Microscopic Theory of Multipole Ordering in f-Electron Systems

A microscopic framework to determine multipole ordering in f -electron systems is provided on the basis of the standard quantum field theory. For the construction of the framework, a seven-orbital Hubbard Hamiltonian with strong spin-orbit coupling is adopted as a prototype model. A type of multipole and ordering vector is determined from the divergence of multipole susceptibility, which is evaluated in a random phase approximation. As an example of the application of the present framework, a multipole phase diagram on a three-dimensional simple cubic lattice is discussed for the case of n = 2, where n denotes the average f -electron number per site. Finally, future problems concerning multipole ordering and fluctuations are briefly discussed.


Introduction
Recently, complex magnetism in rare-earth and actinide compounds has attracted much attention in the research field of condensed matter physics [1][2][3].Since in general, spin-orbit coupling between electrons in 4 f and 5 f orbitals is strong, spin and orbital degrees of freedom are tightly coupled in f -electron materials.Thus, when we attempt to discuss magnetic ordering in f -electron systems, it is necessary to consider the ordering of spin-orbital complex degrees of freedom, that is, multipole.In fact, ordering of higher-rank multipole has been actively investigated both from experimental and theoretical sides in the research field of strongly correlated f -electron systems [2,3].Moreover, due to recent remarkable developments in experimental techniques and measurements, nowadays it has been possible to detect directly and/or indirectly the multipole ordering.Note, however, that only spin degree of freedom often remains, when orbital degeneracy is lifted, for instance, due to the effect of crystal structure with low symmetry.In order to promote the research of multipole phenomena, felectron compounds crystallizing in the cubic structure with high symmetry are quite important.For instance, octupole ordering has been discussed in the phase IV of Ce 0.7 La 0.3 B 6 [4] and NpO 2 [3,[5][6][7][8] with cubic structure.As for NpO 2 , recently, a possibility of dotriacontapole ordering has been also pointed out [9,10].
Here we emphasize that the study of multipole phenomena has been activated due to the focusing research of filled skutterudite compounds LnT 4 X 12 with lanthanide Ln, transition metal atom T, and pnictogen X [11].Since these compounds crystallize in the cubic structure of T h point group, they have provided us an ideal stage for the research of multipole physics.Furthermore, many isostructural materials with different kinds of rare-earth and actinide ions have been successfully synthesized, leading to the development of systematic research on multipole ordering.In fact, recent experiments in close cooperation with phenomenological theory have revealed that multipole ordering frequently appears in filled skutterudites.For instance, a rich phase diagram of PrOs 4 Sb 12 with field-induced quadrupole order has been unveiled experimentally and theoretically [12][13][14].Furthermore, antiferro-Γ 1 -type higher multipole order [2] has been discussed for PrRu 4 P 12 [15,16] and PrFe 4 P 12 [17][18][19].
Now we turn our attention to theoretical research on multipole order.Thus far, theory of multipole ordering has been developed mainly from a phenomenological viewpoint on the basis of an LS coupling scheme for multi-f -electron state.It is true that several experimental results have been explained by those theoretical studies, but we believe that it is also important to promote microscopic approach for understanding of multipole phenomena in parallel with 2 Physics Research International phenomenological research.Based on this belief, the present author has developed a microscopic theory for multipolerelated phenomena with the use of a j-j coupling scheme [1,[20][21][22].In particular, octupole ordering in NpO 2 has been clarified by the evaluation of multipole interaction with the use of the standard perturbation method in terms of electron hopping [6][7][8]23].We have also discussed possible multipole states of filled skutterudites by analyzing multipole susceptibility of a multiorbital Anderson model based on the j-j coupling scheme [24][25][26][27][28][29].
On the other hand, it is still difficult to understand intuitively the physical meaning of multipole degree of freedom due to the mathematically complicated form of multipole operator defined by using total angular momentum.As mentioned above, multipole is considered to be spin-orbital complex degree of freedom.In this sense, it seems to be natural to regard multipole as anisotropic spincharge density.This point has been emphasized in the visualization of multipole order [6][7][8]23].Then, we have defined multipole as spin-charge density in the form of onebody operator from the viewpoint of multipole expansion of electromagnetic potential from charge distribution in electromagnetism [30,31].Due to the definition of multipole in the form of one-electron spin-charge density operator, it has been possible to discuss unambiguously multipole state by evaluating multipole susceptibility even for heavy rare-earth compounds with large total angular momentum [30].
As for the determination of the multipole state, we have proposed to use the optimization of multipole susceptibility on the basis of the standard linear response theory.We have analyzed an impurity Anderson model including seven f orbitals with the use of the numerical renormalization group technique and checked the effectiveness of the microscopic model on the basis of the j-j coupling scheme for the description of multipoles.We have also shown the result for multipole susceptibility of several kinds of filled skutterudite compounds.With the use of the seven-orbital Anderson model, we have discussed field-induced multipole phenomena in Sm-based filled skutterudites, [32] multipole Kondo effect, [33] and multipole state of Yb-and Tm-based filled skutterudites [34].We have also discussed possible multipole state in transuranium systems such as AmO 2 [35] and magnetic behavior of CmO 2 [36].
From our previous investigations on the basis of the multiorbital Anderson model, it has been clarified that the multipole can be treated as spin-orbital complex degree of freedom in the one-electron operator form.However, in order to discuss the ordering of multipole, it is necessary to consider a periodic system including seven f orbitals per atomic site with strong spin-orbit coupling.The validity of the model on the basis of the j-j coupling scheme can be also checked by such consideration.Namely, for the steady promotion of multipole physics, it is highly expected to treat the multipole ordering in a seven-orbital periodic model by overcoming a heavy task to solve the model including 14 states per atomic site.
In this paper, we define a seven-orbital Hubbard model with strong spin-orbit coupling and explain a procedure to define the multipole ordering by the divergence of multipole susceptibility from a microscopic viewpoint.For the evaluation of multipole susceptibility, we introduce a random phase approximation.In principle, we can treat all the cases for n = 1 ∼ 13 on the same footing, but here we focus on the case of n = 2 corresponding to Pr and U compounds.As a typical example of the present procedure, we show a phase diagram including quadrupole ordering in a threedimensional simple cubic lattice.Finally, we also discuss some future problems such as superconductivity induced by multipole fluctuations near the multipole phase.
The organization of this paper is as follows.In Section 2, we explain each part of the seven-orbital Hubbard model with strong spin-orbit coupling.For the reference of readers, we show the list of hopping integrals among f -orbitals along x, y, and z-axes through σ, π, δ, and φ bonds.In Section 3, we define the multipole operator as the complex spincharge degree of freedom in the one-electron form.Then, we explain a scheme to determine the multipole ordering from the multipole susceptibility.Here we use a random phase approximation for the evaluation of the multipole susceptibility.In Section 4, we show the results for the case of n = 2 in a three-dimensional simple cubic lattice.We discuss the phase diagram of the multipole ordering.In Section 5, we discuss some future problems and summarize this paper.Throughout this paper, we use such units as

Model Hamiltonian
The model Hamiltonian H is split into two parts as where H kin denotes a kinetic term and H loc is a local part for potential and interaction.The latter term is further given by where H so is a spin-orbit coupling term, H CEF indicates crystalline electric field (CEF) potential term, and H C denotes Coulomb interaction term.We explain each term in the following.

Local f -Electron
Term.Among the three terms of H loc , the spin-orbit coupling part is given by where f imσ is an annihilation operator of f -electron at site i, σ = +1 (−1) for up (down) spin, m is the z-component of angular momentum = 3, λ is the spin-orbit interaction, and the matrix elements are expressed by and zero for other cases.
Next we consider the CEF term, which is expressed as where B m,m is the CEF potential for f electrons from the ligand ions, which is determined from the table of Hutchings for angular momentum = 3 [37].For the cubic structure with O h symmetry, B m,m is expressed by using three CEF parameters, B 40 and B 60 , as Note the relation of B m,m = B m ,m .Following the traditional notation [38], we define 6) , where W determines an energy scale for the CEF potential, x specifies the CEF scheme for O h point group, and F(4) = 15 and F(6) = 180 for = 3.Finally, the Coulomb interaction term H C is given by where the Coulomb integral I m1m2,m3m4 is expressed by Here F k is the Slater-Condon parameter and c k is the Gaunt coefficient which is tabulated in the standard textbooks of quantum mechanics [39].Note that the sum is limited by the Wigner-Eckart theorem to k = 0, 2, 4, and 6.The Slater-Condon parameters should be determined for the material from the experimental results, but in this paper, for a purely theoretical purpose, we set the ratio among the Slater-Condon parameters as physically reasonable values, given by Note that F 6 is considered to indicate the scale of Hund's rule interaction among f orbitals.

Kinetic Term.
Next we consider the kinetic term of f electrons.When we discuss magnetic properties of felectron materials as well as the formation of heavy quasiparticles, it is necessary to include simultaneously both conduction electrons with wide bandwidth and f electrons with narrow bandwidth, since the hybridization is essentially important for the formation of heavy quasiparticles.In this sense, it is more realistic to construct orbital-degenerate periodic Anderson model for the theory of multipole ordering in heavy-electron systems.However, if we set the starting point of the discussion in the periodic Anderson model, the calculations for multipole susceptibility will be very complicated.Thus, we determine our mind to split the problem into two steps: namely, first we treat the formation of heavy quasiparticles and then, we discuss the effective model for such heavy quasiparticles.If we correctly include the symmetry of f -electron orbital, we believe that it is possible to grasp qualitatively correct points concerning the multipole ordering by using an effective kinetic term for f electrons.
Based on the above belief, we consider the effective kinetic term in a tight-binding approximation for f electrons.Then, H kin is expressed as where t a m,m indicates the f -electron hopping between mand m -orbitals of adjacent atoms along the a direction.The hopping amplitudes are obtained from the table of Slater-Koster integrals, [40][41][42] but, for convenience, here we show explicitly t a m,m on the three-dimensional cubic lattice.The hopping integrals along the z-axis are given in quite simple forms as and zeros for other cases.Here ( f f ) denotes the Slater-Koster integral through bond between nearest neighbor sites.Note that the above equations are closely related to the definitions of ( f f σ), ( f f π), ( f f δ), and ( f f φ).
On the other hand, hopping integrals along the xand y-axes are given by the linear combination of ( where the coefficient E a m,m indicates the two-center integral along a direction between m and m orbitals and runs among σ, π, δ, and φ.In Table 1, we show the values of E a m,m .Other components are zeros unless they are obtained with the use of relation of E a m,m = E a m ,m = E a −m,−m .By using the experimental results concerning the Fermisurface sheets for actual materials, it is possible to determine the Slater-Koster parameters, ( f f σ), ( f f π), ( f f δ), and ( f f φ), so as to reproduce the experimental results.Namely, the hopping integrals should be effective ones for quasiparticles, as mentioned above.Here it is important to include correctly the symmetry of local f orbitals in the evaluation of hopping amplitudes, although the whole energy scale will be adjusted by experimental results and band-structure calculations.
Table 1: Coefficients E a m,m along the x-and y-axes between f orbitals of nearest neighbor sites.Note that in double signs, the upper and lower signs correspond to the value along the x-and yaxes, respectively.

Multipole Ordering
In order to discuss the multipole ordered phase from the itinerant side, we evaluate the multipole susceptibility χ by following the standard quantum field theory.The multipole susceptibility is defined by where X q denotes the multipole operator with momentum q, ν = 2πTn is the boson Matsubara frequency with an integer n, T is a temperature, X q (τ) = e Hτ X q e −Hτ , and • • • indicates the thermal average by using H.In the following, we introduce the multipole operator and explain a method to evaluate the susceptibility.

Multipole Operator.
In any case, first it is necessary to define multipole.As for the definition of multipole, readers should consult with [30,31], but here we briefly explain the definition in order to make this paper self-contained.We define X in the one-electron density-operator form as where k denotes the rank of multipole, γ indicates the irreducible representation for cubic point group, and T (k) γ (q) indicates the cubic tensor operator, expressed in the secondquantized form as Here the matrix elements of the coefficient T (k,γ) are calculated from the Wigner-Eckert theorem as [43] T where = 3, s = 1/2, j = ± s, μ runs between − j and j, q runs between −k and k, G (k) γ,q is the transformation matrix between spherical and cubic harmonics, JM|J M J M denotes the Clebsch-Gordan coefficient, and j T (k) j is the reduced matrix element for spherical tensor operator, given by Note that k ≤ 2 j and the highest rank is 2 j.When we define multipoles as tensor operators in the space of total angular momentum J on the basis of the LS coupling scheme, there appear multipoles with k ≥ 8 for the cases of J ≥ 4, that is, for 2 ≤ n ≤ 4 and 8 ≤ n ≤ 12, where n is local f -electron number.If we need such higher-rank multipoles with k ≥ 8, it is necessary to consider many-body operators beyond the present one-body definition.
Note that when we express the multipole moment as ( 16) and ( 17), we normalize each multipole operator so as to satisfy the orthonormal condition [44] Tr where δ kk denotes the Kronecker's delta.

Multipole
Susceptibility.Now we move to the evaluation of multipole susceptibility.In order to determine the coefficient p k,γ (q) in ( 15), it is necessary to calculate the multipole susceptibility in the linear response theory.The multipole susceptibility is expressed as where the susceptibility matrix is given by Then, χ and p k,γ are determined by the maximum eigenvalue and the corresponding normalized eigenstate of the susceptibility matrix equation (21).
In order to calculate actually the multipole susceptibility, it is necessary to introduce an appropriate approximation.In this paper, we use a random phase approximation (RPA) for the evaluation of multipole susceptibility.For the purpose, we redivide the Hamiltonian H into two parts as where H 0 indicates the one-electron part given by H 0 = H kin + H so + H CEF and H 1 is the interaction part, which is just equal to H C in the present case.Then, we consider the perturbation expansion in terms of the Coulomb interaction.The susceptibility diagrams are shown in Figure 1 and they are expressed in a compact matrix form as where U and J are, respectively, given by U m1σ1m2σ2,m3σ3m4σ4 = I m1m2,m3m4 δ σ1σ4 δ σ2σ3 , and the dynamical susceptibility χ (0) is given by Here G (0) is the one-electron Green's function defined by the noninteracting part H 0 .
In order to determine the multipole ordering, it is necessary to detect the divergence of χ at ν n = 0. We cannot evaluate the susceptibility just at a diverging point, but we find such a critical point by the extrapolation of 1/χ max as a function of U, where U indicates the energy scale of the Slater-Condon parameters and χ max denotes the maximum eigenvalue of susceptibility matrix equation (21) for ν n = 0.When we increase the magnitude of U, 1/χ max is gradually decreased from the value in the weak-coupling limit.In actual calculations, we terminate the calculation when 1/χ max arrives at a value in the order of unity.By using the calculated values of 1/χ max , we make an extrapolation of 1/χ max as a function of U.Then, we find a critical value of U at which 1/χ max becomes zero.As for the type of multipole and ordering vector in the ordered phase, we extract such information from the eignevectors of the susceptibility matrix corresponding to the maximum eigenvalue.By performing the above calculations, it is possible to find the multipole ordered phase from a microscopic viewpoint in principle.

Results
In the previous sections, we have explained the model Hamiltonian and the procedure to determine the type of multipole ordering.We believe that the present procedure can be applied to actual materials, but there are so many kinds of materials and multipole phenomena.Here we show the calculated results for the case of n = 2 concerning Γ 3 non-Kramers quadrupole ordering, in order to see how the present procedure works.The results for actual materials will be discussed elsewhere.

CEF States.
First we discuss the local CEF states in order to determine the CEF parameter.We consider the case of n = 2 corresponding to Pr 3+ and U 4+ ions.Since we discuss the local electron state, the energy unit is taken as F 6 .As for the spin-orbit coupling, here we take λ/F 6 = 0.1.Concerning the value of W, it should be smaller than λ and we set W as W/F 6 = 0.001.
In Figure 2, we show the CEF energies as functions of x.As easily understood from the discussion in the LS coupling scheme, the ground state multiplet for n = 2 is characterized by J = 4, where J is total angular momentum given by J = |L − S| with angular momentum L and spin momentum S. For n = 2, we find L = 5 and S = 1 from the Hund's rules and, thus, we obtain J = 4. Due to the effect of cubic CEF, the nonet of J = 4 is split into four groups as Γ 1 singlet, Γ 3 non-Kramers doublet, Γ 4 triplet, and Γ 5 triplet.In the present diagonalization of H loc , we find such CEF states, as shown in Figure 2. When we compare this CEF energy diagram with that of the LS coupling scheme [38], we find that the shape of curves and the magnitude of excitation energy are different with each other.However, from the viewpoint of symmetry, the structure of the low-energy states is not changed between the LS and j-j coupling schemes [1].Since we are interested in a possibility of Γ 3 quadrupole ordering, we choose the value of x as x = 0.0 in the following.

Energy Bands.
Next we consider the band structure obtained by the diagonalization of H 0 = H kin + H CEF + H so .As for the Slater-Koster integrals, it is one way to determine them so as to reproduce the Fermi-surface sheets of actual materials, but here we determine them from a theoretical viewpoint as where t indicates the magnitude of hopping amplitude.The size of t should be determined by the quasi-particle bandwidth, but here we simply treat it as an energy unit.
In Figure 3, we depict the eigen energies of H 0 along the lines connecting some symmetric points in the first Brillouin zone.As for the spin-orbit coupling and CEF parameters, we set λ/t = 0.1 and W/t = 0.001.First we note that there exist seven bands and each band has double degeneracy due to time-reversal symmetry, which is distinguished by pseudospin.Since the magnitude of λ is not so large, we do not observe a clear splitting between j = 7/2 octet and j = 5/2 sextet bands.Around at Γ point, we find that j = 5/2 sextet is split into two groups, Γ 7 doublet and Γ 8 quartet.Here we note that the energy of Γ 8 quartet is lower than that of Γ 7 .Since the Γ 8 has orbital degeneracy, it becomes an origin of the formation of Γ 3 non-Kramers doublet, when we accommodate a couple of electrons per site.
Note that the Fermi level is denoted by a horizontal line, which is determined by the condition of n = 2, where n is the average electron number per site.When we pay our attention to the band near the Fermi level, we find that the orbital degeneracy exists in the bands on the Fermi surface.For instance, we see the degenerate bands on the Fermi surface around the Γ point.Such orbital degeneracy in the momentum space is considered to be a possible source Energy (units of Figure 2: CEF energy levels obtained by the diagonalization of H loc for λ/F 6 = 0.1 and W/F 6 = 0.001 with F 0 = 10F 6 , F 2 = 5F 6 , and of Γ 3 quadrupole ordering, which will be discussed in the next subsection.Finally, in the present case, we expect the appearance of the large-volume Fermi surface as well as the small-size pocket-like Fermi surface.Such mixture of the Fermi surface sheets with different topology may be an important issue for the appearance of higher-rank multipole ordering.

Phase
Diagram.Now we show the phase diagram of the multipole state.First it is necessary to calculate the susceptibility equation ( 25) at ν n = 0.As for the momentum q, we divide the first Brillouin zone into 16 × 16 × 16 meshes.Concerning the momentum integration in (25), we exploit the Gauss-Legendre quadrature with due care.At low temperatures such as T/t = 0.01, it seems to be enough to divide the range between −π and π into 60 segments along each direction axis.As found in (25), χ (0) has 14 4 components in the spin-orbital space, but it is not necessary to calculate all the components due to the symmetry argument.We have checked that it is enough to evaluate 1586 components of χ (0) .We set the parameters as λ/t = 0.1, x = 0.0, W/t = 0.001, Note that the ratio among the Slater-Condon parameters is the same as that in Figure 2. We also note that the hopping amplitude t is relatively large compared with local potential and interactions, since we consider the multipole ordering from the itinerant side.Here we emphasize that our framework actually works for the microscopic discussion on the multipole ordering.A way to determine more realistic parameters in the model will be discussed elsewhere.
By changing the values of temperature T/t, we depict the phase diagram in the plane of t/F 0 and T/t.Note that t 2 /F 0 corresponds to the typical magnitude of multipolemultipole interaction between nearest neighbor sites.As naively understood, when the temperature is increased, larger value of U is needed to obtain the ordered state.Then, the phase diagram is shown in Figure 4. We evaluate Energy (units of t) 1, and W/t = 0.001.Note that we show the eigen energies along the lines of the maximum eigenvalue of the multipole susceptibility by increasing F 0 /t.One may think that the magnitude of t/F 0 in Figure 4 is too small to obtain reasonable results in the RPA calculations.Here we note that the total bandwidth of the seven-orbital system is in the order of 10t, as shown in Figure 3. Namely, the critical value of the interaction F 0 c at low enough temperatures is considered to be in the order of the total bandwidth.In this sense, we consider that the value of t/F 0 in Figure 4 is not small for the RPA calculations.Note also that when the temperature is increased, the magnitude of noninteracting susceptibility is totally suppressed, leading to the enhancement of F 0 c .Thus, t/F 0 is decreased when T is increased, as observed in Figure 4.
At low temperatures as T/t < 0.3, we obtain that the maximum eigen value of susceptibility matrix is characterized by the multipole with Γ 3 symmetry and the ordering vector Q = (π, π, π).The component of the multipole depends on the temperature, but the 90% of the optimized multipole is rank 2 (quadrupole).Others are rank 4 (hexadecapole) and rank 6 (tetrahexacontapole) components, which are about 10%.Note again that the multipoles with the same symmetry are mixed in general, even if the rank of the multipole is different.Namely, quadrupole is the main component, while hexadecapole and tetrahexacontapole are included with significant amounts.Note also that the phase diagram is shown only in the region of T/t < 1, but the boundary curve approaches the line of t/F 0 = 0. Since the case with very large F 0 is unrealistic, we do not pay our attention to the phase for T > t, although we can continue the calculation in such higher temperature region.
When we increase the temperature, the magnetic phase is observed for T/t > 0.3.The main component is Γ 4 dipole and the ordering vector is Q = (0, 0, 0).Note that the susceptibility for Γ 4 multipole moment does not mean magnetic susceptibility, which is evaluated by the response of magnetic moment L+2S, that is, J+S.At T/t = 0.4, admixture of the multipole is as follows: rank 1 (dipole) 90.7%, rank 3 (octupole) 6.5%, rank 5 (dotriacontapole) 2.1%, and rank  7 (octacosahectapole) 0.7%.The amounts are changed by the temperature, but the main component is always dipole.
We have found the low-temperature antiferroquadrupole state and the high-temperature ferromagnetic phase.Such a combination of nonmagnetic and magnetic phases can be observed in other parameter sets including quadrupole ordering.

Discussion and Summary
We have constructed the microscopic framework to discuss the multipole ordering due to the evaluation of multipole susceptibility in f -electron systems on the basis of the sevenorbital Hubbard model with strong spin-orbit coupling.For the evaluation of multipole susceptibility, we have used the RPA and found the critical point from 1/χ max .As an example of the present scheme, we have shown the results for the case of n = 2 concerning quadrupole ordering on the threedimensional simple cubic lattice.If we specify the lattice structure and determine the hopping parameter from the comparison with the experimental results on the Fermisurface sheets, in principle, it is possible to determine the type of multipole ordering with the use of appropriate local CEF parameters and Coulomb interactions.
Although the microscopic theory of multipole ordering has been proposed, it is necessary to elaborate the present scheme both from theoretical and experimental viewpoints.In order to enhance the effectiveness of the present procedure, we should increase the applicability of the theory.For instance, we have not considered at all the sublattice structure in this paper, but in actuality, the staggered-type multipole ordering has been observed.In order to reproduce the structure, it is necessary to maximize the multipole susceptibility by taking into account the sublattice structure.It is one of future problems from a theoretical viewpoint.
It is also highly expected that the present scheme should be applied to actual materials in order to explain the origin of multipole ordering.For instance, it is interesting to seek for the origin of peculiar incommensurate quadrupole ordering observed in PrPb 3 [45].At the first glance, it seems to be quite difficult to explain the origin of the Γ 3 quadrupole ordering with the ordering vector of Q = (π/2±δ, π/2±δ, 0) with δ = π/8.However, if we use the present scheme, it may be possible to find a solution in a systematic way.Another issue is the revisit to octupole and higher-rank multipole ordering in NpO 2 .The significant amount of dotriacontapole component may be understood naturally in the present scheme.
Another interesting future problem is the emergence of superconductivity near the multipole ordered phase.It has been widely accepted that anisotropic d-wave superconductivity appears in the vicinity of the antiferromagnetic phase, as observed in several kinds of strongly correlated electron materials.In general, near the quantum phase transition, anisotropic superconducting pairs are formed due to the effect of quantum critical fluctuations.Thus, also in the vicinity of multipole ordering, superconductivity is generally expected to occur.Even from purely theoretical interest, it is worthwhile to investigate superconductivity near the antiferroquadrupole phase in Figure 4.When we turn our attention to actual material, in PrIr 2 Zn 20 , superconductivity has been observed and quadrupole fluctuations have been considered to play some roles [46].Within the RPA, it is possible to discuss the appearance of superconductivity in the vicinity of quadrupole ordering in the present scheme.It is another future problem.
In summary, we have proposed the prescription to determine the type of multipole ordering from a microscopic viewpoint on the basis of the seven-orbital Hubbard model.The multipole susceptibility has been obtained in the RPA and the quadrupole ordering has been actually discussed in a way similar to that for the spin ordering in the singleorbital Hubbard model.The application to actual f -electron materials will be discussed elsewhere, but we believe that the present scheme is useful to consider the origin of multipole ordering.In addition, a possibility of superconductivity near the multipole ordering is an interesting future problem.

Figure 1 :
Figure 1: Feynman diagrams for multipole susceptibility in the RPA.The solid curve and broken line denote the noninteracting Green's function G (0) and Coulomb interaction, respectively.

Figure 4 :
Figure 4: Phase diagram of the multipole ordering for n = 2 on the three-dimensional simple cubic lattice.