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Random phase approximation for multi-band Hubbard models

We derive the random-phase approximation for spin excitations in general multi-band Hubbard models, starting from a collinear ferromagnetic Hartree-Fock ground state. The results are compared with those of a recently introduced variational many-body approach to spin-waves in itinerant ferromagnets. As we exemplify for Hubbard models with one and two bands, the two approaches lead to qualitatively different results. The discrepancies can be traced back to the fact that the Hartree-Fock theory fails to describe properly the local moments which naturally arise in a correlated-electron theory.


I. INTRODUCTION
Metallic ferromagnetism is one of the oldest problems in solid-state physics. Despite all progress in solid-state theory over the last century 1-8 , we are still far away from a definite theory for transition metals like iron, cobalt and nickel. It is generally accepted that the magnetic order in metals is a consequence of the interplay between the electrons' kinetic energy and their mutual Coulomb interactions. Therefore, it does not come as a surprise that a complete theory for this many-particle effect is difficult to develop.
Nevertheless, it is commonly believed that at least the ground-state properties of itinerant magnets can be described by (effective) one-particle theories such as Hartree-Fock theory or (spin-)density-functional theory (SDFT). However, serious doubts are in order, as a closer look on available experimental data reveals. For example, SDFT fails to reproduce even qualitatively the electronic band-structure of nickel, one of the simplest itinerant ferromagnets 9 ; for a summary, see Ref. [10]. For iron, the agreement is better but cannot be considered as satisfactory either. In any case, the same physical mechanism is acting in both systems, and a convincing theory should cover all elements of the iron group equally well.
Single-particle theories are not very satisfactory also from a purely theoretical point of view: It is well known that they significantly overestimate the stability of ferromagnetism 1 . For example, Hartree-Fock theory for one-band Hubbard models gives a ferromagnetic ground state above some moderate interaction strength. In contrast, many-body theories show that ferromagnetism in one-band Hubbard models occurs, if at all, only for very large Coulomb interactions or very peculiar choices of the one-particle density of states 11,12 . This indicates that the Stoner mechanism is less prominent in itinerant ferromagnets than implied by single-particle theories.
As already argued by van Vleck 2 and confirmed recently [13][14][15][16] , an essential ingredient for metallic ferromagnetism is the local (Hund's-rule) exchange coupling of electrons, a generic feature of electrons in degenerate bands. In contrast to the one-band case, ferromagnetism occurs for moderate Coulomb-interactions in a generic two-band model provided that local Hund's-rule terms are included. Therefore, even a qualitative understanding of the 3delements requires a careful consideration of their atomic open-shell structures.
In our recent work 13,17 we have developed a variational approach which allows us to examine multi-band Hubbard models with an arbitrary number of correlated orbitals per lattice site. For a realistic description of the 3d-elements a minimal model includes the 3d, 4s, and 4p bands. When applied to nickel, our theory provides results which agree much better with experiments than all previous SDFT-calculation 10,18 . In particular, our approach reproduces the experimental Fermi-surface topology of nickel and removes all other qualitative SDFT deficiencies. Since our variational many-body approach provides a reasonable description of the ground state, it may serve as an appropriate starting point for the study of excitations in itinerant ferromagnets.
As observed in inelastic neutron scattering experiments 19,20 , the spectrum of itinerant ferromagnets displays low-lying spin-wave excitations. These excitations are generally considered to be the driving force for the transition from the ferromagnetic phase into the paramagnetic phase at the Curie temperature. Therefore, their correct description is also very important for our understanding of finite-temperature properties of itinerant ferromagnets. Unfortunately, almost all established theories which are used for the analysis of spin-waves in transition metals are based on effective single-particle theories which do not necessarily provide a satisfactory description of the ground state.
The textbook approach to the problem of itinerant ferromagnetism is a Hartree-Fock theory for the ground state which is then combined with a random-phase approximation (RPA) for the description of spin-excitations 5,7 . For example, in Ref. [21] the RPA-method has been applied to iron and nickel starting from a SDFT calculation for the ground state.
Later, a similar method has been used to analyze data from electron energy-loss spectroscopy for the same materials 22 . Another common approach starts from energy calculations for static spin deformations of the ferromagnetic ground state within the SDFT. The spin-wave energies are identified as the energy difference between these "frozen-magnon" states and the ground state 23 . Despite their conceptual shortcomings both SDFT-based approaches are able to reproduce the experimental data surprisingly well.
Very recently 24 one of us has proposed a spin-wave theory which starts from the variational ground state of multi-band Hubbard models as introduced in Ref. [13]. This theory allows us to determine the spin-wave spectrum for systems with an arbitrary number of correlated orbitals per lattice site. The main advantage of this new approach lies in its foundation on a true many-particle description of the ground state. In this way, we are able to provide a consistent picture of the ferromagnetic Fermi liquid and its spin-wave excitations.
In order to obtain a better understanding of our own but likewise of the SDFT-based methods it is crucial to apply them for a generic but simple model system. To our knowledge such an analysis has not been performed yet. Even a comparison between the two wellestablished SDFT methods is still lacking. In this work we analyze the Hartree-Fock/RPA theory and compare it with our variational many-body approach. Although the RPA is generally considered as the "standard" method for the description of magnetic excitations in itinerant ferromagnets [5] there exists no systematic derivation for multi-band Hubbard models. In Sect. II, we derive the general RPA equations from a decoupling scheme and a standard diagrammatic viewpoint.
In Sect. III we provide numerical results for a generic two-band model and show that both methods lead to qualitatively different results. For generic values of the magnetization, the Hartree-Fock/RPA treatment incorrectly predicts an instability of the collinear ferromagnetic phase against some spiral ordering. In our variational approach the spin-waves always Conclusions in Sect. IV close our presentation.

II. RANDOM PHASE APPROXIMATION FOR MULTI-BAND SYSTEMS
In this section we derive the random phase approximation (RPA) for ferromagnetism in multi-band Hubbard models. Starting from a LDA-calculation, RPA results for iron and nickel have been reported in Ref. [21]. However, in these calculations the relevant Coulomb matrix elements were significantly simplified. In this work we present a more general treatment.

A. General remarks
We will address the following general class of multi-band Hubbard models 13,25 Here,ĉ + i;s creates an electron with combined spin-orbit index s = (b, σ) at the lattice site R i of a solid (b = 1, . . . , N; σ =↑, ↓; N = 5 for 3d electrons). The atomic Hamiltonian is assumed to have site-independent interaction parameters U s 1 ,s 2 ;s 3 ,s 4 .
In order to determine the spin-wave properties, we need to consider the imaginary part χ T ( q, E) of the transverse susceptibility 19 , which is given by the retarded two-particle Green Here, we introduced the q-dependent spin-flip operatorŝ in the Heisenberg picture, where the sum runs over all L lattice sites R l and orbitals b.
Magnetic excitations are found as poles of the Green function G ret ( q, E), or, equivalently, as peaks in at energies E > 0. The Lehmann representation of (3), shows that the weights of the poles in G ret ( q, E) at the energies E n − E 0 > 0 are given by In a ferromagnetic system the state is also a ground state ofĤ for q = 0 because the operatorŜ − q= 0 just flips a spin in the spin multiplet of the ground state |Ψ 0 . Therefore, we can conclude that G T ( 0, E) has one isolated pole for E − E 0 = 0. This is not surprising since the spin-wave is the gapless Goldstone mode in the symmetry-broken ferromagnetic phase. For finite values of q, it is an experimental fact that there are also pronounced peaks in χ T ( q, E) at the spin-wave energies The constant D is usually denoted as the "spin-wave stiffness".

B. Hartree-Fock treatment
The one-particle HamiltonĤ 1 in (1) may be written in momentum space aŝ Here, we introduced the operatorŝ and the respective energy matrix-elements where the elements u α,b ( k, σ) of the unitary matrix u( k, σ) will be identified later. The transformation (11) leads to the following expression for the atomic Hamiltonian, where We choose the matrix u( k, σ) in order to fulfill the corresponding Hartree-Fock equations with Our Hartree-Fock theory is restricted to cases where the translational invariance is conserved but we still allow for a collinear ferromagnetic ground state. The Hartree-Fock Hamiltonian thus readsĤ In the special case where, (i), all orbitals b belong to different representations of the pointsymmetry group of the lattice, and, (ii), the Hartree-Fock ground state Ψ HF 0 is invariant under the respective symmetry operations, Eq. (16a) becomes Under these conditions, the results of Ref. [21] are recovered; see below.

C. RPA from the equation-of-motion technique
We will derive the Green function (3) in the random-phase approximation using the standard equation-of-motion technique 26 . First, we rewrite G ret ( q, E) as The equation of motion for we will consider the more general Green function which allows us to express (19b) as The equation of motion for G k, k ′ , q In RPA we assume that the exact ground state |Ψ 0 of our Hamiltonian (1) may be replaced by the (spin-polarized) Hartree-Fock ground state Ψ HF 0 of (17). Then, the expectation value (22b) becomes The evaluation of (22c) leads to Finally, the commutator in (22d) generates two-particle operators, which we decouple according to the rule This approximation appears to be natural because we replaced |Ψ 0 by the one-particle product state Ψ HF 0 . Note, however, the replacement (25) does not become exact even if we work with Ψ HF 0 instead of |Ψ 0 .
After the application of the decoupling scheme (25), Eq. (22d) may be written as When we use the explicit expression (14), the sum over p can be carried out for C 1 and C 2 because the Green functions do not depend on p.
With the help of the Hartree-Fock equations (15) and relation (21) we may cast the Green function (20) into the form This equation, together with (21), leads to where Here, we added the infinitesimal increment η = 0 + to ensure the properties of a retarded Green function 26 . When we consider Γ ret ( q, E) and U in (29) as matrices with respect to the indices D = (b 1 b 2 ), the solution of Eq. (21) is given by This result, together with Eq. (19a), gives us the Green function G ret ( q, E), whose poles at E RPA ( q ) define the RPA spin-wave dispersion. We will further analyze it in section III.

D. RPA from the diagrammatic approach
The result (31) of the RPA decoupling scheme may also be derived from the standard diagrammatic approach. Here, the matrix (30a) is nothing but the retarded part of the respective causal transversal susceptibility Γ, where the one-particle lines are evaluated within the Hartree-Fock approximation; see below. Using this expression and the vertex for spin-flip excitations U b 3 b 4 b 5 b 6 from (30b), we can calculate the causal two-particle Green function G D,b ′ T ( q, E) as the usual RPA sum of the "bare bubbles" in (32), This equation has the same solution as (31) if we replace Γ ret and G ret by Γ and G T , respectively. The retarded Green function G ret can then be re-derived from G T with the help of the standard relations 26 In order to show the equivalence of both approaches, it remains to calculate Γ in (32) explicitly. To this end we first determine the Hartree-Fock Green function G HF bb ′ ,σ ( k, E). Its diagrammatic evaluation leads to where G 0 bb ′ ,σ ( k, E) is the causal Green function corresponding to the one-particle operatorĤ 1 in (1), Moreover, we introduced the usual Hartree-Fock self-energy whose elements are calculated self-consistently, compare (16). Using a matrix notation in (35) we find where the matrix E σ ( k) is given by From Eqs. (39) and (40) we see that [G HF σ ( k, E)] −1 is diagonalized by the matrix u( k, σ) which solves the Hartree-Fock equations (15). The eigenvalues are given by E HF β ( k, σ). The diagonalization and inversion of (39) give the final result Again, we added a positive or negative increment η β ( k, σ) depending on whether E HF β ( k, σ) is larger or smaller than the Fermi energy. This ensures the analytical properties of a causal Green function. We insert (41) into (32), and find From this equation one easily re-derives (30a) with the help of the general relations (34). In this way, we have shown the complete equivalence of the diagrammatic and the equation-ofmotion derivation of the multi-band RPA equations.

A. Variational spin-wave dispersion
In Ref. [13] we proposed the following Gutzwiller wave-function 3 for a variational examination of the Hamiltonian (1): Here, |Φ 0 is any normalized single-particle product-state and the Gutzwiller correlatorP G allows for a variational adjustment of the occupation of local atomic multiplets. Our choice for the correlatorP G ensures that the wave-function (43) yields the exact ground state ofĤ both in the uncorrelated and the atomic limit. For all other values of correlation parameters the expectation value ofĤ in the wave-function can be determined analytically in the limit of large spatial dimensions, d → ∞. As in our recent work, this limit must be considered as yet another approximation since we apply our general analytical results to real threedimensional systems. Note that 1/d-corrections are expected to be small 10,27 .
Once the optimum variational ground state |Ψ opt G has been found by minimizing the ground-state energy-functional, the following expression for the variational spin-wave dispersion can be evaluated In Ref. [24] a general analytical expression for E var ( q ) has been found in the limit of large spatial dimensions. The numerical evaluation of this result for real materials like iron or nickel is involved but feasible. So far, we have applied our method only to a degenerate two-band model. The results for this model will be compared with the RPA in the next subsection.

B. Two-band model
In Ref. [24] we have discussed the variational spin-wave dispersion for a model with two degenerate e g orbitals (b = 1, 2) on a simple-cubic lattice. In this system the general atomic Hamiltonian (2) becomeŝ In cubic symmetry, the Coulomb and exchange-integrals U, U ′ , J and J C are not independent from each other. Instead we have only two free parameters, because the relations J = J C and U − U ′ = 2J hold.
In Ref. [13] we have discussed the appearance of ferromagnetic order in this model in detail both for the Hartree-Fock and our Gutzwiller theory. As is well known for mean-field theories, the stability of a ferromagnetic solution is drastically overestimated in a Hartree- The density of electrons per band and spin direction in the paramagnetic phase is given by In Fig. 1  ferromagnet (m ≈ n). As can be seen from Fig. 1, the spin-wave dispersion drastically depends on the magnetization. The data can be fitted very well to respectively. This is the right order of magnitude in nickel where D = 0.43 eVÅ 2 . The inset of Fig. 1 shows that the spin-wave dispersion is almost isotropic which also agrees with experimental observations. In our variational many-body approach, the spin-wave stiffness decreases as a function of U. This does not come as a surprise because the effective coupling between the sites decreases when the electron hopping between the sites becomes less effective.
In Fig. 2 the results for the spin-wave dispersion in the RPA are shown for three different values of U. In contrast to our variational approach, the RPA predicts a negative slope of the dispersion for non-saturated ferromagnetism, m < n. Note that for U = 5.2 eV the system has just attained the fully polarized state (compare Fig. 7 of Ref. [13]). But even in this case, a well-defined RPA spin-wave excitation does not yet exist. It requires a further increase of U until the fully polarized collinear Hartree-Fock ground state appears to be stable. Since the RPA probes the 'local' stability of a mean-field state it is conceivable that another ferromagnetic Hartree-Fock state with some spiral order has a lower energy than the collinear state. However, ferromagnetic phases with spiral order are not generic for transition metals. We argue that this prediction of exotic ferromagnetic spin structures is an artifact In some special cases where the fully polarized state is a one-particle product state we find for large couplings. This is also the case in the one-band system; see below.

C. One-band model
The study of the one-band model allows us to elucidate the origin of the qualitatively different results for the spin-wave dispersion. In the case of the one-band model, the results of Sect. II simplify significantly. The Hartree-Fock energies can be written as and the susceptibility (30a) becomes For the spin-wave energies E RPA ( q ) the denominator in vanishes, i.e., To the leading (second) order in q we can expand (50a) in powers of 1/∆ because is fulfilled for our low-lying spin-wave excitations, see (9). This expansion leads to This result is to be compared with our variational spin-wave dispersion. However, such a comparison is somewhat questionable, since the existence of a stable ferromagnetic phase is hard to obtain in our variational approach. Nevertheless, let us assume that a given oneband model with a special density of states may have a fully-polarized ferromagnetic ground state. Then we obtain the following expression for the variational spin-wave dispersion This expression differs from E RPA ( q ) in (54) by the second term, i.e., for n ↓ = 0 we find The energy difference δE is always negative and vanishes for U → ∞, i.e., for ∆ → ∞. It is essentially this contribution which eventually leads to negative values of the RPA spin-wave stiffness for finite U in the two-band model. These findings also explain the qualitatively different behavior of the spin-wave stiffness as a function of U. In the variational approach we find a stable ferromagnetic ground state with well-defined local moments. The interaction of these 'spins' is mediated by hopping processes. Similar to the well-known large-U antiferromagnetic coupling J AF ∝ t 2 /U in the case of the half-filled Hubbard model, we find that the effective spin-coupling J F in ferromagnets is reduced with increasing U because hopping processes are more and more suppressed as they induce charge fluctuations. Note that the coupling J F remains finite in the limit U → ∞ because we are dealing with a non-integer band-filling where electron transfers are never completely suppressed. In contrast to this generic behavior, the Hartree- We conclude that the Hartree-Fock/RPA theory provides an inadequate description for spin-wave properties in correlated electron systems. In our view, the main problem is the mean-field character of the underlying Hartree-Fock theory. In this approach the ferromagnetic ground state lacks well-developed local moments, and, therefore, high-energy longitudinal fluctuations of the local spins are large and relevant. As a consequence, RPA theory predicts an instability of the collinear ferromagnetic ground state against some spiral order.
Our many-body approach reveals quite a different picture. In the region of stable ferromagnetism we find well-defined local moments. Spin excitations in these systems are transversal fluctuations of the local moments whose length is essentially conserved in space and time.
Consequently, the stiffness of the variational spin-wave dispersion is always positive and decreases for increasing interaction strength.
The popular "frozen-magnon" approximation for the examination of spin-wave excitations in itinerant ferromagnets can be considered as a mean-field theory in which the conservation of the local spins is introduced by hand. In this way, the overestimation of longitudinal fluctuations in the Hartree-Fock/RPA approach is eliminated by construction. In fact, it can be shown that the frozen-magnon approximation would lead to the same results as our variational approach if we chose a Hartree-Fock variational wave-function instead of our Gutzwiller wave-function. A more detailed analysis of the frozen-magnon approach will be the subject of a forthcoming study 28 .

ACKNOWLEDGMENTS
We thank Werner Weber and David Logan for helpful discussions. Inset: E var ((q, 0, 0)) and E var ((q/ √ 2, q/ √ 2, 0)) for m = 0.2 and m = 0.28, respectively. The spin-wave dispersion is almost isotropic for strong ferromagnets. Also shown is the spin-wave dispersion for U = ∞. The lattice constant is a = 2.5Å.