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997

The Ferromagnetic Kondo Model for Manganites:

Phase Diagram and Charge Segregation Effects.

E. Dagotto, S. Yunoki, A. L. Malvezzi, A. Moreo, and J. Hu

National High Magnetic Field Lab and Department of Physics,

Florida State University, Tallahassee, FL 32306

S. Capponi, and D. Poilblanc

Laboratoire de Physique Quantique Unite Mixte de Recherche 5626, C.N.R.S.,

Universite Paul Sabatier, 31062 Toulouse, France

N. Furukawa

Institute for Solid State Physics,

University of Tokyo, Roppongi 7-22-1, Minato-ku, Tokyo 106, Japan

February 1, 2008

Abstract

The phase diagram of the ferromagnetic Kondo model for manganites is investigated using computational tech-niques. In clusters of dimensions 1 and 2, Monte Carlo simulations in the limit where the localized spins are classicalshow a rich low temperature phase diagram with three dominant regions: (i) a ferromagnetic phase, (ii) phase separa-tion between hole-poor antiferromagnetic and hole-rich ferromagnetic domains, and (iii) a phase with incommensuratespin correlations. Possible experimental consequences of the regime of phase separation are discussed. Studies usingthe Lanczos algorithm and the Density Matrix Renormalization Group method applied to chains with localized spin1/2 (with and without Coulombic repulsion for the mobile electrons) and spin 3/2 degrees of freedom give results inexcellent agreement with those in the spin localized classical limit. The Dynamical Mean Field (D =∞) approxima-tion was also applied to the same model. At large Hund coupling phase separation and ferromagnetism were identified,again in good agreement with results in low dimensions. In addition, a Monte Carlo study of spin correlations allowedus to estimate the critical temperature for ferromagnetism T

F M

c in 3 dimensional clusters. It is concluded that TF M

c

is compatible with current experimental results.

1 Introduction and Main Results

Materials that present the phenomenon of “colossal”magnetoresistance are currently under much experimen-tal investigation due to their potential technological ap-plications. Typical compounds that have this phe-nomenon are ferromagnetic (FM) metallic oxides ofthe form R1−xXxMnO3 (where R = La, Pr, Nd; X = Sr,Ca, Ba, Pb) [1, 2]. As an example, a decrease in resistiv-ity of several orders of magnitude has been reported inthin films of Nd0.7Sr0.3MnO3 at magnetic fields of 8 Tes-las [3]. The relative changes in resistance for the mangan-ites can be as large as ∆R/R ∼ 100, 000%, while in mag-netic superlattices Co/Cu/Co the enhancement is about100%. This result suggests that manganites indeed havetechnological potential since large changes in resistancecan be obtained at fixed temperature upon the applica-tion of magnetic fields, opening an alternative route fornext generation magnetic storage devices. However, since

the development of La-manganite sensors is still at a veryearly stage, a more fundamental approach to the studyof manganites is appropriate and, thus, theoretical guid-ance is needed. The existence of correlation effects inthe fairly dramatic magnetic, transport, and magneto-transport properties of doped La-manganites reinforcesthis notion.

The early theoretical studies of models for mangan-ites concentrated their efforts on the existence of fer-romagnetism. The so-called “Double Exchange” (DE)model [4, 5] explained how carriers improve their kineticenergy by forcing the localized spins to become ferromag-netically ordered (this phenomenon is quite reminiscentof the Nagaoka phase discussed in models for cuprates).However, in spite of this successful explanation of the ex-istence of ferromagnetism at low temperature several fea-tures of the experimental phase diagram of manganitesremain unclear, and they are likely beyond the DE model.

1

Actually, the phase diagram of La1−xCaxMnO3 is veryrich with not only ferromagnetic phases, but also regionswith charge-ordering and antiferromagnetic correlations atx > 0.5 [6], and a poorly understood “normal” state abovethe critical temperature for ferromagnetism, T FM

c , whichhas insulating characteristics at x ∼ 0.33. Finding an in-sulator above T FM

c is a surprising result since it wouldhave been more natural to have a standard metallic phasein that regime which could smoothly become a ferromag-netic metal as the temperature is reduced. Some theoriesfor manganites propose that the insulating regime aboveT FM

c is caused by a strong correlation between electronicand phononic degrees of freedom [7]. Other proposals in-clude the presence of Berry phases in the DE model thatmay lead to electronic localization [8]. On the other hand,the regime of charge ordering has received little theoreticalattention and its features remain mostly unexplored. Tocomplicate matters further, recent experiments testing thedynamical response of manganites have reported anoma-lous results in the ferromagnetic phase using neutron scat-tering [9], while in photoemission experiments [10] the pos-sible existence of a pseudogap above the critical tempera-ture was reported.

The rich phase diagram of the manganites describedabove, plus the several experimental indications of strongcorrelations in the system, deserves a systematic theoreti-cal study using state-of-the-art computational techniques.These methods are unbiased and can provide useful infor-mation on models for manganites in a regime of couplingsthat cannot be handled perturbatively or exactly. How-ever, the large number of degrees of freedom and associ-ated couplings of a full Hamiltonian model for mangan-ites makes this approach quite cumbersome. In principlethe two eg active orbitals per Mn-ion must be included,in addition to the t2g electrons. Also phonons should beincorporated to fully describe these materials. However,as a first step towards a theoretical understanding of thebehavior of models for manganites, in this paper it wasdecided to work only in the electronic sector (i.e. leavingaside phonons) and with just one orbital per site (i.e. keep-ing only one eg orbital of the two available, which in prac-tice amounts to assuming a static Jahn-Teller distortion).In addition, the t2g degrees of freedom are here assumedto be localized i.e. no mobility is given to these electrons.They basically provide a spin background in which theeg electrons move, with a Hund term that couples all ac-tive electrons per Mn-ion. Under these assumptions a richphase diagram was obtained, as explained in detail in therest of the paper. It is left for future work the analysis ofthe influence of phonons and orbital degeneracy into thefairly complicated phase diagrams reported here.

Some of the main results of the present effort are sum-marized in Figs.1 and 2 where the phase diagrams of theferromagnetic Kondo model, using classical spins to repre-sent the t2g spins, are presented for low dimensional sys-

tems. These figures were reported previously in a shortversion of this paper [11] but here considerable more de-tails, as well as a large set of novel results, are provided.Three dominant regimes have been identified: (1) a ferro-magnetic region in excellent agreement with the DE mech-anism, (2) phase separation between hole-rich ferromag-netic and hole-poor antiferromagnetic regions, and (3) anintermediate phase with short-range incommensurate cor-relations. The regime of phase separation was previouslyconjectured to exist in this model from studies of Hamilto-nians for manganites at large Hund coupling in one dimen-sion [12]. It is important to remark that phase separationis currently widely discussed in the context of high tem-perature superconductors since models for the cuprates,such as the t− J model in two dimensions, present densi-ties that cannot be uniformly stabilized in a given volumeby suitably selecting a chemical potential [13, 14, 15]. Af-ter the introduction of 1/r Coulombic interactions the re-sulting hole-rich regions become unstable against the for-mation of “stripe” configurations [16, 17]. If the tenden-cies towards phase separation reported in this paper arerealized in the manganites, a similar phenomenon maylikely occur i.e. neutron scattering experiments could re-veal evidence of stripe configurations in compounds suchas La1−xCaxMnO3 as it occurs in the cuprates [18]. Thepseudogap features found in photoemission [10] could alsobe related to this phenomenon.

0.00.20.40.60.81.0⟨n⟩

0.0

5.0

10.0

15.0

J H / t

1D

FM

IC

PS

Figure 1: Phase diagram of the ferromagnetic Kondomodel in one dimension obtained with Monte Carlo tech-niques in the limit where the localized spins are classical.PS, FM, and IC denote regions with phase separation, fer-romagnetism, and incommensurate correlations. For de-tails see the text.

2

0.00.20.40.60.81.0⟨n⟩

0.0

5.0

10.0

15.0

J H / t

2D

FM

IC

PS

Figure 2: Same as Fig.1 but for the case of two dimen-sions. The boundary between the PS and IC regimes wasdifficult to obtain numerically and, thus, it is not sharplydefined in the figure (gray region).

The bulk of the paper is devoted to the discussion ofthe numerical evidence that provides support to these pro-posed phase diagrams. Results were not only obtained indimensions 1 (Secs.III and IV) and 2 (Sec.V), but alsoin 3 (Sec.VII) and ∞ (Sec.VI) using a variety of numer-ical techniques. Both classical and quantum mechanicalt2g degrees of freedom were analyzed on chains. Exper-imental consequences of our results are discussed, spe-cially those related with the existence of phase separation(Sec.VIII). There is a clear underlying qualitative univer-sality between results obtained on lattices with differentcoordination number and using different algorithms. Thisuniversality lead us to believe that the results reportedhere contain at least the dominant main features of thephase diagram corresponding to realistic electronic mod-els for manganites that have been widely discussed beforein the literature, but which were not systematically stud-ied using computational methods.

2 Model, Symmetries and Algo-rithm

The ferromagnetic Kondo Hamiltonian [4, 19] studied inthis paper is defined as

H = −t∑

〈ij〉σ

(c†iσcjσ + h.c.) − JH

∑

iαβ

c†iασαβciβ · Si, (1)

where ciσ are destruction operators for electrons at site i

with spin σ, and Si is the total spin of the t2g electrons,assumed localized. The first term is the electron transferbetween nearest-neighbor Mn-ions, JH > 0 is the ferro-magnetic Hund coupling, the number of sites is L, and therest of the notation is standard. The boundary conditionsused in the present study are important for some results,and they will be discussed later in the text. The electronicdensity of eg electrons, denoted by 〈n〉, is adjusted using achemical potential µ. In several of the results shown belowthe spin Si will be considered classical (with |Si| = 1). In1D both classical and quantum mechanical t2g spins willbe studied, the latter having a realistic spin 3/2 but alsoconsidering spin 1/2 for comparison. PhenomenologicallyJH ≫ t, but here JH/t was considered an arbitrary pa-rameter, i.e. both large and small values for JH/t werestudied. Below some calculations were also carried outincluding a large on-site Coulombic repulsion among themobile electrons.

For a one-dimensional chain (open ends), or for aone-dimensional ring with L even and periodic or an-tiperiodic boundary conditions (PBC and APBC, respec-tively) the model is particle-hole symmetric with respect

to 〈n〉 = 1 by simply transforming c†i↑ → (−1)ici↓ and

c†i↓ → −(−1)ici↑ for the mobile electrons. In this case thedensity is transformed as 〈n〉 → 2− 〈n〉. Similar transfor-mations can be deduced for clusters of dimension largerthan one. Then, here it is sufficient to study densities〈n〉 ≤ 1.

The FM Kondo model Eq.(1) with classical spins canbe substantially simplified if the limit JH → ∞ is alsoconsidered. In this situation at every site only the spincomponent of the mobile electrons which is parallel to theclassical spin is a relevant degree of freedom. The best wayto make this reduction in the Hilbert space is by rotatingthe ciσ operators into new operators diα using a 2 × 2rotation matrix such that the transformed spinors pointin the direction of the classical spin. Explicitly, the actualtransformation is:

ci↑ = cos(θi/2) di↑ − sin(θi/2)e−iφi di↓, (2.1)

ci↓ = sin(θi/2)eiφi di↑ + cos(θi/2) di↓. (2.2)

The angles θi and φi define the direction of the classicalspin at site i. After this transformation the new Hamilto-nian becomes

HJH=∞ = −∑

〈m,n〉

(tm,nd†m↑dn↑ + h.c.), (3)

where the down component of the new operators has beendiscarded since the JH → ∞ limit is considered. Theeffective hopping is

tm,n = t [cos(θm/2) cos(θn/2)

+ sin(θm/2) sin(θn/2)e−i(φm−φn)], (4)

3

i.e. it is complex and dependent on the direction of theclassical spins at sites m,n [8]. In the limit JH = ∞ theband for the transformed spinors with up spin has itselfa particle-hole symmetry, which exists for any arbitraryconfiguration of classical spins. This can be shown bytransforming d†i↑ → (−1)idi↑, and noticing that after thistransformation the resulting Hamiltonian matrix is simplythe conjugate of the original. Then, the eigenvalues areunchanged, but 〈d†i↑di↑〉 → 1 − 〈d†i↑di↑〉.

The partition function of the FM Kondo model withclassical spins can be written as

Z =

L∏

i

(∫ π

0

dθi sin θi

∫ 2π

0

dφi

)

trc(e−βH). (5)

For a fixed configuration of angles {θi, φi} the Hamilto-nian amounts to non-interacting electrons moving in anexternal field. This problem can be diagonalized exactlysince the Hamiltonian is quadratic in the fermionic vari-ables. This diagonalization is performed simply by callinga library subroutine in a computer program. If the 2Leigenvalues are denoted by ǫλ the resulting partition func-tion becomes

Z =

L∏

i

(∫ π

0

dθi sin θi

∫ 2π

0

dφi

) 2L∏

λ=1

(1 + e−βǫλ). (6)

The integrand in Eq.(6) is positive and, thus, a MonteCarlo simulation can be performed on the classical spin an-gles without “sign” problems. This was the procedure fol-lowed in our study below. The number of sweeps throughthe lattice needed to obtain good statistics varied widelydepending on the temperature, densities, and couplings.In some cases, such as in the vicinity of phase separationregimes, up to 106 sweeps were needed to collect goodstatistics. Measurements of equal-time spin and chargecorrelations for the mobile electrons were performed bytransforming the operators involved using the basis thatdiagonalizes the Hamiltonian for a fixed configuration ofclassical spins. Dynamical studies of the optical conduc-tivity σ(ω) and the one particle spectral function A(p, ω)could also be performed following a similar procedure, buttheir detailed study is postponed for a future publication.The analysis of the FM Kondo model in the case of quan-tum mechanical t2g degrees of freedom was performed atzero temperature using standard Lanczos [14, 20] and Den-sity Matrix Renormalization Group (DMRG) [21] tech-niques. The study at infinite dimension was carried outwith the Dynamical Mean-Field approach [19].

3 Results in D=1 with Classical

Localized Spins

In this section the computational results that led us to pro-pose Fig.1 as the phase diagram of the FM Kondo model in

one dimension using classical spins are presented. Resultsfor quantum mechanical t2g spins are also provided.

3.1 Ferromagnetism

The boundary of the ferromagnetic region in Fig.1 wasobtained by studying the spin-spin correlations (betweenthe classical spins) defined in momentum space as S(q) =(1/L)

∑

j,m ei(j−m)·q〈Sj · Sm〉, using a standard notation,at the particular value of zero momentum. L is the num-ber of sites. The analysis was performed for couplingsJH/t = 1, 2, 3, 4, 8, 12 and 18. For the last four (strong)couplings and in the region of densities that were foundto be stable (see next section for a discussion) the realspace spin-spin correlations 〈Sj · Sm〉 have a robust tailthat extends to the largest distances available on the clus-ters studied here. This is obviously correlated with thepresence of a large peak in S(q) at zero momentum. Fig.3shows representative results at JH/t = 12 using openboundary conditions (OBC) and L = 24. A robust fer-romagnetic correlation is clearly observed even at largedistances. The strength of the tail decreases as 〈n〉 in-creases in the region 〈n〉 ≥ 0.5, and it tends to vanish at adensity ∼ 0.78 which corresponds to the boundary of theferromagnetic region in Fig.1. A similar behavior is ob-served for smaller values of the coupling. It is interestingto note that the maximum strength of the spin-spin corre-lation tail appears at 〈n〉 ≈ 0.50. This result is compatiblewith the behavior expected at JH = ∞ where large andsmall densities are exactly related by symmetry (as dis-cussed in Sec.II) and, thus, the ferromagnetic correlationsshould peak at exactly 〈n〉 = 0.50. Working at JH/t ≤ 8this qualitative behavior is washed out and in this regimethe correlations at densities 〈n〉 ≤ 0.50 are very similar.

Special care must be taken with the boundary con-ditions (BC). Closed shell BC or open BC are neededto stabilize a ferromagnetic spin arrangement. If otherBC are used the spin correlations at short distances arestill strongly FM (if working at couplings where ferromag-netism is dominant), but not at large distances where theybecome negative since a kink appears separating two re-gions of opposite total spin, with each region having allspins aligned in the same direction. This well-known ef-fect was observed before in a similar context [12, 22] andit does not present a problem in the analysis shown below.Actually results using other boundary conditions and lat-tice lengths up to 60 sites are compatible with the dataof Fig.3. In particular it can be shown that the ferromag-netic correlations persist when the lattice size is increased.Fig.4a shows S(q = 0) versus temperature for several lat-tice sizes. At small temperature the sum over sites of thecorrelations grow with L showing that the FM correlationlength is larger than the size of the chains used here. How-ever, in 1D the Mermin-Wagner theorem forbids the exis-tence of a finite critical temperature and, thus, eventuallyS(q = 0) must tend to saturate at any fixed finite temper-

4

ature T as L increases, as it already occurs for T ≥ 0.02in Fig.4a. In spite of this subtle detail, using closed-shellboundary conditions the numerical data presented in thissubsection supports the presence of strong ferromagneticcorrelations in the one dimensional FM Kondo model, andit is reasonable to expect that at zero temperature themodel in the bulk will develop a finite magnetization inthe ground state. Fig.4b contains the spin-spin correla-

0 4 8 12 l

0.0

0.2

0.4

0.6

0.8

1.0

⟨€S i •

€S i+

l ⟩

: 0.75 : 0.67 : 0.29 : 0.50

1D

JH = 12t

T = t / 75

L = 24

OBC

⟨n⟩

Figure 3: Spin-spin correlations among the localizedspins (assumed classical) obtained with the Monte Carlomethod working on a 1D system, and at the coupling, tem-perature, densities, and lattice size shown. Open bound-ary conditions were used. Clear strong ferromagnetic cor-relations are observed.

tions vs distance parametric with temperature. They showthat even at relatively high temperatures, the correlationsat short distances are clearly ferromagnetic, an effect thatshould influence on transport properties since carriers willreact mainly to the local environment in which they areimmersed. Fig.5 contains information that illustrates theformation of ferromagnetic spin polarons at relatively hightemperatures. Fig.5a shows the spin correlations at lowelectronic density. At T = t/10 and distance 1, the corre-lation is only about 15% of the maximum. However, if thiscorrelation is measured in the immediate vicinity of a car-rier using 〈ne

iSi · Si+l〉, where nei =

∑

σ niσ(1−ni−σ) andniσ is the number operator for eg-electrons at site i andwith spin projection σ, then the correlation is enhanced to40%. Thus, it is clear that FM correlations develop in thevicinity of the carrier [23]. This spin polaron apparentlyhas a size of 3 to 4 lattice spacings in our studies.

0.001 0.01 0.1 1 T / t

0.0

10.0

20.0

30.0

40.0

S(q

=0

)

L = 10 L = 20 L = 30 L = 40

1D

⟨n⟩ = 0.5

JH = 8t

(a)

0 1 2 3 4 5l

0.0

0.5

1.0

⟨Si•S

i+l⟩

: 1/75: 1/50: 1/25: 1/15: 1/10: 1/2.5

T / t(b)

Figure 4: (a) Temperature dependence of the spin cor-relations in the one dimensional FM Kondo model usingclassical spins and the Monte Carlo method. Shown is thezero momentum spin structure factor S(q) (for the classi-cal spins) at the density, coupling, and lattice sizes indi-cated. Closed shell boundary conditions were used (peri-odic boundary conditions (PBC) for L = 10 and 30, andantiperiodic boundary conditions (APBC) for L = 20 and40); (b) Spin-spin correlations at density 〈n〉 = 0.7, usinga 10 site chain, PBC, and JH/t = 8.0. The temperaturesare indicated.

5

0 1 2 3 4 5l

0.00

0.05

0.10

543210l

0.0

0.5

1.0

⟨Si•S

i+l⟩

: 1/75: 1/50: 1/25: 1/10

(a) (b)T / t

⟨nie S

i•Si+

l⟩

Figure 5: (a) Spin-spin correlations at 〈n〉 = 0.1 using achain of 10 sites with PBC. The Hund coupling is JH/t =8.0; (b) Similar as (a) but measuring the spin correlationsnear a conduction electron using 〈ne

iSi · Si+l〉, where nei is

defined in the text.

3.2 Phase Separation

One of the main purposes of this paper is to report theexistence of phase separation in the FM Kondo model. Tostudy this phenomenon in the grand-canonical ensemble,where the Monte Carlo simulations are performed, it isconvenient to analyze 〈n〉 versus µ. If 〈n〉(µ) is discontin-uous then there are densities that can not be estabilized,regardless of the value of µ. The results shown in Fig.6obtained at JH/t = 12 clearly show that indeed phase sep-aration occurs in the FM Kondo model. The discontinuityis between a density corresponding to the antiferromag-netic regime 〈n〉 = 1.0 and ∼ 0.77 where ferromagneticcorrelations start developing, as shown before in Fig.3. Inthe case of the canonical ensemble these results can berephrased as follows: if the system is initially setup witha density in the forbidden band, it will spontaneously sep-arate into two regions having (i) antiferromagnetic (AF)correlations and no holes, and (ii) FM correlations andmost of the holes.

The existence of phase separation can also be deducedfrom the actual Monte Carlo runs since they require largeamounts of CPU time for convergence in the vicinity ofthe critical chemical potential µc. The reason is that inthis regime there are two states in strong competition.Qualitatively this effect can be visualized analyzing 〈n〉as a function of Monte Carlo time. Fig.7 shows such atime evolution when the chemical potential is fine tunedto its critical value µc ∼ −6.69812t at JH/t = 8. Wildfluctuations in 〈n〉 are observed with frequent tunnelingevents covering a large range of densities. A change in µas small as 0.001 or even smaller reduces drastically thefrequency of the tunneling events, and makes the results

−11.0 −10.5 −10.0µ / t

0.7

0.8

0.9

1.0

⟨n⟩

: L = 20, PBC : L = 20, APBC : L = 24, OBC : L = 30, PBC

1D

JH = 12t

T = t / 75

Figure 6: Electronic density 〈n〉 vs the chemical poten-tial µ obtained with the Monte Carlo technique appliedto the one dimensional FM Kondo model with classicalspins. Coupling, temperature, lattice sizes, and bound-ary conditions are indicated. The discontinuity suggeststhat some densities are unstable, signaling the presence ofphase separation.

0 0.5 1 1.5 2MC steps (×€ 10

5)

0.7

0.8

0.9

1.0

⟨n⟩

µ = −6.69812t

Figure 7: Monte Carlo time evolution of the density 〈n〉at the particular value of µ where the discontinuity takesplace working at JH/t = 8, T = t/75, using L = 20 sites,and PBC. Frequent tunneling events are observed showingthe competition between two states as in a first order phasetransition.

more stable although certainly strong fluctuations remainin a finite window near µc. Fig.8 illustrates this effectshowing a histogram that counts the number of times that

6

a density in a given window of density is reached in thesimulation. As µ crosses its critical value the histogramschange rapidly from having a large peak close to 〈n〉 ∼ 1to a large peak at 〈n〉 ∼ 0.75. At densities that are stableaway from the phase separation region these histogramspresent just one robust peak.

The qualitative behavior exemplified in Fig.6 was alsoobserved at other values of JH/t. For instance, Fig.9a con-tains resuls for JH/t = 4 which are very similar to thosefound at a larger coupling. In a weaker coupling regimethe discontinuity is reduced and now the competition is be-tween the AF state and a state with incommensurate cor-relations rather than ferromagnetism. As example, Fig.9bcontains results for JH/t = 2. The discontinuity is lo-cated near µc ∼ −1.35t and the two states competing inthis regime actually have very similar properties. Analyz-ing, for instance, just S(q) would have not provided anindication of a sharp discontinuity in the density, unlikethe case of a large Hund coupling where the peak in thespin structure factor jumps rapidly from q = π to 0 at thecritical chemical potential.

12 13 14 15 16 17 18 19 20Total number of electrons

0

10000

20000

30000

4000012 13 14 15 16 17 18 19 20

0

10000

2000012 13 14 15 16 17 18 19 20

0

10000

20000

30000

(a)

(b)

µ = −6.698125t(c)

µ = −6.698t

µ = −6.69812t

Figure 8: Hystogram for the total number of electronsobtained at JH/t = 8, T = t/75, using periodic boundaryconditions on a 20 sites chain. The chemical potentialsare shown. (a) corresponds to a case where the averagedensity is close to 〈n〉 = 1, (b) is at the critical chemicalpotential, and at (c) the electronic density is close to 0.77.

−4.0 −3.5 −3.0 −2.5µ / t

0.5

0.6

0.7

0.8

0.9

1.0

: L = 20, PBC : L = 20, APBC : L = 20, OBC : L = 30, PBC : L = 30, APBC

−1.8 −1.6 −1.4 −1.2µ / t

0.6

0.7

0.8

0.9

1.0

: L = 20, OBC : L = 30, OBC : L = 40, OBC

⟨n⟩

1D 1DJH = 4t

T = t / 75JH = 2t

T = t / 100

(a) (b)

Figure 9: Electronic density vs µ for the FM Kondomodel with classical spins in one dimension using MonteCarlo methods. A clear discontinuity signals the existenceof phase separation. (a) corresponds to JH/t = 4, while(b) is for JH/t = 2. Temperatures, chain sizes, and bound-ary conditions are indicated. The apparent second discon-tinuity located at µ/t ∼ −1.55 in (b) is expected to be justa rapid crossover.

3.3 Incommensurate Correlations

The tendency to develop a spin pattern with incommen-surate characteristics can be easily studied in our calcu-lations observing the behavior of S(q) as couplings anddensities are varied. While at large JH/t the spin struc-ture factor is peaked only at the momenta compatiblewith ferro or antiferromagnetic order, a different resultis obtained as JH/t is reduced. Fig.10 shows S(q) atJH/t = 1.0 for a variety of densities. The AF peak close to〈n〉 = 1 smoothly evolves into a substantially weaker peakwhich moves away from q = π in the range 0.5 ≤ 〈n〉 ≤ 1.0.The peak position is close to 2kF = 〈n〉, and since the spin-spin correlations for the mobile electrons show a similarbehavior this is compatible with Luttinger liquids predic-tions [24]. In this density range, and for the lattice sizesand temperatures used here, the peak in S(q) is broadand, thus, it cannot be taken as an indication of long-range incommensurate (IC) correlations but rather of thepresence of IC spin arrangements at short distances. In theregion of low densities 〈n〉 ≤ 0.4, S(q) is now peaked atzero momentum which is compatible with the presence ofrobust ferromagnetic spin-spin correlations at the largestdistances available in the studied clusters. The transitionfrom one regime to the other in the intermediate smallwindow 0.4 ≤ 〈n〉 ≤ 0.5 is very fast and was not studiedin detail here, but for a second order phase transition it isexpected to be continuous. In Fig.11 results for a slightlylarger coupling JH/t = 2 are shown. Here the patternis more complicated since apparently the AF peak does

7

not evolve smoothly into the peak at q ∼ π/2 observed at〈n〉 = 0.733, suggesting an interesting interplay betweenspin and charge.

0.0 0.2 0.4 0.6 0.8 1.0q / π

0.0

2.0

4.0

6.0

S(q

)

: 0.97 : 0.83 : 0.62 : 0.51 : 0.38

1D⟨n⟩

JH = t

T = t / 75

Figure 10: S(q) (Fourier transform of the spin-spin cor-relations between classical spins) at JH = t and T = t/75,on a chain with 30 sites and PBC. The densities are in-dicated. The positions of the peaks indicate the tendencyto have incommensurate correlations in the FM Kondomodel.

0.0 0.2 0.4 0.6 0.8 1.0q / π

0.0

2.0

4.0

6.0

S(q

)

: 0.83 : 0.80 : 0.73 : 0.66 : 0.53

1D⟨n⟩ JH = 2t

T = t / 100

Figure 11: Same as Fig.10, but working at JH = 2t andT = t/100, with 30 sites and open boundary conditions(OBC).

Note that the presence of IC correlations in models for

manganites was predicted theoretically using a Hartree-Fock approximation [25]. Our results are compatible withthese predictions although, once again, it is not clear ifthe IC pattern corresponds to long-range order or simplyshort distance correlations. More work is needed to clarifythese issues. Nevertheless, the present effort is enough toshow that the tendency to form IC spin patterns exists inthe FM Kondo model at small JH/t.

3.4 Influence of a Direct Coupling Amongthe Localized Spins

The results of Sec.III.B show the presence of phase sep-aration near half-filling, but not in the opposite extremeof low conduction electron density. This is a consequenceof the absence of a direct coupling among the localizedspins in Eq.(1). This coupling may be caused by a smallhybridization between the t2g electrons. If a Heisenbergterm J ′

∑

〈ij〉 Si · Si coupling the classical spins is added,

then at 〈n〉 = 0 an antiferromagnetic state is recoveredsimilarly as at 〈n〉 = 1. This term was already considered

1.0 0.8 0.6 0.4 0.2 0.0⟨n⟩

0.00

0.01

0.02

0.03

0.04

0.05

J’/

t

FMPS PS

Figure 12: Phase diagram in the plane J ′/t-〈n〉 obtainedstudying the density vs µ calculated using the Monte Carlotechnique with classical localized spins. The coupling isfixed to JH/t = 8 and the temperature is low. Note thatthe regime of low density and J ′/t = 0 is somewhat diffi-cult to study due to the influence of van Hove singularitieswhich produce a rapid change of the density with µ.

in the study of the strong coupling version of the Kondomodel [12]. Although a detailed study of the influence of J ′

on the phase diagram Fig.1 is postponed for a future pub-lication [26], here the effects of this new coupling into theexistence of phase separation are reported. Following thesame procedure described in Sec.III.B to obtain unstabledensities, the phase diagram shown in Fig.12 was found.Note that at J ′ 6= 0 phase separation occurs at large and

8

small electronic densities. In the latter the separation isbetween electron-rich ferromagnetic and electron-poor an-tiferromagnetic regions. Then, the experimental searchfor phase separation discussed below in Sec.VIII should becarried out at both large and small hole densities. Moredetails will be given elsewhere [26].

4 Results in D=1 with QuantumLocalized Spins

4.1 Quantum Mechanical t2g S=3/2 spins

The use of classical spins to represent the t2g degrees offreedom is an approximation which has been used since theearly days of the study of manganites [4, 5]. While such anapproach seems reasonable it would be desirable to havesome numerical evidence supporting the idea that usingspin operators of value 3/2 (denoted by Si) the results aresimilar as those obtained with classical spins. Althoughnumerical unbiased calculations with spins 3/2 are diffi-cult in dimensions larger than 1, at least this issue can beaddressed numerically with 1D chains using the Lanczosand Density Matrix Renormalization Group techniques.The Hamiltonian is the same as in Eq.(1) but now withquantum mechanical degrees of freedom normalized to 1(i.e. replacing Si by Si/(3/2)) to simplify the comparisonof results against those obtained using classical spins. Cal-culating the ground state energy in subspaces with a fixedtotal spin in the z-direction, it is possible to study thetendency to have a ferromagnetic state in the model usingthe Lanczos technique. The results indicate that there isa robust region of fully saturated ferromagnetism, as indi-cated in Fig.13. The actual boundary of this region agreesaccurately with the results obtained using classical spinsshown in Fig.1. This reinforces the belief that S=3/2 andclassical spins produce similar results, at least regardingferromagnetism. Finite size effects are apparently smallfor the chains accessible with the DMRG method, as ex-emplified in Fig.14a where the ground state energy forJH/t = 6.0 using a variety of chain lengths is presented.

To study other important features of the phase diagram,such as phase separation, the compressibility is needed.This quantity is defined as

κ−1 =N2

e

L

E(Ne + 2, L) + E(Ne − 2, L) − 2E(Ne, L)

4,

(7)where E(Ne, L) is the ground state energy correspondingto a chain with L sites and Ne electrons. If κ−1 becomesnegative at some fixed density (remember that here thenumerical study is in the canonical ensemble), the sys-tem is unstable and phase separation occurs. The DMRGresults for the compressibility in Fig.14b for several cou-plings led us to conclude that the spin 3/2 model also hasphase separation in the ground state near half-filling, simi-

0.40.50.60.70.80.91.0⟨n⟩

0.0

5.0

10.0

15.0

20.0

8 10 12

IC

PSFM

LJH−t

1DS=3/2

Figure 13: Phase diagram of the FM Kondo model withS = 3/2 localized t2g spins obtained with the DMRG andLanczos techniques applied to finite chains as indicated.The notation is as in Figs.1 and 2, i.e. ferromagnetism,incommensurability, and phase separation appear both forclassical and quantum mechanical localized spins.

0.00.20.40.60.81.0⟨n⟩

−6.5

−5.5

−4.5

−3.5

−2.5

−1.5

8 10 12 14 16

0.750.800.850.90 ⟨n⟩

−0.5

0.0

0.5

1.0

1.5

5.85 6.00 8.25 10.50

(a) (b)JH / t

L

1_κ

E0−L

Figure 14: (a) Ground state energy per site vs density atJH/t = 6.0 for the S = 3/2 FM Kondo model in 1D usingDMRG techniques keeping m = 48 states. Results for avariety of chain lengths are shown; (b) Inverse compress-ibility vs density for the same model and chain lengths asin (a), calculated using several couplings. A negative 1/κsignals an unstable density.

9

larly as in the case of classical spins. Indeed Fig.14b showsthat 1/κ becomes negative for densities in the vicinity of〈n〉 ∼ 0.85 and larger, for the range of couplings shown.Thus, at least qualitatively there are tendencies to phaseseparate in the same region suggested by the simulationsusing classical localized spins.

However, a difference exists between results obtainedwith classical and quantum mechanical t2g degrees of free-dom: apparently there is a finite window in density be-tween the phase separated and FM regimes in the phasediagram of Fig.13. This window could be a finite size ef-fect, but the lack of a strong dependence with the chainlengths in the results of Fig.14a led us to believe that itmay actually exist in the bulk limit. In addition, pre-vious studies using the strong JH/t coupling version ofthe FM Kondo model have also reported an intermediatewindow between phase separation and FM [12], and theanalysis below for localized spins 1/2 suggests a similarresult. Studying the spin of the ground state in this inter-mediate region here and in Ref.[12] it has been observedthat it is finite i.e. apparently partial ferromagnetism ap-pears immediately at any finite stable density in the modelwith spins 3/2, at least working at intermediate and largeHund couplings [27]. The transition from phase separationto FM appears to be smooth in the spin quantum num-ber, which is somewhat reminiscent of the results in theclassical limit where the tail of the spin-spin correlationgrows with continuity from small to large as the densitydiminishes in the stable region.

0.0 0.2 0.4 0.6 0.8 1.0q / π

0.0

10.0

20.0

30.0

S(q

)

: 1.00 : 0.66 : 0.33 : 0.16

1D

⟨n⟩

JH = 0.75 tS = 3/2

Figure 15: S(q) for the quantum FM Kondo model usinglocalized spins 3/2 on a chain of 12 sites. The techniqueis DMRG, keeping 48 states in the iterations. Densitiesand coupling are indicated. In the spin correlations usedto obtain S(q), full spins 3/2 (i.e. not normalized to 1)were used.

Finally, let us analyze whether incommensurate corre-lations exist in the spin 3/2 model as it occurs in theclassical model. Fig.15 shows S(q) obtained with DMRGworking on a chain of 12 sites. Although momentum is nota good quantum number on a system with OBC, neverthe-less using the same definition of the Fourier transform ofthe spin correlation as in the study of the classical systemwith periodic and antiperiodic boundary conditions (PBCand APBC, respectively), qualitative information aboutthe tendency to form incommensurate structures can begathered. The results of Fig.15 obtained at very smallJH/t indeed show that strong IC correlations exist in theground state of this model, in agreement with the resultsof Figs.10 and 11 for classical spins. Then, it is concludedthat the three dominant features of the phase diagramFig.1 (PS, FM, and IC) have an analog in the case of thespin 3/2 quantum model. This agreement gives supportto the believe that the results presented below in this pa-per for dimensions larger than one using classical spinsshould be qualitatively similar to those corresponding toa model with the proper quantum mechanical t2g degreesof freedom.

4.2 Quantum Mechanical t2g S=1/2 spins

For completeness, in this paper the special case of local-ized spins 1/2 in 1D has also been studied. The anal-ysis has relevance not only in the context of manganitesbut also for recently synthesized one dimensional materialssuch as Y2−xCaxBaNiO5 which have a mobile and a local-ized electron per Ni-ion. This compound has been studiedexperimentally [28] and theoretically [29], and upon dop-ing interesting properties have been observed including ametal-insulator transition. As discussed below, the con-clusion of this subsection will be that the results for local-ized spins 1/2 are qualitatively similar to those obtainedwith classical and spin 3/2 degrees of freedom, i.e. fer-romagnetism, incommensurability, and phase separationappear clearly in the phase diagram. The Hamiltonianused for this study is as defined in Eq.(1) but now withSi replaced by a spin 1/2 operator (not normalized to 1).The technique used to obtain ground state properties isthe finite-size DMRG method on chains with up to 40sites, a variety of densities, and typical truncation errorsaround 10−5. Some calculations were also performed withthe Lanczos algorithm on lattices with up to 12 sites. Thedata obtained with the DMRG and Lanczos methods arequalitatively similar.

The main result is contained in the phase diagramshown in Fig.16. Three regimes were identified [30].The region labeled FM corresponds to saturated ferro-magnetism i.e. the ground state spin is the maximum.This regime was found using the Lanczos technique sim-ply searching for degeneracies between the lowest energystates of subspaces with different total spin in the z-direction. For the IC regime, S(q) was calculated using

10

0.40.50.60.70.80.91.0⟨n⟩

0.0

20.0

40.0

60.0

80.0

8 10 12

IC

PSFM

LJH−t

1DS=1/2

Figure 16: Phase diagram of the FM Kondo model withS = 1/2 localized states obtained with the DMRG andLanczos methods. The length of the chains is indicated.The notation FM, PS, and IC is as in previous figures.Note the similarity with the results of Fig.13, up to anoverall scale.

the same definition as in subsection III.A but now withspin 1/2 operators instead of classical spins of length 1.The results shown in Fig.17 suggest the presence of incom-mensurate correlations at small JH/t, at least at short dis-tances. Similarly as for the cases of larger localized spins(Figs.10 and 15), the peak in the spin structure factormoves smoothly from q = π near 〈n〉 = 1 to q = 0 in theFM region. The position of the peak is at 2kF . Our studyalso showed that correlated with this behavior the chargestructure factor N(q) has a cusp at the same position. Itis important to clarify that in the IC regime of Fig.17 atlow temperatures, the ground state has a finite spin but itis not fully saturated. Within the accuracy of our studythe spin varies smoothly as the couplings and density arechanged reaching its maximum value at the FM boundary.

The tendency to phase separate in this model can bestudied calculating 〈n〉 vs µ. In the study of this section,carried out in the canonical ensemble where 〈n〉 is fixed,the procedure to obtain µ involves (i) the calculation ofthe ground state energy for a variety of densities, and (ii)the addition of −µN to the Hamiltonian, where N is the

0.0 0.2 0.4 0.6 0.8 1.0q / π

0.0

1.0

2.0

3.0

S(q

)

: 1.0 : 0.8 : 0.6 : 0.4 : 0.2

1D⟨n⟩

JH = 4 tS = 1/2

Figure 17: S(q) for the quantum FM Kondo model usinglocalized spins 1/2 on a chain of 20 sites. The technique isDMRG, keeping up to 60 states in the iterations. Densitiesand coupling are indicated. In the spin correlations usedto obtain S(q), full spins 1/2 (i.e. not normalized to 1)were used.

total number operator. As µ varies, different density sub-spaces become the actual global ground state. If there aredensities that cannot be stabilized for any value of µ, thensuch a result is compatible with phase separation in themodel (for more details see Ref. [31]). Results are pre-sented in Fig.18.a,b,c for a chain with 20 sites: at smallJH/t = 1 all densities are accessible tuning µ, and the cur-vature of the ground state energy EGS vs 〈n〉 is positive.However, working at JH/t = 30 the density 〈n〉 = 0.9becomes unstable, and now EGS vs 〈n〉 has less curva-ture. Fig.18d shows again density vs chemical potentialbut now on a larger chain of 40 sites. Here densities be-tween 1 and 0.8 are unstable, in agreement with Fig.18c.A similar method to calculate the region of phase separa-tion is to evaluate the inverse compressibility since a neg-ative value for this quantity indicates phase separation, asexplained in Sec.IV.A. Using this procedure once again aregion of κ−1 < 0 was identified signaling unstable den-sities in the system. From the combination of these typeof analysis performed for several chains, the boundaries ofphase separation were estimated as shown in Fig.16. Al-though the error bars are not negligible, the presence ofphase separation is a robust feature of the calculation andit is in excellent agreement with the conclusions of previ-ous sections. Then, irrespective of the actual value of thespin corresponding to the t2g degrees of freedom the phasediagram presents universal features, specially robust fer-romagnetism, unstable densities, and short-range incom-

11

0.40.50.60.70.80.91.0⟨n⟩

−3.5

−3.0

−2.5

−2.0

−1.5

−1.0

−0.5

1 t 30 t

−2.2 −1.4 −0.6µ

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

−15.7 −14.9 −14.1µ

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

−12.8 −12.6 −12.4µ

0.75

0.80

0.85

0.90

0.95

1.00

1.05

(a) (b)

(c) (d)

⟨n⟩

⟨n⟩

⟨n⟩

E0−L JH

JH = 1 t

JH = 30 t JH = 28 t

L = 20

L = 20 L = 40

L = 20

Figure 18: Results for the FM Kondo model with S = 1/2localized spins using the DMRG method. (a) contains 〈n〉vs µ for a chain of 20 sites at JH/t = 1, showing that allavailable densities are accessible; (b) Ground state energyper site vs 〈n〉 for JH/t = 1 and 30. The energies for thelatter were divided by 5 to make both curves of comparablemagnitude; (c) Same as (a) but for JH/t = 30. Now〈n〉 = 0.90 appears unstable; (d) Results on a 40 siteschain at JH/t = 28. The densities available between 1.00and 0.80 (i.e. 0.95, 0.90, and 0.85) are unstable.

mensurate correlations. Below in Secs.V and VI, it will beshown that this universality can actually be extended toinclude higher dimensional clusters.

4.3 Influence of an On-site Coulomb Re-pulsion

The on-site Coulombic repulsion among electrons in theconduction band has been neglected thus far. Althoughit is not expected to produce qualitative changes in thephysics described before (since a large JH/t prevents dou-ble occupancy) it would be desirable to have some indica-tions of its quantitative influence on the phase diagramsdiscussed in previous sections. Unfortunately, the additionof a Hubbard repulsion U complicates substantially themany-body numerical studies. The Monte Carlo methodin the classical limit can only proceed after the HubbardU -term is decoupled using Hubbard-Stratonovich (HS)variables. If such a procedure is followed the simulationwould run over both angles and HS degrees of freedom,and likely a “sign” problem would occur. Such a cumber-some approach will not be pursued here. Instead the largeU/t limit for the case of localized spins 1/2 will be inves-tigated using the Lanczos method, which can be appliedwithout major complications to the study of Hubbard-likesystems. The study will be limited to chains due to re-strictions on the size of the clusters that can be analyzednumerically. In addition, in the strong coupling limit theHubbard model becomes the t − J model, and, thus, the

actual Hamiltonian studied here is defined as

H = J∑

i

[si · si+1 − (1/4)nini+1]

− t∑

iσ

(c†iσ ci+1σ + h.c.) − JH

∑

i

si · Si, (8)

where c is a destruction fermionic operators which includesa projector operator avoiding double occupancy. Both Si

and si are spin-1/2 operators representing the spin of thelocalized and mobile degrees of freedom, respectively. Therest of the notation is standard.

Once again studying the energy of the ground state forsubspaces with different total spin projection in the z-direction, the existence of ferromagnetism can be studied.The boundary of the region with fully saturared ferro-magnetism obtained using clusters with 8 and 10 sites isshown in Fig.19a. Strong ferromagnetic correlations ap-pear even for small values for JH/t, suggesting that as U/tgrows the tendency to favor ferromagnetism increases, asnoticed also in Ref. [12]. It is interesting to observe that inthe region not labeled as FM in Fig.19a a finite spin existsin the ground state (see Fig.19b), in excellent agreementwith results for the U/t = 0 case reported in Sec.IV.A andIV.B for quantum mechanical t2g degrees of freedom (seealso Ref.[12]). In addition, a tendency towards incom-mensurate correlations is observed in this phase, also inagreement with previous results (see Fig.19c). Regardingthe issue of phase separation, the results of Fig.16 sug-gests that this regime should appear for densities between〈n〉 = 1.0 and 〈n〉 = 0.90. Unfortunately these densitiesare not accessible on the clusters studied in this subsec-tion, and, thus, the confirmation of the existence of phaseseparation at large U/t will still require further numericalwork [32]. Nevertheless, Figs.16 and 19a, which containresults with and without the Coulomb interaction, haveclear qualitative common trends. Regarding the quanti-tative aspects, the Coulomb interaction changes substan-tially the scales in the phase diagram specially regardingferromagnetism which now appears at smaller values ofJH/t [33].

5 Results in D=2

In this section the computational results that led us to pro-pose Fig.2 as the phase diagram of the FM Kondo modelin two dimensions are presented. Results in 2D are partic-ularly important since manganite compounds with layeredstructure have been recently synthesized [34].

5.1 Ferromagnetism

The search for ferromagnetism in the ground state of the2D clusters was carried out similarly as in 1D for the caseof classical t2g spins. The real space spin-spin correlations

12

1.0 2.0 3.0 4.0 5.00.0

2.0

4.0

6.0

8.0

S

0.00.20.40.60.81.00.0

1.0

2.0

3.0

4.0

5.0

JH / t

⟨n⟩

JH / t

(a)

(b)

FM

IC

J = 0.2t

J = 0.2t

⟨n⟩ = 0.75

0.0 0.2 0.4 0.6 0.8 1.0q / π

0.0

0.1

0.2

0.3

S(q

)

: 1.0 1.0 : 0.8 1.0 : 0.6 0.8 : 0.6 1.0

(c)

⟨n⟩ JH / t

Figure 19: Results corresponding to the FM Kondo modelwith localized spins 1/2 and using the t − J model forthe mobile electrons (Eq.(8)), obtained with the Lanczosmethod. The boundary conditions were such that a fullysaturared ferromagnetic states is stable. J/t = 0.2 wasused. (a) shows the boundary of the FM region using 8 and10 sites clusters. IC correlations were observed at smallJH/t in the non-fully saturated ferromagnetic region; (b)the spin of the ground state as a function of JH for thecase of 2 holes on the 8 sites cluster; and (c) S(q), theFourier transform of the spin-spin correlations between thelocalized spins, vs momentum for the case of a 10 sitescluster. The densities and couplings are indicated.

(between those classical spins) was monitored, as well asits Fourier transform S(q) at zero momentum. CouplingsJH/t = 1, 2, 4, 8, and 16 were particularly analyzed. Typ-ical results are presented in Fig.20 where the spin corre-lations are presented at a fixed coupling, parametric withthe electronic density. It is clear that the tails of the corre-lations are very robust, and the ferromagnetic correlationlength exceeds the maximum distance, dmax, available onthe 6 × 6 cluster. Plotting the spin correlation at dmax

vs 〈n〉, an estimate of the critical density for ferromag-netism can be obtained. Combining results from a varietyof clusters and boundary conditions, the FM boundary ofFig.2 was constructed. Note that for this analysis the useof open boundary conditions seem to be the optimal i.e.using other boundary conditions kept the ferromagneticcharacter of the system at short distances but modify thecorrelations at large distances, as it occurs for non-closed-shell BC in 1D.

0.0 1.0 2.0 3.0 4.0 5.0 l

0.0

0.2

0.4

0.6

0.8

1.0

⟨€S i •

€S i+

l ⟩

2D

JH = 8t

T = t / 50

L = 6×6

PBC

Figure 20: Spin-spin correlations (among the classicalspins) vs. distance obtained at JH/t = 8, T = t/50, andusing periodic boundary conditions on a 6×6 cluster. Fullcircles, open circles, full squares, open squares, and fulltriangles correspond to densities 〈n〉 = 0.807, 0.776, 0.750,0.639, and 0.285, respectively.

The influence of the lattice size can be estimated bystudying S(q) vs temperature for several clusters. Repre-sentative results are shown in Fig.21. They were obtainedat JH/t = 16 and a density close to the PS-FM bound-ary. At low temperature S(q = 0) clearly grows with thelattice size due to the strong ferromagnetic correlations.

13

0.01 0.1 1 T / t

0.0

10.0

20.0

30.0

40.0

50.0

S(q

=0

)

L = 4×4 L = 6×6 L = 8×8

2D

⟨n⟩ = 0.72(3)

JH = 16t

Figure 21: S(q) at zero momentum vs. temperature,working at 〈n〉 = 0.72 ± 0.03, JH/t = 16, using openboundary conditions, and for the two-dimensional clustersizes shown.

However, as in the case of 1D, the Mermin-Wagner the-orem forbids long-range ferromagnetism in 2D at finitetemperature and, thus, S(q) should converge to a finiteconstant if the lattice sizes are further increased beyondthose currently accessible, working at a fixed finite tem-perature. The verification of this subtle detail is beyondthe scope of this paper, and should not confuse the read-ers: the presence of very strong ferromagnetic correlationsat low temperature in the FM Kondo model is clear in thepresent numerical study and it is likely that small cou-plings in the direction perpendicular to the planes wouldstabilize ferromagnetic order at a finite temperature.

5.2 Phase Separation

The computational analysis carried out in this subsectionshow that the phenomenon of phase separation occurs notonly in one dimension but also in two dimensions (andhigher). This result indicates that the unstable densitiesfound in Secs.III and IV are not a pathology of 1D clustersbut its existence is generic of the FM Kondo model.

Typical numerical results in 2D clusters at low temper-ature are shown in Fig.22a at JH = 8t. Data from avariety of cluster sizes and boundary conditions are pre-sented. Although the results corresponding to the lowestdensity in the discontinuity in 〈n〉 are somewhat scattereddue to size effects, the existence of such discontinuity isclear from the figure. Fig.22b contains similar results butnow for JH/t = 4. Once again, 〈n〉 is discontinuous sig-

−6.5 −6.3 −6.1µ / t

0.8

0.9

1.0

⟨n⟩

: 6×6, PBC : 6×6, APBC : 6×6, OBC : 8×8, PBC : 10×10, PBC

−3.5 −3.0 −2.5µ / t

0.7

0.8

0.9

1.0

: 6×6, PBC : 6×6, OBC

JH = 8t JH = 4t

(a) (b)

Figure 22: 〈n〉 vs µ on two-dimensional clusters andat temperature T = t/50, to illustrate the presence ofphase separation in the 2D ferromagnetic Kondo modelwith classical spins. (a) contains results at JH/t = 8 anda variety of clusters and boundary conditions. (b) cor-responds to JH/t = 4, a 6 × 6 cluster, and using bothperiodic and open boundary conditions.

naling the presence of phase separation in the low tem-perature regime of the FM Kondo model. Calculating thediscontinuity in 〈n〉 for several couplings the boundary ofthe phase separated regime in the 2D phase diagram ofFig.2 was established. Note that the scales in computertime needed to achieve convergence are very large near thecritical chemical potential where frequent tunneling eventsbetween the two minima slow down the simulations. ForJH/t < 4, it becomes difficult to distinguish between anactual discontinuity in 〈n〉 and a very rapid crossover and,thus, in the 2D phase diagram (Fig.2) the boundary ofphase separation at small Hund coupling is not sharplydefined.

5.3 Incommensurate Correlations

The regime of small coupling JH/t is not ferromagnetic,according to the behavior of S(q) at zero momentum, anddoes not correspond to phase separation since all densi-ties are stable. It may occur that robust incommensuratespin correlations exist here, as it occurs in 1D. S(q) ispresented in Fig.23 for two representative couplings anda large range of densities. The antiferromagnetic peak at(π, π) is rapidly suppressed as 〈n〉 decreases, and the posi-tion of the maximum in S(q) moves along the (π, π)–(π, 0)line (and also along (π, π)–(0, π) by symmetry). The in-tensity of the peak is rapidly reduced as it moves awayfrom (π, π), and the IC character of the correlations is ap-parently short-range. However, the numerical study of ICphases are notoriously affected by lattice sizes, and thus at

14

0.0

3.0

6.0

S(q

)

: 0.83 : 0.76 : 0.62 : 0.40 : 0.34

0.0

2.0

4.0

S(q

)

: 1.00 : 0.70 : 0.63 : 0.50 : 0.18

JH = t

T = t / 25

(0,0) (π,0) (π,π)

q

JH = 2t

T = t / 50

(a)

(b)

⟨n⟩

⟨n⟩

Figure 23: S(q) vs momentum on a 6 × 6 cluster andat (a) JH/t = 1 and T = t/25, and (b) JH/t = 2 andT = t/50. In both cases open boundary conditions wereused. The densities are indicated.

this stage it can only be claimed that a tendency to formIC spin patterns has been detected in 2D clusters, with-out a firm statement regarding their short- vs. long-rangecharacter. As 〈n〉 is further reduced, the peak positionreaches (π, 0) and (0, π) with substantial intensity. Re-ducing further the density, a rapid change into the ferro-magnetic phase was observed. At a larger JH/t couplingsuch as 3, a more complicated IC pattern was detectedwith regions where S(q) peaked simultaneously at (π, 0)-(0, π) and (0, 0). Further work is needed to clarify thefine details of the spin arrangements in this regime, butnevertheless the results given here are enough to supportthe claim that a tendency to form incommensurate cor-relations exists in the ground state of the 2D FM Kondomodel [35].

6 Results in D=∞

The existence of phase separation and ferromagnetism inthe ground state of the FM Kondo model can also bestudied in the limit of D = ∞. The Dynamical MeanField equation [19] is solved iteratively starting from arandom spin configuration, and as a function of tempera-

ture and density three solutions have been observed hav-ing AF, FM, and paramagnetic character. Efforts wereconcentrated on a particular large coupling JH/W = 4.0studying the temperature dependence of the results, whereW is the half-width of the semicircular density of statesD(ǫ) = (2/πW )

√

1 − (ǫ/W )2 for the eg electrons. Par-tial results are contained in Fig.24a. The presence of fer-romagnetism at finite doping and antiferromagnetism athalf-filling are quite clear in the calculations. Close tohalf-filling and at low temperature, the density 〈n〉 as afunction of µ was found to be discontinuous, in excellentagreement with the results already reported in D = 1and 2. Fig.24b provides a typical example obtained atT/W = 0.0003 (results at T/W = 0.002 were already pre-sented in Ref. [11]). The phase separation in this figureis between antiferromagnetic and ferromagnetic regions.However, in Fig.24b note that at a slightly larger temper-ature the separation occurs between hole-poor antiferro-magnetic and hole-rich paramagnetic regions.

For completeness, in Fig. 25 the density of states A(ω)for the AF and FM phases is shown at JH/W = 2 andT/W = 0.005 (for details of the calculation see Ref.[19]).The critical chemical potential where the AF and FMphases coexist is µc ∼ −1.40W . A(ω) in Fig.25 is cal-culated at µ = µc for both phases. In the two cases thedensity of states splits into upper and lower bands dueto the large Hund coupling. The width of the upper andlower bands is wider for the FM phase, which causes anarrower gapped region centered at ω ∼ 0. Let us nowconsider the process of hole doping starting at 〈n〉 = 1 anddecreasing µ. In the AF phase at µ ≥ µc, the chemicalpotential lies in the gap. However, at µ ≤ µc the chemicalpotential is located already inside the lower band of theFM phase, since this band is wider than in the AF phase.This suggest that before the lightly doped AF phase isrealized in the system by decreasing µ, the FM phase isinstead stabilized. Thus, the discontinuous change fromthe AF to FM phases at µ = µc also causes a jump in thecarrier number. This discontinuity occurs only when thebandwidth of the AF phase is considerably narrower thanthat of the FM phase. For this reason phase separationexists only in the large JH region.

Summarizing, at very low temperature there is a re-markable agreement between the D = ∞ and D = 1, 2results which were obtained using substantially differentnumerical techniques. Such an agreement give us confi-dence that the phase separation effect discussed here isnot pathological of low dimensions or induced by approx-imate algorithms but it is intrinsic of the physics of theFM Kondo model, and likely it exists in dimension D = 3as well.

15

−3.43 −3.42 −3.41 −3.40µ / W

0.85

0.90

0.95

1.00

1.00 0.95 0.90 0.85⟨n⟩

0.00

0.01

0.02

T / W ⟨n⟩(a) (b)

D = ∞ D = ∞

PM

FM

PS(AF−FM)

JH/W = 4T/W = 0.0003

PS(AF

−PM)

Figure 24: (a) Phase diagram in the D = ∞ limit work-ing at JH/W = 4.0. The “PS(AF-PM)” region denotesphase separation (PS) between a hole-poor antiferromag-netic (AF) region, and a hole-rich paramagnetic (PM) re-gion. The rest of the notation is standard; (b) Density 〈n〉vs µ/W obtained in the D = ∞ limit, JH/W = 4.0, andT/W = 0.0003. The discontinuity in the density is clear.

-4 -3 -2 -1 0 1 2 3 4ω / W

Den

sity

of

Sta

tes

(Arb

. un

it)

Figure 25: Density of states in the D = ∞ limit corre-sponding to the antiferromagnetic (solid line) and ferro-magnetic (dotted line) solutions, working at JH/W = 2.0and T/W = 0.005. The chemical potential is tuned to beat its critical value µc ∼ −1.40W .

7 Results in D=3

Results in three dimensions for large enough clusters aredifficult to obtain with the Monte Carlo algorithm usedin this paper. The reason is that the CPU time neededto diagonalize exactly the problem of an electron movingin a fixed spin background grows rapidly with the num-ber of sites. Nevertheless, studies using 43 clusters carriedout as part of this project have shown clear indications ofstrong ferromagnetic correlations in a region of parameter

space compatible with those found in one and two dimen-sions where FM dominates, and thus it is reasonable toassume that there are ferromagnetic phases in the FMKondo model in all dimensions from 1 to ∞. Regardingphase separation, the studies on 43 clusters cannot provideconclusive evidence due to the presence of intrinsic gaps in〈n〉 vs µ caused by size effects. But once again by simplecontinuity between D = 1, 2 and D = ∞, the existenceof phase separation in D = 3 is strongly suggested by ourresults.

In spite of the size limitations of studies in three di-mensions, it is possible to obtain useful information aboutthe actual value of the critical temperature in the limitof JH = ∞ i.e. working in a region which correspondsto a fully saturated ferromagnetic state at zero tempera-ture. In this limit the problem is simplified since for themobile electrons only the spin component in the directionof the classical spin survives, as discussed in Sec.II wherethe effective model (“complex double exchange”) was de-scribed. The absence of a spin index reduces substantiallythe CPU time for diagonalization in the numerical algo-rithm, and allowed us to study clusters with 63 sites usingthe Hamiltonian Eq.(3). Measuring the spin-spin correla-tion (among the classical spins) in real space and reducingthe temperature, it is possible to study at what temper-ature, T ∗, such correlation becomes nonzero at the maxi-mum distance available in a 63 cluster. For temperatureshigher than T ∗ the spin correlations can be accommodatedinside the cluster and, thus, the ferromagnetic correlationlength, ξFM , is finite. Using this idea, upper bounds onthe critical temperature can be obtained using T ∗ > T FM

c .In addition, it is reasonable to assume that in a 3D sys-tem the growth of ξFM with T is very rapid once it starts,and T ∗ itself may actually provide a good estimate of thecritical temperature in the bulk.

Figs.26.a-d contain the spin-spin correlations at JH =∞ on the 63 cluster parametric with temperature. Re-sults for four densities are given, and only some repre-sentative temperatures are shown. The spin correlationson a 43 cluster (not shown) are in good agreement withthose provided in Fig.26. Based on this information thetemperature where the ferromagnetic correlations reachthe boundary with a nonzero value can be obtained withreasonably small error bars. The results are shown inFig.27a. Once again, assuming a rapid increase of ξFM

as the temperature is reduced, the results of Fig.27a canbe considered as a rough estimate of the actual criticaltemperature. As anticipated in Sec.III.A, T FM

c is max-imized at 〈n〉 = 0.50. In Fig.27b, S(q) at zero momen-tum is presented as a function of temperature for the fourdensities used in Fig.26. A rapid growth is observed atparticular temperatures which are compatible with thoseobtained in Fig.27a using the tail of the real space correla-tions. The present results are qualitatively similar to thoseobtained using high temperature expansions (HTE) [36],

16

0.0 2.0 4.0 6.0 8.0 10.0|i−j|

−0.1

0.1

0.3

0.5

0.7

0.9

<S iS

j>

T/t = 1/8T/t = 1/9T/t = 1/10T/t = 1/15

0.1

0.3

0.5

0.7

0.9

<S iS

j>T/t = 1/7T/t = 1/8.5T/t = 1/10T/t = 1/20

(a)

(b)

<n>=0.5

<n>=0.34

0.0 2.0 4.0 6.0 8.0 10.0|i−j|

−0.1

0.1

0.3

0.5

0.7

0.9

<S iS

j>

T/t = 1/10T/t = 1/16T/t = 1/21T/t = 1/30

0.1

0.3

0.5

0.7

0.9

<S iS

j>

T/t = 1/13T/t = 1/15T/t = 1/20T/t = 1/30

(c)

<n>=0.14

(d)<n>=0.08

Figure 26: Spin-spin correlations (among the classi-cal spins) vs distance for several temperatures (as indi-cated) working on a 63 cluster and using the Monte Carlotechnique. (a) contains results for 〈n〉 = 0.50, (b) for〈n〉 = 0.34, (c) for 〈n〉 = 0.14, and (d) for 〈n〉 = 0.08.

although our estimates for the critical temperatures aresmaller. For instance, at 〈n〉 = 0.50 the HTE predictiongives T FM

c ∼ 0.16t, about a factor 1.5 larger than ourresult.

0.00 0.05 0.10 0.15 0.20 0.25T/t

0.0

0.5

1.0

S(0

)/L

<n> = 0.5<n> = 0.34<n> = 0.14<n> = 0.08

0.00.20.40.60.81.0<n>

0.00

0.05

0.10

0.15

TC/t

(a)

(b)

FM

PM

Figure 27: (a) Upper bound for the critical temperaturevs density deduced from spin correlations on 63 clusters,working at JH = ∞; (b) S(q = 0)/L vs temperature atseveral densities, obtained with the Monte Carlo methodon a 63 cluster at JH/t = ∞.

To obtain the critical temperature in degrees, an esti-mate of t is needed. Results in the experimental litera-ture for the eg electrons bandwidth range from BW ∼1 eV [37] to BW ∼ 4eV [38]. Assuming a dispersionǫp = −2t(cospx + cospy + cospz), the hopping amplitudeshould be t = BW/12 i.e. between 0.08 and 0.33 eV .With this result our estimate for the critical temperatureis roughly between T FM

c ∼ 100 K and 400 K, which arewithin the range observed experimentally. Then, in ouropinion it is possible to obtain realistic values for T FM

c

using purely electronic models, in agreement with othercalculations [19, 36].

17

8 Conclusions

In this paper the phase diagram of the ferromagneticKondo model for manganites was investigated. Usinga wide variety of computational techniques that includeMonte Carlo simulations, Lanczos and DMRG methods,and the Dynamical Mean Field approach, regions in pa-rameter space with (i) robust ferromagnetic correlations,and (ii) phase separation between hole-poor antiferromag-netic and hole-rich ferromagnetic regions were identified.In addition, incommensurate spin correlations were ob-served in dimension 1 and 2 at small JH/t. The criti-cal temperature towards ferromagnetism for the case ofa three dimensional lattice was also estimated, and theresults are compatible with experiments for manganites.The agreement between the results obtained with differ-ent computational techniques, working at several spatialdimensions, and using both classical and quantum me-chanical localized spins in the case of chains lead us to be-lieve that the conclusions of this paper are robust and theyrepresent the actual physics of the ferromagnetic Kondomodel.

The novel regime of phase separation is particularly in-teresting, and possible consequences of its existence inmanganites can be envisioned. Experimentally, phase sep-aration can be detected using neutron diffraction tech-niques if the two coexisting phases have different latticeparameters as it occurs in La2CuO4+δ, a Cu-oxide withhole-rich and hole-poor regions [39]. NMR and NQR spec-tra, as well as magnetic susceptibility measurements, canalso be used to detect phase separation [40, 41] since asplitting of the signal appears when there are two dif-ferent environments for the ions. Note also that in theregime where AF and FM coexist S(q) presents a two peakstructure, one located at the AF position and the other atzero momentum. This also occurs in a canted ferromag-netic state and, thus, care must be taken in the analysisof the experimental data. Actually, recent experimentalresults by Kawano et al. [42] are in qualitative agreementwith the results of Fig.24a since these authors observeda reentrant structural phase transition accompanied by“canted ferromagnetism” below T FM

c , at 0.10 < x < 0.17in La1−xSrxMnO3. In addition the polaron-ordered phasereported by Yamada et al. [43] can be reanalyzed usingthe results of this paper since it is known that the AFphase in 3D manganites is orthorhombic while the FMis pseudo-cubic. The formation of a lattice superstructuremay stabilize the magnetic tendency to phase separate andminimize lattice distortions.

Note, however, that phase separation may manifest it-self as in “frustrated phase separation” scenarios [15]:Since the Coulombic interaction between holes was notexplicitly included it is possible that in realistic situa-tions phase separation may be replaced by the forma-tion of complex structures, such as the stripes observedin cuprates [16, 17, 18]. Thus, it is reasonable to specu-

late that these stripes could also appear in the insulatingregime of the manganites and they should be detectableusing neutron scattering techniques.

On the theoretical side, future work will be directed tothe analysis of the influence of phonons and orbital de-generacy into the phase diagram observed in the presentpaper, specially regarding phase separation, as well as thecalculation of dynamical properties for the models investi-gated here. Work is in progress along these fronts to com-plete a qualitative understanding of the phase diagramcorresponding to models for the manganites beyond thedouble exchange model.

9 Acknowledgments

We thank K. Hallberg, J. Riera and S. Kivelson for use-ful conversations. E. D. and A. M. are supported bythe NSF grant DMR-9520776. S. Y. is supported by theJapanese Society for the Promotion of Science. J. H. issupported by the Florida State grant E&G 502401002. A.L. M. acknowledges the financial support of the ConselhoNacional de Desenvolvimento Cientıfico e Tecnologico(CNPq-Brazil)

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20