Multiband Gutzwiller wave functions for itinerant ferromagnetism^{1}^{1}1Dedicated to Martin C. Gutzwiller on the occasion of his 75th birthday.
Abstract
Multiband Gutzwillercorrelated wave functions reconcile the contrasting concepts of itinerant band electrons versus electrons localized in partially filled atomic shells. The approximate evaluation of these variational ground states becomes exact in the limit of large coordination number. The result allows the identification of quasiparticle band structures for correlated electron systems. As a first application, we summarize a study of itinerant ferromagnetism in a twoband model, thereby elucidating the cooperation of the Coulomb repulsion and the Hund’srule exchange. Then, we present results of calculations for ferromagnetic nickel, using a realistic 18 spinorbital basis of , and valence electrons. Good agreement with the experimental groundstate properties of nickel is obtained. In particular, the quasiparticle energy bands agree much better with the photoemission and Fermi surface data than the band structure obtained from spindensity functional theory. Finally, we present results for the variational spinwave dispersion for our twoband model.
pacs:
PACS numbers: 75.10.Lp, 75.50Cc, 71.10.Fd, 71.20BeI Introduction
How well do we understand the itinerant ferromagnetism of iron, nickel, and other transition metals and their compounds?
More than 50 years ago two basically different scenarios had emerged from early quantummechanical considerations on electrons in metals with partly filled bands. In scenario I, suggested by Slater [1] and Stoner [2], band theory alone was proposed to account for itinerant ferromagnetism: due to the Pauli principle, electrons with parallel spins cannot come arbitrarily close to each other (“Pauli” or “exchange hole”), and, thus, a ferromagnetic alignment of the electron spins reduces the total Coulomb energy with respect to the paramagnetic situation (“exchange energy”). This scenario, equivalent to HartreeFock theory, was criticized by van Vleck [3]. In an independentelectron theory electrons are distributed statistically over the lattice which implies strong charge fluctuations on the transition metal atoms.
Consequently, as a scenario II, van Vleck [3] emphasized the importance of electron correlations: due to the strong electronelectron interactions, charge fluctuations in the atomic shells are strongly suppressed (“minimum polarity model”), and, thus, atomic rather than band aspects are decisive for itinerant ferromagnetism. In scenario II, the atomic magnetic moments arise due to the local Coulomb interaction (in particular, Hund’srule couplings), and they may align because of the electrons’ motion through the crystal.
In principle, such a dispute can be resolved in natural sciences. The corresponding theories have to be worked out in detail, and their results and predictions have to be compared to experiments. This was indeed done for scenario I [4]. The (spin)density functional theory is a refined band theory which describes simple metals with considerable success. Unfortunately, progress for scenario II was much slower. It calls for a theory of correlated electrons, i.e., a genuine manybody problem has to be solved. It was only recently that reliable theoretical tools became available which allow to elucidate scenario II in more detail [5, 6, 7, 8, 9].
A first step in this direction was the formulation of appropriate model Hamiltonians which allowed to discuss matters concisely. In 1963/1964 Gutzwiller [10, 11], Hubbard [12], and Kanamori [13] independently introduced the Hubbard model which since then has become the standard model for correlated electrons on a lattice. This model covers both aspects of electrons on a lattice: they can move through the crystal, and they strongly interact when they sit on the same lattice site. The model is discussed in more detail in Sec. II.
Even nowadays, it is impossible to calculate exact groundstate properties of such a model in three dimensions. In 1963/1964 Gutzwiller introduced a trial state to examine variationally the possibility of ferromagnetism in such a model [10, 11]. His wave function covers both limits of weak and strong correlations and should, therefore, be suitable to provide qualitative insights into the magnetic phase diagram of the Hubbard model. Gutzwillercorrelated wave functions for multiband Hubbard models are defined and analyzed in Sec. III.
As envisaged by van Vleck [3], Gutzwiller [10] found that ferromagnetism is nongeneric in the oneband Hubbard model, in contrast to the predictions of the Stoner theory. The reason is simple: the Gutzwiller wave function not only describes the exchange hole between electrons of the same spin species but also the “correlation hole” between electrons of different spin species which forms when the electronelectron interaction is taken into account consistently. Therefore, the energy difference between the correlated paramagnetic and ferromagnetic states is very small, and paramagnetism wins in almost all oneband cases. In the multiband situation [11], itinerant ferromagnetism is favored by two concomitant effects: (i) the suppression of atomic charge fluctuations such that, (ii), the local exchange interactions give rise to a local magnetic moment as in isolated atoms (Hund’s rule). Hence, as proposed by van Vleck, small charge fluctuations and local exchange interactions generate large local moments which are present in both the paramagnetic and ferromagnetic phases. At low temperatures the electrons’ motion through the crystal may then lead to longrange order of these preformed local moments. In this way, Gutzwiller’s approach unified and substantiated van Vleck’s early ideas on a correlatedelectron theory of itinerant ferromagnetism.
Unfortunately, the evaluation of multiband Gutzwiller wave functions itself poses a most difficult manybody problem. Perturbative treatments [14, 15] are constrained to small to moderate interaction strengths. The region of strong correlations could only be addressed within the “Gutzwiller approximation” [10, 11, 16] and its various extensions [17, 18]. Thus, an application to ferromagnetism of real materials was not in sight, and Gutzwiller’s qualitative findings were not appreciated much in the sixties.
About 25 years later, a major technical advancement opened a new way to analyze Gutzwillercorrelated wave functions. The oneband Gutzwiller wave function was evaluated exactly in one dimension [19, 20], and the Gutzwiller approximation was found to become exact for the oneband Gutzwiller wave function in the limit of infinite spatial dimensions, [19, 21, 22]. Based on a diagrammatical approach, Gebhard [23] developed a compact formalism which allows to calculate the variational groundstate energy in infinite dimensions without the evaluation of a single diagram. He found that corrections are quantitatively small for the Gutzwiller wave function, i.e., the result in infinite dimensions is a reliable approximation to three dimensions.
Recently, Gebhard’s approach was generalized by us to the case of multiband Gutzwiller wave functions [9]. Thereby, earlier results by Bünemann and Weber [24], based on a generic extension of the Gutzwiller approximation [25], were found to become exact in infinite dimensions [26]. Half a century after van Vleck’s call for new manyparticle methods and more than thirty years after Gutzwiller’s first paper on the subject, techniques are at hand to carry out the program outlined above as scenario II.
Our results on a twoband toy model in Sect. IV confirm Gutzwiller’s insights on the fundamental requirements for itinerant ferromagnetism. Substantial atomic Coulomb interactions suppress local charge fluctuations and thereby allow the generation of local moments due to the atomic Hund’srule exchange couplings. Our quantitative results from a fullscale calculation for nickel in Sect. V confirm these qualitative findings. The quasiparticle properties of this material (e.g., exchange splittings, density of states, topology of the Fermi surface) are reproduced quantitatively from our Gutzwillercorrelated wave functions. In particular, strong electron correlations are required to describe nickel appropriately.
In Sect. VI, we briefly discuss a new method to calculate spinwave spectra of correlated electron systems based on the Gutzwiller variational scheme [27]. In this way, the dispersion relation of the fundamental lowenergy excitations can be derived consistently. Albeit the description is based on itinerant electrons, the results for strong ferromagnetism in a generic twoband model resemble those of a Heisenberg model for localized spins. Again, the Gutzwiller approach provides a comprehensive scheme for a unified description of localized and itinerant aspects of transition metals.
In this brief review we skip all technical details and refer the reader to the original literature for further information.
Ii Hamilton operator
Our multiband Hubbard model [12] is defined by the Hamiltonian
(1) 
Here, creates an electron with combined spinorbit index ( for 3 electrons) at the lattice site of a solid.
The most general case is treated in Ref. [9]. In this work we assume for simplicity that different types of orbitals belong to different representations of the point group of the respective atomic state (e.g., , , , ). In this case, different types of orbitals do not mix locally, and, thus, the crystal field is of the from . Consequently, we use normalized singleparticle product states which respect the symmetry of the lattice, i.e.,
(2) 
We further assume that the local interaction is siteindependent
(3) 
This term represents all possible local Coulomb interactions.
As our basis for the atomic problem we choose the configuration states
(4)  
which are the “Slater determinants” in atomic physics. The diagonalization of the Hamiltonian is a standard exercise [28]. The eigenstates obey
(5) 
where are the elements of the unitary matrix which diagonalizes the atomic Hamiltonian matrix with entries . Then,
(6)  
(7) 
The atomic properties, i.e., eigenenergies , eigenstates , and matrix elements , are essential ingredients of our solidstate theory.
Iii Multiband Gutzwiller wave functions
Gutzwillercorrelated wave functions are written as a manyparticle correlator acting on a normalized singleparticle product state ,
(8) 
The singleparticle wave function contains many configurations which are energetically unfavorable with respect to the atomic interactions. Hence, the correlator is chosen to suppress the weight of these configurations to minimize the total energy in (1). In the limit of strong correlations the Gutzwiller correlator should project onto atomic eigenstates. Therefore, the proper multiband Gutzwiller wave function with atomic correlations reads
The variational parameters per site are real, positive numbers. For and all other all atomic configurations at site but are removed from . Therefore, by construction, covers both limits of weak and strong coupling. In this way it incorporates both itinerant and local aspects of correlated electrons in narrowband systems.
The class of Gutzwillercorrelated wave functions as specified in (III) was evaluated exactly in the limit of infinite dimensions in Ref. [9]. The expectation value of the Hamiltonian (1) reads
Here, is the local particle density in . The local factors are given by [9]
(11)  
where () is the probability to find the configuration (the atomic eigenstate ) on site in the singleparticle product state . The fermionic sign function
(12) 
gives a minus (plus) sign if it takes an odd (even) number of anticommutations to shift the operator to its proper place in the sequence of electron creation operators in .
Eqs. (III) and (11) show that we may replace the original variational parameters by their physical counterparts, the atomic occupancies . They are related by the simple equation [9]
(13) 
The probability for an empty site () is obtained from the completeness condition,
(14) 
The probabilities for a singly occupied site () are given by
(15a)  
(15b) 
The parameters and must not be varied independently. All quantities in (III) are now expressed in terms of the atomic multiparticle occupancies (), the local densities , and further variational parameters in .
It is seen that the variational groundstate energy can be cast into the form of the expectation value of an effective singleparticle Hamiltonian with renormalized electron transfer amplitudes ,
(16) 
Therefore, is the ground state of whose parameters have to be determined selfconsistently from the minimization of with respect to and . For the optimum set of parameters, defines a band structure for correlated electrons. Similar to densityfunctional theory, this interpretation of our groundstate results opens the way to detailed comparisons with experimental results.
Applications are discussed in the next two sections.
Iv Results for a generic twoband model
The atomic Hamiltonian for a twoband model () can be cast into the form
For two orbitals, exhausts all possible twobody interaction terms.
We assume that the model describes two degenerate orbitals which leads to the following restrictions enforced by the cubic symmetry [28]: (i) , and (ii) . Therefore, there are two independent Coulomb parameters, the local Coulomb repulsion and the local exchange coupling . Since are of the same order of magnitude the relation (ii) shows that is of the typical strength of the atomic Hund’s rule couplings.
For the oneparticle part we use an orthogonal tightbinding Hamiltonian with first and second nearest neighbor hopping matrix elements. We apply the twocenter approximation for the hopping matrix elements and exclude any spinflip hopping. The resulting band structure (width ) is discussed in detail in Ref. [9].
In the following we concentrate on two bandfillings , the number of electrons per lattice site. Alternatively, we use , the particle density per band and per spin direction. For , the noninteracting density of states has a maximum which is most favorable for ferromagnetism according to the Stoner (i.e, HartreeFock) theory. Moreover, we study , where the densityofstates has a positive curvature.
In Fig. 1 we display the  phase diagram for both fillings. It shows that Hartree–Fock theory always predicts a ferromagnetic instability. In contrast, the correlatedelectron approach strongly supports the ideas of van Vleck [3] and Gutzwiller [11]: (i) a substantial onsite exchange is required for the occurrence of ferromagnetism if, (ii), realistic Coulomb repulsions are assumed. In 1964 Gutzwiller wrote “…one may hope to show some day that the terms (ii) can never induce ferromagnetism at all by themselves …”. Fig. 1 strongly supports Gutzwiller’s ideas on the importance of the local Hund’srule couplings. At the same time the comparison of Figs. 1a and 1b shows the importance of bandstructure effects which are the basis of the Stoner theory. The ferromagnetic phase in the  phase diagram is much bigger when the density of states at the Fermi energy is large. Therefore, the Stoner mechanism for ferromagnetism is well taken into account by Gutzwiller’s correlatedelectron approach.
In Fig. 2, we display the energy differences between the paramagnetic and ferromagnetic ground states (“condensation energy”, ) as a function of the interaction strength for . This quantity should be of the order of the Curie temperature which is in the range of in real materials. The HartreeFock–Stoner theory yields such small condensation energies only in the range of eV; for larger , is of order . In any case, the interaction parameter has to be tuned very precisely to give condensation energies which concur with experimental Curie temperatures [1]. In contrast, for the Gutzwillercorrelated wave function, we find relatively small condensation energies even for interaction values as large as twice the bandwidth (eV). Moreover, the dependence of the condensation energy on is rather weak such that uncertainties in do not drastically influence the estimates for the Curie temperature.
These toymodel studies show that the Gutzwiller variational approach contains all the fundamental elements for a successful theory of itinerant ferromagnetism. In the next section we show how to quantify this statement for ferromagnetic nickel.
V Correlated bandstructure of nickel
v.1 Discrepancies between experiment and SDFT
Of all the iron group magnetic metals, nickel is the most celebrated case of discrepancies between the results from experiment and from SDFT. From very early on, the photoemission data have indicated that the width of the occupied part of the bands is approximately [29] whereas all SDFT results yield values of or larger [4, 29]. Similarly, the low temperature specific heat data [30] give a much larger value of , the quasiparticle density of states at the Fermi energy ( versus states/(eV atom)), which indicates a quasiparticle mass enhancement by a factor of approximately . Here, the Sommerfeld formula is used to convert the specific heat data; the theoretical value follows directly from the quasiparticle band structure. Furthermore, very detailed photoemission studies at symmetry points and along symmetry lines of the Brillouin zone show discrepancies to SDFT results for individual bandstate energies which are of similar magnitude as seen in the overall bandwidth.
The studies also revealed even bigger discrepancies in the exchange splittings of majority spin and minority spin bands. The SDFT results give a rather isotropic exchange splitting of about [4, 29, 31]. In contrast, the photoemission data show small and highly anisotropic exchange splittings between for pure states such as and for pure states, the latter value estimated from the exchange splitting of states along to [32, 33]. The much larger and much too isotropic exchange splitting of the SDFT results has further consequences.

the state of the minority spin bands lies below [35], whereas all SDFT results predict it to lie above the Fermi level [4, 36, 37]. As a consequence, the SDFT Fermi surface exhibits two hole ellipsoids around the point of the Brillouin zone while in the deHaas–vanAlphen experiments only one ellipsoid has been found [36, 38].
In the late 70’s and early 80’s various authors have investigated in how far manybody effects improve the agreement between theory and experiment, see, e.g., Refs. [40, 41]. For example, Cooke et al. [40] introduced an anisotropic exchange splitting as a fit parameter.
v.2 Parameterization and minimization
For the singleparticle Hamiltonian in (1) we use a nineorbital , , basis. The hopping matrix elements are determined from a leastsquare fit to energy bands obtained from a densityfunctionaltheory calculation for nonmagnetic nickel. The rootmeansquare deviation of the band energies is about . Details of this calculation will be published elsewhere [42].
The atomic Hamiltonian in (3) is restricted to only orbitals. As a consequence, there are three independent interaction parameters (we use the sphericalatom approximation), which can be expressed either as Slater integrals () or as Racah parameters , , [28]. We neglect any cubic crystalfield corrections to the interaction parameters and, furthermore, put , as suggested by the results of ligandfield theory [28]. We find that and are the choice which best reproduces the experimental results; see below.
The number of multielectron states is ; because of the cubic site symmetry, the number of independent (internal) variational parameters reduces to approximately 200 for paramagnetic and to approximately 400 for ferromagnetic solutions. For the magnetic solutions, there exist a certain number of (external) variational parameters in the oneparticle wave function . In particular, the influence the spinorbital densities (). Among the are the “exchange splittings” of the individual orbitals and , i.e., these parameters determine the degree of magnetization in . Most important are the exchange splittings for and orbitals, of minor importance are those for and orbitals. Furthermore, the vs. crystalfield splitting has to be included in the set of . Technically, each , which is a functional of the external variational parameters , defines a band structure and, consequently, a Fermi surface. Therefore, calculating the particle densities and the expectation values implies momentumspace integrations up to the respective Fermi surface. It should be noted that the introduction of the variational parameters causes a charge flow between the , and and states which has to be compensated by using appropriate “chemical potentials”.
The optimum for a given set of , is the ground state of the effective singleparticle Hamiltonian (16) with renormalized hopping matrix elements which, through the factors, are functions of the internal variational parameters . Thus, can be obtained from a selfconsistent procedure, starting with , i.e., with . After minimization of the internal variational parameters, the resulting values for define a new effective Hamiltonian from which a new is constructed. In this step, the Fermi surface integrations have to be repeated. Selfconsistency is usually reached rather quickly, i.e., is found after three to five iterations.
The global minimum, is found by a search through the dimensional space of the external variational parameters . This search can be sped up by first optimizing with respect to the most important external variational parameter which is the isotropic exchange splitting . In a second step, the anisotropy of the exchange splitting is investigated, i.e., we introduce and , keeping the average close to the value of . The searches for and for and can be carried out starting with . Only then the selfconsistency procedure for has to be launched.
Typical energy gains are (in meV):
(18a)  
(18b)  
(18c) 
The energy gains from the variations of and are of the order of .
At present, our description does not contain the spinorbit coupling which could be relevant in some transition metals. This interaction contributes to the orbital moment to the total magnetic moment. The inclusion of the spinorbit coupling poses no principle difficulties [9] but it considerably enhances the numerical complexity of the problem. Moreover, we may want to extend our basis set beyond the nine bands , , and to allow for a further relaxation of the oneparticle wave functions. Again, this is feasible but numerically much more costly. Lastly, one may want to include some atomic  and  interactions. All these extensions are left for future studies.
v.3 Comparison to experiments
Typical results of our calculations for Ni are summarized in table 1. They are obtained from the multiband Gutzwiller method utilizing the nineorbital tightbinding model based on DFT energy band calculations for nonmagnetic Ni, and employing values of and , with .
The width of the bands is predominantly determined by (essentially the Hubbard), via the values of the hopping reduction factors . The exchange splittings and, consequently, the magnetic moment are strongly influenced by and only moderately by . The Racah parameter causes the Hund’srule splitting of the multiplets; in the hole picture, is the only manyparticle configuration which is significantly occupied (by electrons), while electrons are in , electrons are in , and electrons have or character.
Generally, the Gutzwiller results agree much better with experiment than the SDFT results. This is the case for, (i), the bandwidth , (ii), the value for , (iii), the positions of individual quasiparticle energies, (iv), the values of the exchange splittings, (v), their  anisotropy, and, (vi), the ratio of the part of the magnetic moment. As a consequence of the small exchange splitting, the state lies below and, thus, the Fermi surface exhibits only one hole ellipsoid around , in nice agreement with experiment.
Exp.  Ref.  this work  SDFT  
0.55  [34]  0.53  0.59  
4.3  [34]  5.1  2.8  
3.3 0.2  [29]  3.14  4.86  
3.5 0.3  [29]  3.29  5.20  
3.0  [30]  2.6  1.9  
2.8 0.2  [29]  2.94  4.06  
0.24 0.02  [43, 35]  0.22  0.56  
0.06 0.02  [43, 35]  0.04  0.09  
0.10 0.02  [43, 44]  0.14  0.22  
0.23 0.04  [43, 44]  0.25  0.34  
0.4 0.1  [29]  0.64  1.02  
1.1 0.2  [29]  1.45  2.14  
1.3 0.1  [29]  1.49  2.23  
0.21 0.02  [33]  0.30  0.60 
The large anisotropy of the exchange splittings is a result of our groundstate energy optimization, which allows and to be independent variational parameters. We find . Note that these values enter and are renormalized by factors , , when is reached. This also implies that the width of the majority spin bands is about 10% bigger than that of the (higher lying) minority spin bands. It causes a further reduction of the exchange splittings of states near , especially for those with strong character. Note that this band dispersion effect causes larger exchange splittings near the bottom of the bands, e.g., splitting of and splitting of . There, however, the quasiparticle linewidths have increased to and , respectively [29], so that an exchange splitting near the bottom of the bands could, so far, not be observed experimentally. When we investigate the quasiparticle energies in the middle of the bands, such as , , or , we observe that our theoretical values lie consistently below the experimental ones. If we assume that these states can be measured more reliably than those near the bottom of the band, we may conclude that the value of could still be somewhat bigger than . Studies with larger values are presently carried out.
We think that the large anisotropy originates from peculiarities special to Ni with its almost completely filled bands and its fcc lattice structure. Near the top of the bands, the states dominate because they exhibit the biggest hopping integrals to nearest neighbors, . The states have to nearest neighbors, and to nextnearest neighbors; the latter are small because of the large lattice distance to second neighbors. The states also mix with the nearestneighbor states with type coupling. Therefore, the system can gain more band energy by avoiding occupation of antibonding states in the minority spin bands via large values of , at the expense of allowing occupation of less antibonding states via small values. Note that this scenario is special to nickel and should not apply to materials with a bcc lattice structure which have almost equal nearest and nextnearest neighbor separations. Since the bands in nickel are almost completely filled, the suppression of charge fluctuations actually reduces the number of atomic configurations where the Hund’srule coupling is active. It is also in this respect that nickel does not quite reflect the generic situation of other transition metals with less completely filled bands; see Sect. IV.
The agreement between theory and experiments on nickel shows that our multiband Gutzwiller theory is a useful tool for the theoretical description of the groundstate properties of transition metals and their compounds. In the next section we describe a way to calculate consistently lowenergy excitations in itinerant ferromagnets within our variational approach.
Vi Spinwaves in itinerant ferromagnets
As in magnetic insulators, the elementary excitations in itinerant ferromagnets are spinwaves. Since groundstate properties are well described by Gutzwillercorrelated wave functions it is highly desirable to derive spinwave dispersions from the same variational approach.
In fact, the variational principle can also be used to calculate excited states [45]. If is the ferromagnetic, exact ground state with energy , the trial states
(19) 
are necessarily orthogonal to , and provide an exact upper bound to the first excited state with momentum and energy
(20) 
Here, flips a spin from up to down in the system whereby it changes the total momentum of the system by . In this way, the famous BijlFeynman formula for the phononroton dispersion in superfluid Helium was derived [46]. In the case of ferromagnetism the excitation energies can be identified with the spinwave dispersion if a welldefined spinwave exists at all [27]. Experimentally this criterion is fulfilled for small momenta and energies .
Unfortunately, we do not know the exact ground state or its energy in general. However, we may hope that the Gutzwiller wave function is a good approximation to the true ground state. Then, the states
(21) 
will provide a reliable estimate for ,
(22) 
Naturally, does not obey any strict upperbound principles.
The Hamiltonian (1) commutes with the operator for the total spin. Thus, for , generates a state from the same spin multiplet. Therefore, the spinwave at zero momentum is the Goldstone mode of the ferromagnetic system, . Within the variational approach is guaranteed if we choose as an eigenstate of the total spin operator. This is most easily accomplished by a minor restriction of the variational space for the atomic occupancies [27] with negligible influence on the variational groundstate properties.
The actual calculation of the variational spinwave dispersion is rather involved even in the limit of large dimensions. We note, however, that explicit formulae are available [27] which can directly be applied once the variational parameters have been determined from the minimization of the variational groundstate energy. As a first application, we present results for the twoband toymodel of Sect. IV.
In Fig. 3 we show the variational spinwave dispersion in direction, , for the model parameters , , and the four different values which correspond to a magnetization per band of . This quantity is defined as . Note that our last case corresponds to an almost complete ferromagnetic polarization. The data fit very well the formula
(23) 
in qualitative agreement with experiments on nickel [47]. The corresponding values and for and , respectively, are of the right order of magnitude for nickel where . As lattice constant of our simplecubic lattice we chose .
As shown in the inset of Fig. 3, the dispersion relation is almost isotropic for values up to half the Brillouin zone boundary [27], in particular for large magnetizations. This is in contrast to the strong dependence of the electrontransfer amplitudes on the lattice direction. This implies for strong ferromagnets that the collective motion of the local moments is similar to that of localized spins in an insulator [48]. Such ferromagnetic insulators are conveniently described by the Heisenberg model with exchangeinteraction between neighboring sites on a cubic lattice,
(24) 
For such a model one finds . The length of the effective local spins can be calculated from as () for [9]. Therefore, , which gives the typical value . For an estimate of the Curie temperature we use the result from quantum MonteCarlo calculations [49]
(25) 
for spins on a simplecubic lattice. In this way we find . This is the same order of magnitude as the condensation energy for these values of the interaction, , see Sect. IV. Given the arbitrariness in the relation between and , and the application of the Heisenberg model to our itinerantelectron system, we may certainly allow for difference of a factor two in these quantities. Nevertheless, the results of this section clearly show that, (i), gives the right order of magnitude for , and that, (ii), the spinwave dispersion of strong itinerant ferromagnets resemble the physics of localized spins.
Vii Conclusions
Which scenario for itinerant ferromagnetism in transition metals is the correct one?
For a long time this questions could not be answered conclusively. Band theory along the lines of Slater and Stoner could be worked out in much detail whereas a correlatedelectron description of narrowband systems was lacking until recently. In fact, van Vleck stated in the summary to his conference contribution in 1953: “The gist of this paper is that it would be highly desirable if good methods of computing with (c) (minimum polarity model) could be developed.” In 1963, Gutzwiller laid the foundations for a concise treatment of the problem, and today we are able to draw definite conclusions based on Gutzwillercorrelated multiband wave functions.
Our results for nickel indicate that the vanVleck–Gutzwiller scenario is valid. Band theory does not appreciate the strong electronic correlations present in the material which lead to the observed renormalization of the effective mass, exchange splittings, bandwidths, and Fermi surface topology. Moreover, as also covered by our approach, charge fluctuations are small, and large local moments are present both in the paramagnetic and the ferromagnetic phases. Roughly we may say that the electrons’ motion through the crystal leads to a ferromagnetic coupling of preformed moments which eventually order at low enough temperatures. In this way, strong itinerant ferromagnets resemble ferromagnetic insulators as far as their lowenergy properties are concerned: spinwaves exist which destroy the magnetic longrange order at the Curie temperature.
It is thus seen that today computational difficulties no longer obscure “the recognition in principle of the situation which confirms closest to physical reality”(van Vleck, 1953 [3]): transition metals and their compounds are strongly correlated electron systems. More than thirtyfive years after their introduction by Gutzwiller [10], our studies clearly show that Gutzwillercorrelated multiband wave functions successfully describe the lowenergy physics of these materials.
Acknowledgments
We gratefully acknowledge helpful discussions with P. van Dongen, H. Eschrig, and D. Vollhardt. This project is supported in part by the Deutsche Forschungsgemeinschaft under WE 1412/81.
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