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Microscopic conditions favoring itinerant ferromagnetism

J. Wahle,∗ N. Blumer, J. Schlipf, K. Held, and D. VollhardtTheoretische Physik III, Elektronische Korrelationen und Magnetismus, Universitat Augsburg, D-86135 Augsburg, Germany

(February 1, 2008)

A systematic investigation of the microscopic conditions stabilizing itinerant ferromagnetismof correlated electrons in a single-band model is presented. Quantitative results are obtained byquantum Monte Carlo simulations for a model with Hubbard interaction U and direct Heisenbergexchange interaction F within the dynamical mean-field theory. Special emphasis is placed on theinvestigation of (i) the distribution of spectral weight in the density of states, (ii) the importanceof genuine correlations, and (iii) the significance of the direct exchange, for the stability of itinerantferromagnetism at finite temperatures. We find that already a moderately strong peak in the densityof states near the band edge suffices to stabilize ferromagnetism at intermediate U -values in a broadrange of electron densities n. Correlation effects prove to be essential: Slater–Hartree-Fock resultsfor the transition temperature are both qualitatively and quantitatively incorrect. The nearest-neighbor Heisenberg exchange does not, in general, play a decisive role. Detailed results for themagnetic phase diagram as a function of U , F , n, temperature T , and the asymmetry of the densityof states are presented and discussed.

71.27.+a,75.10.Lp

I. INTRODUCTION

In contrast to conventional superconductivity and an-tiferromagnetism, metallic ferromagnetism is in generalan intermediate or strong coupling phenomenon. Sincethere do not exist systematic investigation schemes tosolve such types of problems the stability of metallic fer-romagnetism is still not sufficiently understood. Thisis true even within the simplest electronic correlationmodel, the one-band Hubbard model,1 in spite of signifi-cant progress made recently. The Hubbard interaction isvery unspecific, i.e., does not depend on the lattice struc-ture or dimension. Hence the dispersion, and therebythe shape of the density of states (DOS), is of consider-able importance for the stability of ferromagnetism. Thiswas recognized already by Gutzwiller,2 Hubbard,3 andKanamori4 in their initial work on the Hubbard model.However, the approximations used in the early days ofmany-body theory were usually not reliable enough toprovide definite conclusions. An exception are the exactresults by Nagaoka5 on the stability of ferromagnetismat U = ∞ in the case of one electron above or below halffilling. They show an important lattice sensitivity but,unfortunately, are not applicable in the thermodynamiclimit.

Over the years the stability of metallic ferromagnetismhas turned out to be a particularly difficult many-bodyproblem whose explanation requires subtle nonperturba-tive techniques. There has been an upsurge of inter-est in this topic most recently6–13,15,14. These investi-gations confirm that ferromagnetism is favored in sys-tems with (i) frustrated lattices (which suppress antifer-romagnetism) and (ii) high spectral weight near the bandedge closest to the Fermi energy (which improve the ki-netic energy of the polarized electrons). Taken together,these properties imply a strongly asymmetric DOS of theelectrons. Ferromagnetism on bipartite lattices having a

symmetric DOS may still be possible, but seems to re-quire very large values of U .16 With the exception ofRefs. 13,15 all previous calculations refer to the groundstate. It is therefore of interest to obtain an answer to thequestion: How does the distribution of spectral weight in

the DOS influence the stability regime of ferromagnetismat finite temperatures?

It should be noted that a strongly peaked, asymmet-ric DOS is a considerably more complex condition forferromagnetism than the Stoner criterion. The lattermerely asserts that, at T = 0, the critical interaction forthe instability is determined by the inverse of the DOSprecisely at the Fermi energy EF , Uc = 1/N(EF ), thusneglecting antiferromagnetism and the structure of theDOS away from EF . Stoner (i.e., Hartree-Fock17) theoryis a purely static mean-field theory which ignores correla-tion effects, e.g., the correlation-induced redistribution ofmomentum states and the dynamic renormalizations ofthe band shape and width. So the question remains: How

essential are genuine correlation effects for the stabilityof itinerant ferromagnetism at finite temperatures?

A third question concerns the suitability of the Hub-bard model itself as a model for ferromagnetism. Indeedthere is no compelling a priori reason why the Hubbardmodel should be a good model for ferromagnetism atall. Not only does it neglect band degeneracy, a featureobserved in all ferromagnetic transition metals (Fe, Co,Ni), it also ignores the (weak) direct Heisenberg exchangeinteraction which is equivalent to a ferromagnetic spin-spin interaction and hence favors ferromagnetism in themost obvious way.18–23 The proposition by Hirsch andcoworkers19,20,22 that this interaction plays a key rolein metallic ferromagnetism was disputed by Campbell etal.18 So the controversial question is:

How important is the direct Heisenberg exchange inter-

action for the stability of itinerant ferromagnetism in theone-band Hubbard model at finite temperatures?

1

In this paper quantitative answers to the three ques-tions formulated above are given within the dynamicalmean field theory (DMFT). The DMFT, a nonperturba-tive approach, becomes exact in the limit of large coordi-nation numbers.24–29 When applied to d = 3, where thecoordination number is O(10), the DMFT has proven toyield accurate and reliable results, especially in the con-text of long-range magnetic order.30,13 It treats local cor-relations exactly while spatial fluctuations are neglected.In this situation the momentum integral entering in thelocal propagator will be replaced by an energy integralinvolving only the DOS of the noninteracting electrons.The latter may be viewed as an input parameter. In ourinvestigation the question concerning the importance ofthe distribution of spectral weight within the band for thestability of ferromagnetism will therefore be studied us-ing a model DOS of the noninteracting electrons whoseshape can be changed continuously from symmetric tostrongly asymmetric by varying an asymmetry parame-ter.

The paper is structured as follows: in Sec. II we presentthe model under investigation, the dynamical mean-fieldequations, and the analytical and numerical steps neededto construct magnetic phase diagrams. The model DOSis introduced in Sec. III. The results of our investigationand quantitative answers to the questions posed aboveare presented in Sec. IV. A discussion where these resultsare put into perspective (Sec. V) closes the presentation.

II. MODEL AND METHODS OF SOLUTION

A. Hubbard model with nearest-neighbor exchange

The minimal model allowing one to treat an asymmet-ric DOS, electronic on-site correlations, and the nearest-neighbor Heisenberg exchange interaction is given by

H = HHub − 2F∑

〈i,j〉

Si · Sj , where (1)

HHub = −∑

i,j,σ

tij(c†iσ cjσ + h.c.) + U

∑

i

ni↓ni↑ (2)

Here Si = 12

∑

σσ′ c†iστσσ′ ciσ′ with the vector of Pauli

matrices τ .We note that there are three other nearest-neighbor

contributions of the Coulomb interaction which mightalso effect the stability of the ferromagnetic phase (seeappendix).

B. Dynamical mean-field theory

Within the DMFT the coupling constants in (1) and(2) have to be scaled with the lattice coordination num-

ber Z as24,25 t = t∗/√Z,F = F ∗/Z, where we consider

nearest-neighbor hopping t only. By analogy to classical

spin models33 the Hartree-Fock approximation yields theexact result for the F -term in high dimensions.

In the following we investigate the influence of thedirect exchange term on the properties of the Hub-bard model in d → ∞. Since the Hubbard model isSU(2) spin-symmetric we can, without loss of general-ity, assume a magnetization parallel to the z-axis. TheHartree34 decoupling then takes the form Si · Sj →〈Sz

i 〉Szj + Sz

i 〈Szj 〉 − 〈Sz

i 〉〈Szj 〉. In terms of the magnetiza-

tion m =∑

i mi/N and its expectation value m = 〈m〉,where mi = 2 Sz

i = ni↑ − ni↓ and N is the number oflattice sites, the Hamiltonian (1) can be written as35

H = HHub − NF ∗

2mm+

NF ∗

4m2. (3)

Apparently the influence of the exchange term in thelimit d → ∞ is that of a (Weiss) magnetic field whichvanishes in the paramagnetic phase (m = 0). Therefore,in this phase, all one-particle properties of the system arethose of the pure Hubbard model. However, two-particlefunctions, especially the ferromagnetic susceptibility, aremodified (see Sec. II C).

In d = ∞, the Hubbard model (2) is equivalent toan Anderson impurity model complemented by a self-consistency condition.36,29 Written in terms of Matsub-ara frequencies ωn, self energy Σσ, the DOS of the non-interacting electrons N0(ε), and thermal average 〈ψψ∗〉Athe resulting coupled equations for the Green function inthe homogeneous phase read:

Gσ(iωn) =

∞∫

−∞

dεN0(ε)

iωn + µ− Σσ(iωn) − ε(4)

Gσ(iωn) = −〈ψσnψ∗σn〉A. (5)

The solution of the k–integrated Dyson equation (4)is straightforward and can be performed analytically forthe DOS used in this paper (see Sec. IV). By contrastthe solution of Eq. (5) is highly nontrivial (for detailsof the notation see Ref. 37). It is achieved using theauxiliary-field quantum Monte Carlo (QMC) algorithmby Hirsch and Fye,38 where a discretization of imaginarytime, ∆τ = β/Λ, is introduced. Here, Λ denotes thenumber of independent Matsubara frequencies. Physicalquantities are obtained in the ∆τ → 0 limit.

C. Calculation of susceptibilities, extrapolation and

error handling

The second order phase transition from a para- to aferromagnetic phase occurs at the zero of the inversesusceptibility χ−1

f , calculated37,39 in the paramagneticphase. It is sufficient to perform all simulations for thepure Hubbard model since the influence of F ∗ on thesusceptibility is given by the following random-phase ap-proximation (RPA)-like expression:

2

χf (U,F ∗, . . .) =χf (U, 0, . . .)

1 − F∗

2 χf (U, 0, . . .), (6)

where χf (U, 0, . . .) is the susceptibility of the pure Hub-bard model. This type of relation holds for pairs of two-particle interactions (here U and F ∗) in arbitrary dimen-sions, whenever one interaction (here F ∗) is treated inHartree-Fock approximation and the other one (here U)exactly.40 In general, since the calculation of the suscep-tibility involves the derivative of the self-energy ΣU,F∗

with respect to some field h,37,39 this follows from thefact that the self-energy of the full Hamiltonian can beexpressed as

ΣU,F∗ [GU,F∗ ] = ΣU,0[GU,F∗ ] + Σ(1)0,F∗ [GU,F∗ ]. (7)

Here Σ[G] refers to the diagrammatic skeleton expan-sion of Σ, where all lines are fully dressed propagators

G. Since the Hartree-Fock term Σ(1)0,F∗ only contributes

in the symmetry-broken phase, all F ∗-renormalizationsof GU,F∗ vanish in the symmetric phase. Evaluat-ing dΣU,F∗/dh in the symmetric phase, the first termin (7) leads to the same contributions as without F ∗-interaction, while the second term introduces the RPA-like term proportional to F ∗. For the t-J model in DMFTthe analogue of (6) was derived by Pruschke et al.41

Since F ∗ only enters the calculations via Eq. (7) we areleft with four physical parameters of the pure Hubbardmodel: Hubbard interaction U , electron density n, tem-perature T and an asymmetry parameter a for the kineticenergy (see Sec. IV). For each set of these five parame-ters Eqs. (4) and (5) are iterated with typically 6 × 104

Monte Carlo sweeps until convergence is reached, i.e., thedifference between two consecutive values of (Gσ)−1 =(Gσ)−1− Σσ is smaller than 5 × 10−4 (measured by thenorm (2Λ)−1

∑

σn |(Gnewσn )−1− (Gold

σn )−1|; the energy scaleis defined in Sec. IV). Subsequently eight measurementsof the susceptibility are performed with a reduced num-ber of 2 × 104 Monte Carlo sweeps. Thus the result foreach parameter set consists of an averaged susceptibilityχf (∆τ) and its statistical error ∆χf (∆τ). We neglectthe propagation of the error in (Gσ)−1 since it is alwaysan order of magnitude smaller than ∆χf (∆τ). The ex-trapolation to ∆τ = 0 is performed by a quadratic leastsquares fit of χf (∆τ), using at least six different valuesof χf for ∆τ ∈ [0.09, 0.5]. Further details regarding thetechnical treatment can be found in Refs. 37,42.

For mean-field theories like the DMFT a linear behav-ior of the inverse susceptibility, i.e., a Curie-Weiss law, isexpected and observed in the vicinity of the transition.Thus the Curie temperature TC can be obtained as thezero of a linear fit of χ−1

f (T ) drawn from values of χf

for four to six different temperatures (see, e.g., Figs. 3and 5, where F ∗

c = 2χ−1f (6) is plotted). The error of TC

is obtained from the errors ∆χf (∆τ) by error propaga-tion and therefore denotes only statistical, not system-atic errors (e.g., due to the extrapolation schemes used).

However, we checked the accuracy of our results by vary-ing the procedure, e.g., extrapolating χ−1

f (∆τ) instead of

χf (∆τ).

III. MODEL SPECTRAL FUNCTION

Due to the vanishing of spatial fluctuations within theDMFT the topology of the underlying lattice enters theself-consistency equation (4) only via the noninteractingDOS, at least for homogeneous phases. The choice of aparticular model spectral function thus represents a spe-cial (not unique) set of hopping elements tij in the Hamil-tonian (1) which characterize the structure of the under-lying lattice. Contributions to the kinetic energy by, e.g.,next-nearest-neighbor hopping can lead to an asymmet-rically shaped DOS, which is apparently favorable forthe stability of ferromagnetism. To investigate this sta-bilizing effect quantitatively, we propose a model DOSwith a shape-controlling parameter. This parameter al-lows us to change smoothly from a noninteracting DOSwith (i) a symmetric shape (mimicking nearest-neighborhopping on a bipartite lattice) to (ii) an asymmetricallypeaked DOS (similar to a cubic lattice with next-nearest-neighbor hopping) to (iii) a DOS with a square-root di-vergence at the band edge (e.g., a fcc lattice with next-nearest-neighbor hopping t′ = t/2). The shape of themodel DOS thus qualitatively captures key features ofreal lattices.

The spectral function which we use throughout the pa-per is given by

N0(ε) = c

√D2 − ε2

D + aε(8)

with c = (1 +√

1 − a2)/(πD) and bandwidth 2D. Thewell-known semielliptic DOS of the Bethe lattice withinfinite number of nearest neighbors is recovered fora = 0. By increasing the parameter a spectral weightis shifted towards the lower band edge (Fig. 1). Fora = 1 the DOS diverges at the lower band edge like aninverse square-root. The particular choice of the modelDOS (8) has the advantage that the numerical effort ofsolving the self-consistency equation (4) is rather smallsince the Hilbert transform can be calculated analyti-cally. In the following we set the variance to unity,∫

dεN0(ε) ε2 − [∫

dεN0(ε) ε]2 = 1, thereby fixing the en-ergy scale. This leads to D = 2 for all values of a. Fora = 0 it is equivalent to choosing t∗ = 1 on the Bethelattice.

While for the study of ferromagnetism within theDMFT the lattice structure only enters via the DOS itis possible to construct (infinitely many) correspondingdispersion relations ε(k) or, equivalently, sets of hoppingelements tij . A realization in d = 1 that is symmetric,ε(k) = ε(−k), and monotonous, dε/dk > 0, for ε > 0 isgiven by

3

0

0.2

0.4

0.6

0.8

1

1.2

1.4

−2 −1 0 1 2

N0(ε)

ε

a=0a=0.5a=0.9a=0.98a=1

0

0.7

1.4

−2 −1.8 −1.6

N0(ε)

ε

a=0.97a=0.98a=0.99a=1

FIG. 1. Model spectral function (8) shown for different val-ues of the asymmetry parameter a. By increasing a spectralweight is shifted towards the lower band edge. The energyscale is fixed by setting the variance of the DOS equal to 1.

ε∫

εmin

dε′N0(ε′) =

k∫

0

dk′

π=k(ε)

π, (9)

where an inversion yields ε(k). Generalizations to otherdimensions are possible with, e.g., ε(k) = ε(|k|)43 or

ε(k) =∑d

i=1 ε(ki).44

Although in principle one could thus choose a latticecorresponding to the DOS (8), this will not be done here,since we only study homogeneous phases. Antiferromag-netism or incommensurate phases are not expected to beimportant far away from half filling. Only for the casea = 0 (Bethe lattice) at n = 1 the stability of an antifer-romagnetic phase is investigated.

IV. RESULTS

A. The importance of the direct exchange

interaction

On a bipartite lattice with perfect nesting, the groundstate of the pure Hubbard model at half filling is antifer-romagnetic for all U > 0, at least in dimensions d ≥ 3. Inthis situation a ferromagnetic state is strongly disfavoredalso in the general model, Eq. (1). At large U , how-ever, when the model reduces to an effective Heisenbergmodel (which, in high dimensions, is exactly describedby Weiss mean-field theory), already a small value ofthe direct exchange interaction, F ∗ > 2(t∗)2/U , is suf-ficient to stabilize a ferromagnetic ground state.45,46,21

Indeed, the Heisenberg model well describes the F ∗–Uground state phase diagram at half filling down to U ≈ 4;

this is evident from Fig. 2, where a comparison with ourQMC results is shown for a symmetric DOS [Eq. (8) witha = 0]. At small U the phase boundary between a param-agnetic and a ferromagnetic state is correctly reproducedby Hartree-Fock theory (but only for U < 1). This is notsurprising since in d = ∞ the F -term is treated exactlywithin this approximation. Also included in Fig. 2 is theline below which a fully saturated ferromagnetic statebecomes unstable against single spin flips as first com-puted by Hirsch47 for cubic lattices. For the Bethe lat-tice this line is given exactly by the Hartree-Fock resultF ∗ = 4 − U for U ≤ Uc = 3 and

F ∗ − U = − 8

(F ∗ + U)(

1 −√

1 − 16/(F ∗ + U)2) (10)

for U > 3. This can be seen from Eqs. 5 and 7 inRef. 47 and the known analytic expression for the Hilberttransform of the semielliptic DOS. The remarkable agree-ment between the QMC results and this curve for U ≥ 3suggests that (at zero temperature) the region of par-tial polarization is very narrow already at intermediateinteraction strength U .

The F ∗–T phase diagram for a strongly peaked DOS(a = 0.98) at filling n = 0.6 is shown in Fig. 3. TheQMC results for the ferromagnetic phase boundary canbe extrapolated linearly to zero temperature leading toa ground state phase diagram. Clearly the values of F ∗

necessary to stabilize ferromagnetism are significantly re-duced in comparison to the bipartite case. In particular,for U = 6 and U = 8 the extrapolation lines cross theordinate at positive temperatures. Thus, an asymmetricDOS stabilizes ferromagnetism even in the pure Hubbard

0

0.5

1

1.5

2

2.5

3

0 2 4 6 8 10 12

F*

U

F

P

AF

QMC (P−F)QMC (P−AF)Hartree−FockWeisssingle spin flip

FIG. 2. Phase diagram for a symmetric DOS (a = 0): di-rect exchange coupling F ∗ vs. Hubbard interaction U at halffilling (n = 1) extrapolated to T = 0. Open triangles andfilled circles correspond to the instability of the paramagneticphase (P) against the ferromagnetic (F) and antiferromag-netic (AF) order, respectively. Solid line: Hartree-Fock the-ory; dashed line: Weiss mean-field theory; dotted line: sinlespin flip instability for the saturated ferromagnetic state.

4

model (F ∗ = 0) above a critical interaction strength Uc

with 4 < Uc < 6.Figs. 3 and 4 prove that already a small direct ex-

change coupling F ∗ can significantly enhance ferromag-netic tendencies and thus give the final “kick” towardsferromagnetism for systems that are close to an instabil-ity. This influence is stronger at larger densities whenthe local magnetic moments are enhanced (Fig. 4). Thelower critical densities are very small, but larger thanthose predicted by Hartree-Fock theory, since Hartree-Fock always overestimates the size of the ferromagneticregime.

0

0.2

0.4

0.6

0.8

0 0.05 0.1 0.15 0.2

F*

T

F

P

U=4U=6U=8

FIG. 3. F ∗–T phase diagram for different values of U for astrongly peaked DOS (a = 0.98) at a filling n = 0.6. The lin-ear extrapolation shows that there exists a critical Uc abovewhich ferromagnetism is stable even without the direct ex-change coupling.

0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.2 0.4 0.6 0.8 1

T

n

F P

F*=0F*=0.05F*=0.1

FIG. 4. T–n phase diagram for different values of the directexchange coupling F ∗ for a strongly peaked DOS (a = 0.98)at U = 6. A small direct exchange coupling is seen to en-large the stability regime of the ferromagnet, especially fordensities close to half filling. The lines are guide to the eyeonly.

Quite generally, the value of the exchange interac-tion F in metals can be expected to be rather small.Note, however, that for three-dimensional lattices thescaled quantity F ∗ = ZF is an order of magnitude largerthan the exchange coupling F itself. Hubbard’s crudeestimate3 of F/U ≈ 1/400 would therefore imply that,e.g., at U = 6 a (scaled) exchange interaction as large asF ∗ = 0.15 is not completely unrealistic.

B. The importance of the asymmetry of the DOS

The dependence of the phase boundary on the asym-metry parameter a is systematically studied in Fig. 5 atU = 4 for a relatively small electron density n = 0.3. Fora symmetric or slightly asymmetric DOS (a = 0, a = 0.5)the system only becomes ferromagnetic for F ∗ > 1 evenat T = 0. For a = 0.9, when the shape of the DOSis roughly triangular, the critical F ∗ is considerably re-duced. But only when a marked peak develops (i.e., fora > 0.95) does the critical F ∗ drop to zero; ferromag-netism is then stable even in the pure Hubbard model.From now on we restrict our studies to this case (F ∗ = 0).

The T vs. n phase diagram is shown in Fig. 6 forU = 4 and three different shapes of the DOS rangingfrom strongly asymmetric (a = 0.97) to divergent at thelower band edge (a = 1). Evidently the ferromagneticphase is largest at a = 1. We want to stress however,that the divergence does not change the physics qualita-tively (except at n≪ 1). A moderately strong peak nearthe band edge is all that is needed to stabilize ferromag-netism.

0

0.5

1

1.5

2

2.5

3

3.5

0 0.1 0.2 0.3 0.4

F*

T

F

P

a=0a=0.5a=0.9a=0.95a=1.0

FIG. 5. F ∗–T phase diagram for different shapes of theDOS at U = 4 and n = 0.3. For values of 0.95 < a ≤ 1the ferromagnetic phase is stable even without the direct ex-change coupling. The lines show a quadratic least-squares fitin T .

5

0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.2 0.4 0.6 0.8 1

T

n

F

P

a=1a=0.98a=0.97

FIG. 6. T–n phase diagram for different shapes of the DOSat U = 4. By shifting spectral weight towards the lower bandedge, the region of stability of the ferromagnetic phase is en-larged. The lines are guide to the eye only.

For symmetric densities of states at U < 12 (Bethelattice and hypercubic lattice) we find suppression of theferromagnetic (as well as the antiferromagnetic) suscep-tibility away from half filling.48 This does not excludethe possibility for ferromagnetism on bipartite lattices atmuch larger values of U . Indeed, very recently ferromag-netism was found within the noncrossing approximationfor the t–J model on a hypercubic lattice in the limitd → ∞ for U > 30 away from half filling.16 Apparently,at least for moderate U , the bipartite lattice with onlynearest-neighbor-hopping is not a natural “environment”for ferromagnetism – the asymmetry of the noninteract-ing DOS is crucial.

C. The importance of correlations

In Fig. 7 the T –n phase diagram is shown for differentvalues of the on-site interaction U . Evidently the fer-romagnetic phase becomes more favorable for increasingU : both the maximal Curie temperature and the (upper)critical density rise. This effect is seen to be qualitativelysimilar to an increase of the exchange interaction F ∗ orthe asymmetry of the DOS a (Figs. 4 and 6, respectively).

Our QMC results are compared with Hartree-Fock the-ory in Figs. 8–10. We note that, applied to the Hubbardmodel, the DMFT includes Hartree-Fock theory as itsstatic limit and is thus superior in any dimension. Fig. 8shows the vast overestimation of the ferromagnetic phasewithin Hartree-Fock. The maximal Curie temperatureobtained in this approximation is more than an order ofmagnitude too large. At such high temperatures detailsof the DOS are averaged out and consequently the densitydependence (e.g., the position of the maximum) is com-pletely artificial. In Fig. 9 the U vs. n ground state phasediagram is shown and compared to the Stoner criterion.17

At low n the Stoner curve clearly approaches the QMCcurve. Since the DOS vanishes smoothly at the lower

band edge for a < 1 both curves diverge for n→ 0. Fig-ure 10 focuses on the limit of large U . The weak couplingHartree-Fock theory fails again: it predicts an unboundedlinear increase of TC with U , TC ∼ Un(2−n)/4, whereasQMC shows that TC has a finite limit for U → ∞. It isexpected that such a finite limit exists for all densities.A saturation is also suggested by the curves in Fig. 7. Itarises from the suppression of double occupancies by cor-relations. In contrast to the Hartree-Fock prediction theinteraction energy goes to zero for U → ∞, thus only thebandwidth remains as an energy scale. In the special casea = 0.98, n = 0.4 one finds49 TC(U = ∞) = 0.07 ± 0.02.

0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.2 0.4 0.6 0.8 1

T

n

F

P

U=8U=6U=4

FIG. 7. T–n phase diagram for a strongly peaked DOS(a = 0.98) for different values of U . With increasing Hubbardinteraction the stability regime of the ferromagnetic phasebecomes larger, especially at higher densities. The lines areguide to the eye only.

0

0.2

0.4

0.6

0.8

0 0.2 0.4 0.6 0.8 1 1.2 1.4

T

n

F

P

Hartree−FockQMC

FIG. 8. T–n phase diagram for a strongly peaked DOS(a = 0.98) at U = 4. The comparison between Hartree-Focktheory (solid line) and DMFT (QMC, circles) reveals the im-portance of correlation effects. The dashed line is guide tothe eye only.

6

0

2

4

6

8

0 0.2 0.4 0.6 0.8

U

n

F

P

Hartree−FockQMC

FIG. 9. U–n phase diagram at T = 0 for a strongly peakedDOS (a = 0.98). The DMFT data (QMC, circles) is ex-trapolated from finite temperature calculations. The Stonercriterion (solid line) underestimates the critical Uc(n) for fer-romagnetism, but becomes better at lower densities. Thedashed line is guide to the eye only.

0

0.05

0.1

0.15

0 2 4 6 8 10

T

U

P

F

Hartree−FockQMC

0

0.5

1

1.5

0 5 10

T

U

FIG. 10. T–U phase diagram for a strongly peaked DOS(a = 0.98) at n = 0.4. The comparison between Hartree-Focktheory (solid line) and DMFT (QMC, circles) shows the for-mer can describe the Curie temperature TC(U) neither quan-titatively, nor qualitatively. The dashed line is guide to theeye only.

One might argue that comparison of the DMFT re-sults should not be made with Hartree-Fock itself butwith Hartree-Fock plus quantum corrections, since thelatter are known to reduce many of the deficiencies ofHartree-Fock. Such corrections have been discussed byvan Dongen50 and Freericks and Jarrell.51 The latterauthors showed how quantum fluctuations modify theStoner criterion by subtracting the particle-particle sus-ceptibility. Evaluating these corrections in the case of

Fig. 10 we find that the ferromagnetic phase is com-pletely suppressed (as previously observed in Ref. 51 fora symmetric DOS). At a = 0.98 this holds for all densi-ties n >∼ 0.3. Thus the “corrections” to Hartree-Fock areseen to underestimate the ferromagnetic region by far.

V. DISCUSSION AND OUTLOOK

After more than three decades of research it has be-come clear at last11–13 that the Hubbard model can de-scribe itinerant ferromagnetism even on regular latticesand at moderate U -values for a wide range of electronicdensities n. Since ferromagnetism is an intermediateto strong coupling problem the question concerning its“mechanism” has, in principle, no straightforward an-swer. This is in contrast to weak coupling phenomena,e.g., conventional superconductivity, which can be ex-plained within perturbation theory. Nevertheless a goodstarting point for an understanding of the origin of itin-erant ferromagnetism can be obtained in the strong cou-pling limit. At U = ∞ doubly occupied sites are ex-cluded and the Hubbard model reduces to a (compli-cated) kinetic energy. To avoid doubly occupied sitesin a paramagnetic phase the DOS is then necessarilystrongly renormalized compared with the noninteractingcase, whereas for the saturated ferromagnetic phase theinteracting DOS is the same as the noninteracting oneexcept for a shift between the spin-up and -down bands.In this situation details of the structure of the noninter-acting DOS become relevant in selecting the state withthe lowest energy. This physical picture is, in principle,similar to that underlying the Nagaoka mechanism; how-ever, the latter only addresses the kinetic energy of asingle hole and it was so far not possible to generalize itto thermodynamically relevant densities. Our investiga-tions within the DMFT explicitly show that a moderatelystrong peak at the band edge closest to the Fermi energyis sufficient to stabilize ferromagnetism. Furthermore astrong asymmetry of the DOS implies a nonbipartite lat-tice which frustrates the competing antiferromagnetismnear half filling.

The mechanism described above is completely differ-ent from the mere band shift of the Hartree-Fock the-ory. This weak coupling approach does not take into ac-count the dynamical renormalization of the DOS in theparamagnetic phase and thus predicts ferromagnetism forany DOS, even at relatively small values of U and forhigh temperatures. The comparison with DMFT clearlyshows that Hartree-Fock (i) overestimates transition tem-peratures by more than an order of magnitude, (ii) ren-ders the dependence of TC on U qualitatively incorrect,and (iii) predicts ferromagnetism for the symmetric DOS,where (at least for U < 12) this is not found. Theseshortcomings of Hartree-Fock theory are due to the ne-glect of dynamical fluctuations which are at the heart ofthe correlation problem.

7

The Heisenberg exchange interaction, not consideredin the pure Hubbard model, provides another mechanismthat may order the fluctuating local moments arising bythe suppression of double occupancies. We found thatfor a symmetric DOS rather large values of F are neededto stabilize ferromagnetism. However, for an asymmet-ric DOS with a peak near the band edge already smallvalues of the exchange interaction may provide the final“kick” towards ferromagnetism. In any case it reducesthe critical on-site interaction and increases the criticaltemperatures of the ferromagnetic phase boundary.

While the DMFT correctly describes the dynamic fluc-tuations of the interacting many-body system, it neglectsspatial fluctuations and short-range order. Hence oneshould suspect that this approach overestimates the tran-sition temperatures TC . Within DMFT Ulmke13 esti-mated TC for a three-dimensional fcc lattice to be of theorder of 500 to 800 K which is in the range of realistictransition temperatures. We may expect spatial fluctua-tions to reduce these temperatures. On the other hand,band degeneracy, not considered in our model so far, isexpected to increase TC . Indeed, band degeneracy andHund’s rule couplings which are clearly present in re-alistic systems can be rigorously shown to improve thestability of ferromagnetism at least for special parame-ter values.52,14 The incorporation of band degeneracy, forwhich the DMFT also provides a suitable framework, isthe most important feature that has to be included infuture investigations of the Hubbard model.53 The addi-tional nearest-neighbor interactions discussed in the ap-pendix may provide yet another mechanism for ferromag-netism and will be studied in the future.

Acknowledgments

We are very grateful to M. Kollar and M. Ulmke for nu-merous helpful discussions. One of us (N.B.) thanks theFulbright Commission for support and the Department ofPhysics at the University of Illinois for hospitality. Com-putations were performed on the Cray T90 of the HLRZJulich.

APPENDIX: NEAREST-NEIGHBOR

INTERACTIONS

In Wannier representation the Coulomb interactiongives rise to the purely local interaction U as well asto four nearest-neighbor interactions.18–20,14 Besides theHeisenberg exchange interaction these are the density-density interaction, the pair-hopping term, and the off-diagonal “bond-charge–site-charge”18 interaction. Thelatter effectively describes a density-dependent hopping20

and leads to a narrowing of the band. Hence this termis expected to stabilize saturated ferromagnetism. Sincethe quantum dynamics of this term make a systematic

investigation difficult – even within the DMFT – its de-tailed study has to be postponed to the future. Thepair-hopping term also weakly enhances ferromagnetictendencies.20,32

Among all nearest-neighbor interactions the density-density term

HVNN = V

∑

〈i,j〉

ninj, (A1)

is largest and is thus investigated explicitly in the fol-lowing. In the case of d-electrons Hubbard roughly es-timated this term to V=2-3 eV, an order of magnitudesmaller than the Hubbard interaction U .3 However, sincethere are Z neighbors contributing, the total energy ofthe nearest-neighbor density-density interactions may insome materials even surpass that of the Hubbard inter-action. This raises the question of the importance of theV -term, in particular its influence on the ferromagneticphases investigated in the present paper.

It was already pointed out by Muller-Hartmann25 thatin the limit d → ∞ and with the proper scaling V =V ∗/Z the nearest-neighbor density-density interactionreduces to its Hartree contribution which may then beviewed as a simple, site-dependent shift of the chemicalpotential. In the absence of broken translational symme-try the chemical potential must compensate this shift tokeep the electron density fixed. Then there is no effectat all.

On bipartite lattices translational symmetry can bebroken by a charge density wave (CDW) with differentelectron densities on A and B sublattices, i.e., with orderparameter nCDW = (nA −nB)/2. To study this possibleordering we analyze the instability towards a CDW inthe following.

Similar to the exchange term F the Hartree contri-bution of the interaction V leads, even in the presenceof other interactions, to an RPA-like pole in the CDWsusceptibility [cf. Eq. (6)]:40

χCDW (U, V ∗, . . .) =χCDW (U, 0, . . .)

1 − V ∗ χCDW (U, 0, . . .).

Thus a second order phase transition to a CDW occursat V ∗

c = 1/χCDW (U, 0, . . .).Since next-nearest-neighbor hopping frustrates CDW

order, the maximal instability towards a CDW is ex-pected for the symmetric DOS with a = 0 in Eq. (8).Half filling is optimal in this case. For these parameterswe determined the phase diagram Fig. 11 employing theQMC technique (for details concerning the calculationof the CDW susceptibility see Ref. 54). Within DMFTa CDW ordering occurs for V ∗ >∼ U (at not too hightemperatures). Compared to the Hartree-Fock approx-imation the CDW phase boundary of the full model isonly slightly moved towards larger values of V ∗. A sim-ilar deviation from Hartree-Fock was found by means ofQMC simulations in d = 1 by Hirsch,55 in d = 2 by Zhangand Callaway,56 and within perturbation theory for both

8

0

1

2

3

4

0 1 2 3 4

V*

U

AF

CDW

P

Hartree−FockQMC

FIG. 11. V ∗–U phase diagram for the semielliptic DOS(a = 0) at T = 0.125. For V ∗ > U a regime with charge den-sity wave order (CDW) is established. The antiferromagneticphase vanishes for U > 7.7 ± 0.5 (not shown). The dashedline is guide to the eye only.

weak and strong coupling (in arbitrary dimensions) byvan Dongen.50,57

All these studies demonstrate that the CDW is stablefor V ∗ = V Z > U . While this relation may in principlehold for some transition metals, ferromagnets apparentlydo not show spatial charge ordering. Therefore the ad-equate correlated electron model for a ferromagnet ap-pears to have parameters in the range V ∗ ≤ U . Thenthe nearest-neighbor density-density interaction V ∗ hasno influence on the phase diagram, especially on the bor-der of the ferromagnetic phase, at least in d = ∞. Evenin d = 3 the Hartree diagram gives the main contributionof the interaction V ∗ since spatial fluctuations, leading togenuine correlations, are suppressed as 1/Z. Moreover ind = 1 and at half-filling the effect of V ∗ on the ferromag-netic phase boundary is still small.22 Therefore, over anextended range of parameters the nearest-neighbor termV ∗ has almost no influence and thus its importance isseen to be much smaller than its value suggests.

∗ Present address: Theoretische Physik, Universitat Duis-burg, Lotharstr. 1, D-47048 Duisburg, Germany.

1 It is known, however, that in the one-dimensional Hubbardmodel with only nearest-neighbor hopping ferromagnetismis unstable [E. Lieb and D. C. Mattis, Phys. Rev. 125, 164(1962)].

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17 For ferromagnetism in the Hubbard model, Hartree-Focktheory is equivalent to Stoner theory.

18 D. K. Campbell, J. T. Gammel, and E. Y. Loh Jr., Phys.Rev. B 38, 12043 (1988).

19 J. E. Hirsch, Phys. Rev. B 40, 2354 (1989).20 J. E. Hirsch, Phys. Rev. B 40, 9061 (1989); S. Tang and

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9

32 J. E. Hirsch, Phys. Rev. B 43, 705 (1991).33 See, e.g., C. Itzykson and J. M. Drouffe, Statistical Field

Theory, Cambridge University Press, Cambridge, 1989.34 The Fock contribution of the F term vanishes as 1/Z since

it is given by two Green functions connecting neighboringsites. In d = 3 the Fock term renormalizes the band-width:D(E(ǫ)) = D(ǫ)/{1 − F ∗/(Zt∗2)

∫

dǫD(ǫ)(−2ǫ)f [E(ǫ)]}(see Eq. (29) in Ref. 19, which readily yields this equationfor the paramagnetic phase). The renormalization is max-imal at half-filling and reduces the critical value of F ∗ forthe onset of ferromagnetism, e.g., for the half-filled cubiclattice from F ∗ = 1.17t to F ∗ = 0.84t within the Hartree-Fock approximation (F ∗(D(ǫF ) = 1). This effect of order1/Z is neglected in the DMFT.

35 The definition of generalized (e.g., staggered) susceptibili-ties for bipartite (AB-) lattices requires the introduction ofan index α ∈ {A, B} in Eq. (3) that indicates the sublat-tice. The corresponding generalizations are straightforward(see, e.g., Ref. 39).

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39 K. Held, M. Ulmke, N. Blumer, and D. Vollhardt, Phys.Rev. B 56, 14469 (1997).

40 J. Schlipf and M. Kollar, unpublished.

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42 M. Ulmke, Phasenubergange in stark korrelierten Elektro-

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49 The error mainly reflects an uncertainty in the extrap-olation law. It was determined by comparing linear andquadratic least-squares fits in 1/U .

50 P. G. J. van Dongen, Phys. Rev. Lett. 67, 757 (1991).51 J. K. Freericks and M. Jarrell, Phys. Rev. Lett 74, 186

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14016 (1994).

10