2 we plot, for x=0.1, x=0.25 and x=0.3, Fig. 5, of the kinetic electronenergy which overestimate theabsolutevalueofthemagnetization,M,asafunction the value of C+C , and therefore the value of T . h i ji0 c ofthetemperature,T,forthecasewheretheinternalen- From Kubo and Ohata model (4), it is clear that T c ergy is obtained by using the approximated DE energy, scaleswitht C+C andthemagnetizationversusT/T h i ji0 c equation (22), for comparing with the case in which it is curveisindependentoftheelectronconcentrationx. But obtained by diagonalizing the DE Hamiltonian. The dif- there is no reason to think that this must be the case ferencebetweenbothcurvesis verysmall,andthe corre- when the second term, E , of our approach is included 2 sponding critical temperatures are practically the same, in the calculations. However, we have found that the M see figure 1. Thereforewe conclude that equation(22) is versusT/T curvesalmostcoincide(seeFig.6). Thisfact c a good approximation for the energy of the DE model. isnotonlyfoundintheframeofourapproximationtothe Usingequation(22),fortheDEenergy,wecanperform DE energybut alsowhenthe DEandcDE Hamiltonians MCsimulationsinmuchbiggersystems,andobtainmore are diagonalized. precisevaluesofthe criticaltemperatures. Infigure6we The presented results are in qualitatively agreement plot the magnetization as a function of the temperature with those obtained by Dagotto et al [16]. However our for a sample of size N = 20. For this size the critical estimationsofT are 1.5timesbigger. Thediscrepancy c ∼ temperature is obtained with a big precision. We have isduetothe differenceinthecriterionusedtoobtainT . c checked that for N = 20 the values of the critical tem- In reference [16] the T is defined as the temperature c peratures are accurate in two digits and that it is not wherethespin-spincorrelationfunctioninrealspacebe- worthwhile to increase more the size of the system. We comeszeroatthemaximumdistanceavailableintheunit haveperformedMCsimulationsfordifferentvaluesofthe cell. Theyusedaunitcellofsize6 6 6. Ourcriterium × × electron concentration and, in figure 1, we have plotted is based in the change of sign of the second derivative the magnetic phase diagram. of M with respect to T. We have checked that both Kubo and Ohata [7] have developed an expression for criteria give the same T in big unit cells, but the cri- c theDEenergywhichcoincideswithE . Inordertoknow terium based on the correlation function underestimate 0 theimportanceofE inthecalculationofthevalueofthe T in small unit cell calculations. This is clear in Fig.8 2 c criticaltemperature,we havealsocomputed the value of where we plot the spin-spin correlation function versus 4 distance for several temperatures and for two unit cells [1] R.M. Kusters et al., Physica (Amsterdam) 155B, 362 with N = 6 and N = 10. The electron concentration is (1989); K. Chahara et al. Appl. Phys. Lett. 63, 1990 x = 0.14 to compare with the results presented in refer- (1993); R. von Helmolt et al. Phys. Rev. Lett. 71, 2331 ence [16]. In the case of N =6 it seems that the critical (1993); S.Jin et al.Science 264, 413 (1994). temperature is smaller than T = 1/15, because the cor- [2] Seeforexample, A.P.Ramirez, J.Phys.: Condens.Mat- relation function is zero at a separation of 8a. However, ter 9, 8171-8199 (1997); J.M.D. Coey, M. Viret and S. when N = 10 it is clear that the ferromagnetic correla- von Molnar, Adv.in Phys., in press. tion at T = 1/15 is not zero at 8a, and in fact it seems [3] C. Zener, Phys.Rev. 82, 403 (1951). [4] P.W. Anderson and H. Hasegawa, Phys. Rev 100, 675 thatatthistemperaturethesystemisstillferromagnetic. (1955). Onthe otherhand,ourcriteriumoverestimatetheT for c [5] P.G. deGennes, Phys.Rev. 118, 141 (1960). small unit cells and the possibility of increasing the size [6] E. Mu¨ller-Hartmann and E. Dagotto, Phys. Rev. B 54, leads to more realisticvalues of the criticaltemperature. R6819 (1996). To compare with the experimental critical tempera- [7] K.KuboandN.Ohata,J.Phys.Soc.Jpn.33,21(1972). tures,itisnecessaryanestimationoft¯. Theexperimental [8] A.J. Millis, P.B. Littlewood and B.I. Shraiman, Phys. bandwidth is around 1-4eV [2], and therefore the value Rev. Lett.74, 5144 (1995). of t¯is around 0.08-0.3eV. With this result our estimate [9] H.R¨oder,J.ZangandA.R.Bishop,Phys.Rev.Lett.76, of the criticaltemperature is between 150Kand 500Kat 1536 (1996); J. Zang, A.R. Bishop and H. R¨oder, Phys. x 0.2. This value of Tc is in the range of the observed Rev. B 53, R8840 (1996). ∼ experimentally. [10] A.J. Millis, B.I. Shraiman and R. Mueller, Phys. Rev. Lett. 77 175 (1996); A.J. Millis, R. Mueller and B.I. Shraiman, Phys.Rev.B 54 5405 (1996). VI. CONCLUSIONS [11] S.Yunoki,J.Hu,A.L.Malvezzi, A.Moreo,N.Furukawa and E. Dagotto, Phys.Rev. Lett.80, 845 (1998). Using Monte Carlo techniques, we have obtained the [12] D. Arovasand F. Guinea, cond-mat/9711145. magnetic phase diagram of the double exchange Hamil- [13] Monte Carlomethods inStatistical Physics,editedbyK. tonian. Comparing the results obtained from the double Binder (Springer-Verlag, Berlin, 1979). [14] P.A. Serena, N. Garc´ıa and A. Levanyuk, Phys. Rev. B, exchange Hamiltonian with a complex hopping (cDE) 47, 5027 (1993). and with its absolute value (DE) we have found that [15] Q. Li, J. Zang, A.R. Bishop and C.M. Soukoulis, Phys. the Berry phase of the hopping has a negligible effect in Rev. B, 56, 4541, (1997). the magnetic critical temperature. This implies that it [16] E. Dagotto, S. Yunoki, A.L. Malvezzi, A. Moreo, J. is possible to choose wave functions phase factors that Hu, S. Capponi, D. Poilblanc and N. Furukawa, cond- counteract the phase of the hopping such that it is not matt/9709029. relevantintheMCcalculations. Toavoidthelimitations on the size of the systems studied we have developed a secondorderperturbativeapproachtocalculatetheelec- FIG. 1. Critical temperature of the ferro-paramagnetic tron kinetic energy without diagonalizing the Hamilto- transitionversusconcentrationofconductionelectronsfordif- nian. Within this approachwe have calculated the criti- ferentapproaches. SquarescorrespondtotheHamiltonian(1) cal temperature for a bigger size of the system (N =20) with the hopping given by (2); circles to hopping (3). The and,consequently,moreaccurately. The valuesofT ob- curves are related to the calculations made with an approxi- c tained are in the range of the experimental ones and in mated expression obtained in order to avoid diagonalization qualitativeagreementwith thoseinref.[16]. Neither the in the MC simulations (ADEE: approximation to the double average of the electron density of states nor the kinetic exchange energy). This fact allows us to increase the size of the system. The line is only a guide to the eye. The critical energy reveal any strange behavior near the magnetic temperatures obtained are in the range of the experimental critical temperature. We conclude that, in order to un- ones. derstand the metal-insulator transition which occurs in the manganites,anothereffects,notincludedinthe dou- ble exchangeHamiltonian,shouldbe takenintoaccount. FIG. 2. Absolute value of the magnetization versus the Acknowledgments. ThisworkwassupportedbytheCI- temperature for concentrations x = 0.1, 0.25 and 0.3. The CyTofSpainunderContractNo. PB96-0085,andbythe size of the unit cell is N = 4. In order to compare we have Fundaci´onRamo´nAreces. Helpful conversationswith F. plottedalsothecurvesobtainedwiththecDEandDEHamil- Guinea, R. Ram´ırez, S. Das Sarma and J.A. Verg´es are tonians as well as with the perturbative approach developed gratefully acknowledged. here(ADEE). FIG.3. Doubleexchangeenergyversustemperature. The lack of difference in these curves implies the result in Fig. 2. The corresponding critical temperatures are pointed with an arrow. 5 FIG.4. Bandwidth (a) and chemical potential (b) versus temperature. In the disordered case, T → ∞, W should be equal to 8. Here, in theparamagnetic phase, W >8 because thetransitionissecondorder(seetextfordetails). Notethat the curves are continuous near Tc which is pointed with an arrow in each case. FIG.5. Coefficients used in the perturbativeapproach to theDE Hamiltonian versus concentration (see eq. (22)). FIG.6. Magnetization as a function of T/Tc as obtained using the 0th order approximation (dashed line), E0 in eq. (22), and the 2nd order approximation (continuous line) to the double exchange energy (ADEE), eq. (22). The critical temperature taken for reference is the one that corresponds totheADEEmodel. Inthecase ofthe0th orderapproxima- tion, theM versusT/Tc curveis independentof theelectron concentration. IntheADEEmodel,thereisnoreasonforthis independence;however,wehavefoundthatM(T/Tc)isprac- tically independent of x. To show this, the M(T/Tc) curve hasbeenplottedforvariousconcentrations(x=0.1, 0.2,0.3, 0.4 and 0.5). FIG.7. Averageoftheabsolutevalueofthehoppingam- plitude, h|ti,j|i, versus temperature for N = 20 and concen- trations x = 0.1, 0.25 and 0.3. This average is proportional to the bandwidth. We recover the continuous behavior that we found on Fig.4. FIG.8. The spin-spin correlation versus distance is plot- ted for different temperatures. In (a) N = 6 and in (b) N =10. Forthesecalculationswehaveusedtheperturbative approach. If the criterium to obtain Tc is that the spin-spin correlation becomes zero, a smaller size of the system leads to an underestimation of thecritical temperature. 6 0.20 0.15 t / c T 0.10 0.05 DE N=4 cDE N=4 ADEE N=4 ADEE N=20 0.00 0.00 0.10 0.20 0.30 0.40 0.50 x Fig.1 M.J. Calderon and L. Brey 1 N=4 x=0.1 DE cDE 0.5 ADEE 0 1 n N=4 x=0.25 o i t a z i t 0.5 e n g a m 0 1 N=4 x=0.3 0.5 0 0 0.05 0.1 0.15 0.2 0.25 T/t Fig.2 M.C. Calderon and L. Brey 1 N=4 x=0.1 DE 0.9 cDE 0.8 0.7 1 N=4 x=0.25 0.9 0 E / E 0.8 0.7 1 N=4 x=0.3 0.9 0.8 0.7 0 0.05 0.1 0.15 0.2 0.25 T/t Fig. 3 M.J. Calderon and L. Brey 12.0 N=4 DE x=0.1 11.0 x=0.25 x=0.3 t 10.0 / W 9.0 (a) 8.0 4.0 3.0 t / µ 2.0 (b) 1.0 0.00 0.05 0.10 0.15 0.20 0.25 T/t Fig.4 M.J. Calderon and L. Brey