Classification of atomic-scale multipoles under

PHYSICAL REVIEW B 98, 165110 (2018) Editors’ Suggestion

Classification of atomic-scale multipoles under crystallographic point groups and application to linear response tensors Satoru Hayami,1 Megumi Yatsushiro,1 Yuki Yanagi,2 and Hiroaki Kusunose2 1

Department of Physics, Hokkaido University, Sapporo 060-0810, Japan 2 Department of Physics, Meiji University, Kawasaki 214-8571, Japan

(Received 30 May 2018; revised manuscript received 26 July 2018; published 8 October 2018) Four types of atomic-scale multipoles (electric, magnetic, magnetic toroidal, and electric toroidal multipoles) give a complete set to describe arbitrary degrees of freedom for coupled charge, spin, and orbital of electrons. We here present a systematic classification of these multipole degrees of freedom towards the application in condensed matter physics. Starting from the multipole description under the rotation group in real space, we generalize the concept of multipoles in momentum space with the spin degree of freedom. We show how multipoles affect the electronic band structure and linear responses, such as the magnetoelectric effect, magnetocurrent (magnetogyrotropic) effect, spin conductivity, piezoelectric effect, and so on. Moreover, we exhibit a complete table to represent the active multipoles under 32 crystallographic point groups. Our comprehensive and systematic analyses will give a foundation to identify enigmatic electronic order parameters and a guide to evaluate peculiar cross-correlated phenomena in condensed matter physics from the microscopic point of view. DOI: 10.1103/PhysRevB.98.165110 I. INTRODUCTION

The multipole moments characterize electric charge and current distributions, whose concept has been widely developed in various fields of physics at different length scales, such as classical electromagnetism [1–4], nuclear physics [5– 8], solid-state physics [9–13], and metamaterials [14–18]. It was well-known that there are four types of fundamental multipoles according to their spatial inversion and timereversal properties [4,19–21]: electric (E: polar/true tensor with time-reversal even), magnetic (M: axial/pseudotensor with time-reversal odd), magnetic toroidal (MT: polar/true tensor with time-reversal odd), and electric toroidal (ET: axial/pseudotensor with time-reversal even) multipoles. A recent study has shown that the four fundamental multipoles in atomic-scale constitute a complete set to span the Hilbert space under the space-time inversion group. They can be applied to not only a classical but also a quantum-mechanical picture [21]. The mutual relationship between four multipoles and their schematic pictures in the quantum-mechanical representation are shown in Fig. 1. In condensed matter physics, the multipoles have been recognized as important quantities to describe multiple degrees of freedom of electrons in solids, e.g., charge, spin, and orbital, in a unified way. Especially, the atomic-scale multipoles have been extensively studied in f -electron systems, as the interplay between the Coulomb interaction and spin-orbit coupling due to its strongly localized nature gives rise to

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. 2469-9950/2018/98(16)/165110(35)

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anisotropic charge and spin distributions [9,11,12]. In fact, higher-rank multipole orders beyond the conventional E and M dipole orders have been established by mutual interplay between experimental and theoretical investigations, for instance, E quadrupole in PrT2 X20 (T = Ir, Rh, V, Ti, Nb and X = Al, Zn) [22–30] and M octupole in Ce1−x Lax B6 [31–34]. On the other hand, such a multipole concept can be applied to a cluster consisting of several atomic sites, which is so-called a cluster multipole [35–46]. The complex electric and magnetic orderings in cluster are regarded as the ferroic arrangement of higher-order cluster multipoles from the symmetry aspect. For example, the all-in/all-out ordering of magnetic moments on the pyrochlore structure is regarded as an M octupole order (precisely speaking, M pseudoscalar in the point group) [47], while a staggered antiferromagnetic ordering on the zigzag chain is regarded as a ferroic order of MT dipole and M quadrupole [35,39,48]. More recently, the concept of multipoles is extended to multiple hybrid orbitals [21], and to the momentum space relevant with topologically nontrivial excitations through the Berry curvature [49–53]. In this way, studies of the multipoles are useful to cover various unconventional order parameters in a systematic manner and understand/expect physical phenomena from the symmetry viewpoint. For example, the nematic order in iron-based superconductors [54–59] and magnetic insulators [60–63], Pomeranchuk instability [64–69], anisotropic density wave including staggered flux phases [70–76], excitonic insulators [77–81], and spin chirality accompanying Berry phase [82–84] are also described by multipole terminology. It is advantageous for the multipole description that once a type of the multipoles is identified, it is easy to understand/predict how the band structure is deformed and what types of cross-correlated couplings and transport phenomena occur. For example, the magnetoelectric effects in Published by the American Physical Society

HAYAMI, YATSUSHIRO, YANAGI, AND KUSUNOSE

E multipole

PHYSICAL REVIEW B 98, 165110 (2018)

ET multipole

reverse

reverse

reverse

reverse

MT multipole

M multipole

FIG. 1. Four types of multipoles under the space-time inversion group. The E (M) multipole transforms the MT (ET) multipole by reversing the time-reversal property, while the E (M) multipole transforms ET (MT) multipole by reversing the spatial-inversion property. Schematic pictures of the wave functions for the E dipole, M dipole, MT dipole, and ET quadrupole are shown in each panel. The shape and color map of the pictures represent the electric charge density and the z component of the orbital angular-momentum density, respectively.

Cr 2 O3 [85–87] and Co4 Nb2 O9 [88–92] are understood by the M quadrupole, magnetocurrent effect in metallic UNi4 B by the MT dipole [93–95], valley splitting in the electronic band structure by the MT octupole [96], nonreciprocal magnon excitations in BiTeBr [97] and α-Cu2 V2 O7 [98–102] by the E and MT dipoles, orbital Edelstein effect in Te by the ET monopole and quadrupole [103,104], and large anomalous Hall effect, Nernst effect, and Kerr rotation in Mn3 Sn by the M octupole [40,105–108]. Moreover, unconventional superconductivities are also discussed in the language of multipoles [109–112]. Since the microscopic quantum-mechanical definition of multipoles has been obtained recently [21], it enables us to describe a variety of these order parameters in a systematic way by means of the atomic-scale multipoles, and gives further understanding in cross-correlated responses. In the present paper, we push forward these multipole studies for a comprehensive classification of multipoles in real and momentum spaces and to elucidate the physical properties brought by the multipole degrees of freedom. In order to deduce an unknown order parameter, a comprehensive classification is also very useful to narrow down a candidate. Starting from the multipole description under the rotation group in real space, we generalize the concept of multipoles in momentum space with the spin degree of freedom. We show how multipoles affect the electronic band structure and linear responses, such as the magnetoelectric and magnetocurrent (magnetogyrotropic) effects. Moreover, we demonstrate

a complete table to display the active multipoles under 32 crystallographic point groups. Our systematic study will encourage a direction for material design based on multipole (electronic) degrees of freedom. The organization of this paper is as follows. In Sec. II, we give a definition of four multipoles in both real and momentum spaces. In Sec. III, we show how the emergent multipoles affect the electronic band structure and linear responses including magnetoelectric and magnetocurrent (magnetogyrotropic) phenomena. After presenting the crystallineelectric-field potential under all the point groups, we clarify what types of multipoles are active and become potential order parameters under the point-group irreducible representation in Sec. IV. Section V is devoted to a summary of the present paper. In Appendix A, we show the atomic basis wave functions for s, p, d, and f orbitals. We discuss a possible extension of the momentum-space multipoles to multiorbital system in Appendix B. In Appendix C, we show transformation properties of linear response tensors under the spatial inversion and timereversal operations. We present which multipoles are relevant with rank-3 linear response tensors, such as the spin conductivity tensor and piezoelectric(current) tensor in Appendix D. In Appendix E, we show the active multipoles for atomic basis functions for the cubic Oh and its subgroups. Appendix E contains various tables for the hexagonal D6h and its subgroups to complete all the point groups. For completeness, we use here some of our results previously reported in Ref. [21]. II. DEFINITION OF MULTIPOLES

The concept of multipoles is useful to describe electronic degrees of freedom in a systematic way, and it is capable to express arbitrary type of order parameters in phase transition and physical responses to external fields at microscopic level [113]. As mentioned in Introduction, there are four types of multipoles, E (electric: Qlm ), M (magnetic: Mlm ), MT (magnetic toroidal: Tlm ), and ET (electric toroidal: Glm ), according to the time-reversal and spatial inversion properties [1,19,20]. In this section, we introduce microscopic expressions of four types of multipoles. First, we give the expressions in real space in Sec. II A, and then we give them in momentum space in Sec. II B. With these prescriptions, we summarize in Table III, which type of multipoles are activated in the atomic basis functions with the spin degree of freedom. A. Multipoles in real space

Based on the standard multipole expansion of the electromagnetic potentials φ(r ) and A(r ) in the presence of the source electric current j e (r ) = c∇ × M(r ) and the magnetic current j m (r ) = c∇ × P (r ) (M and P are the magnetization and electric polarization) [114], and by inserting the quantum-mechanical expressions of j e and j m including spin contributions, two of the present authors have shown the quantum-mechanical expressions of multipoles as [21] ˆ lm = −e Q Olm (r j ), (1)

165110-2

j

Mˆ lm = −μB

j

ml (r j ) · ∇Olm (r j ),

(2)

CLASSIFICATION OF ATOMIC-SCALE MULTIPOLES …

Tˆlm = −μB

t l (r j ) · ∇Olm (r j ),


(3)

j

ˆ lm = −e G

x,y,z j

αβ

gl ∇α ∇β Olm (r j ),

(4)

αβ

αβ

where ml , t l , and gl are defined in terms of the dimensionless orbital and spin angular-momentum operators l j and σ j /2 of an electron at r j as 2l j + σj, l+1 2l j rj × + σj , t l (r j ) = l+1 l+2 ml (r j ) =

αβ

β

gl (r j ) = mαl (r j )tl (r j ). We have introduced

(5) (6) (7)

4π l ∗ r Y (ˆr ), (8) 2l + 1 lm where Ylm (ˆr ) are the spherical harmonics as a function of angles rˆ = r/r [8,9] with the azimuthal and magnetic quantum numbers, l and m (−l m l). We adopt the phase ∗ (ˆr ). The inconvention so as to satisfy Ylm (ˆr ) = (−1)m Yl−m dex l represents the rank of multipoles: l = 0 (monopole), 1 (dipole), 2 (quadrupole), 3 (octupole), 4 (hexadecapole), and so on. Strictly speaking, the rank is meaningful as the irreducible representation only if rotational symmetry is preserved, otherwise, the same irreducible representations appear in different ranks in general, and they mix with each other. Hereafter, we omit “elementary charge” for multipoles, i.e., −e and −μB , for notational simplicity. In the rotation group, arbitrary linear combinations within the same rank can be taken as an irreducible basis set. It (c) (r ) is often useful to introduce the tesseral harmonics Olm (s) and Olm (r ), which correspond to the real expressions for the spherical harmonics [115]: Olm (r ) =

Ol0(c) (r ) ≡ Ol0 (r ),

where L1 ||Xˆ l ||L2 and L1 M1 |L2 M2 ; lm are the reduced matrix element independent of M1 and M2 , and the ClebschGordan coefficient, respectively. Note that the reduced matrix ˆ lm and Tˆlm vanish when those for Mˆ lm and elements for Q ˆ Glm are nonzero, and vice versa (see Table III, for example). Therefore the matrix elements for Tˆlm are proportional to ˆ lm and the proportional coefficient, Rl (L1 , L2 ), is those of Q independent of M1 and M2 , i.e., ˆ lm |L2 M2 . (13) L1 M1 |Tˆlm |L2 M2 = Rl (L1 , L2 )L1 M1 |Q

(10)

ˆ lm and Mˆ lm as Similar proportionality holds between G ˆ lm |L2 M2 = Rl (L1 , L2 )L1 M1 |Mˆ lm |L2 M2 . (14) L1 M1 |G

(11)

The proportional coefficients turn out to be common as

m

(−1) ∗ (r )], √ [Olm (r ) + Olm 2 (−1)m (s) ∗ (r ) ≡ √ [Olm (r ) − Olm (r )], Olm 2i

(c) Olm (r ) ≡

(9)

ˆ lm and l and σ are odd under the time-reversal operation. Q ˆ and Tlm in Eqs. (1) and (3) (see also the left panels in Fig. 1) represent the polar E and MT multipole operators, ˆ lm in Eqs. (2) and (4) (see also the right while Mˆ lm and G panels in Fig. 1) represent the axial M and ET multipole operators. Note that M monopole Mˆ 0 ≡ Mˆ 00 , MT monopole ˆ0 ≡ G ˆ 00 , and ET dipole G ˆ 1m ≡ Tˆ0 ≡ Tˆ00 , ET monopole G ˆ ˆ ˆ ˆ G = (Gx , Gy , Gz ) vanish in Eqs. (2)–(4) owing to the identity ∇O00 (r ) = 0 and ∇α ∇β O1m (r ) = 0. In the spinless basis functions under the rotation group, (L, M ) [L = 0(s), 1(p), 2(d ), 3(f ) and M = −L, −L + 1, . . . , L] shown in Appendix A, the multipoles given by Eqs. (1)–(4) without spin contributions constitute a complete set to express an arbitrary degree of freedom [21]. In other words, the even-rank E and odd-rank M multipoles are active in nonhybrid (intra) orbitals, such as d-d and f -f orbitals. Meanwhile, the odd(even)-rank E and MT and even(odd)-rank M and ET multipoles are active in odd(even)-parity hybrid (inter) orbitals, such as p-d (s-d) and d-f (p-f ) orbitals [21]. The total number of these independent active multipoles equals to the number of matrix elements in the relevant Hilbert space. Since Xˆ lm transforms like Olm (r ) by the rotational operation, the matrix elements of four multipoles are related with each other. According to the Wigner-Eckart theorem, the matrix elements can be divided into the purely angular part and common part as L1 M1 |Xˆ lm |L2 M2 = L1 ||Xˆ l ||L2 L1 M1 |L2 M2 ; lm, (12)

for m = 1, 2, . . . , l. Since the multipole operator Xˆ lm (X = Q, M, T , G) transforms like Olm (r ) by the rotational operation, similar linear combinations are applied to Xˆ lm as well. In what follows, the symbols Olm and Xˆ lm are used to represent the set of harmonics in the real expression, either the tesseral harmonics or cubic harmonics, or hexagonal harmonics, and so on, with rank l. For later convenience, the even- and odd-parity cubic harmonics in Oh group are summarized in Tables I and II, respectively. The hexagonal harmonics in the D6h group are also summarized in Tables XXII and XXIII in Appendix F. These four multipole operators are clearly independent with each other under the space-time inversion operations as shown in Fig. 1. This is readily confirmed by using the facts that Olm (r ) has parity (−1)l under the spatial inversion,

ˆ l ||L2 L1 ||G L1 ||Tˆl ||L2 = ˆ L1 ||Ql ||L2 L1 ||Mˆ l ||L2 L1 (L1 + 1) − L2 (L2 + 1) i. = (l + 1)(l + 2)

Rl (L1 , L2 ) =

(15)

The result is consistent with the fact that the toroidal ˆ lm , vanish for nonhybrid orbitals, i.e., multipoles, Tˆlm and G L1 = L2 . For the spinful basis, as discussed below, a similar proportionality also holds, in which the spinless basis (L, M ) is replaced by a spinful one (J, Jz ). In other words, we can define arbitrary type of MT and ET multipoles in the total (orbital) angular-momentum basis through the proportionality, and it is sufficient to calculate the matrix elements of the E and M multipoles. Next, let us extend to the situation where the total angularmomentum basis (J, Jz ) is appropriate rather than (L, M ) by

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TABLE I. Even-parity cubic harmonics (E multipoles in unit of −e in the cubic Oh group). The linear combinations of the tesseral (c) (s) (r ) and Olm (r ), respectively. In the irreducible representation (irrep.), the harmonics are shown where (lm) and (lm) correspond to Olm subscript represents the spatial parity (even:g, odd:u), while the superscript represents the time-reversal property (even: +, odd: −). A1 , A2 , E, T1 , and T2 correspond to 1 , 2 , 3 , 4 , and 5 , respectively, in the Bethe notation. rank

irrep.

symbol

0

+ A1g

Q0

2

Eg+

Qu , Qv

+ T2g

Qyz , Qzx , Qxy

+ A1g

Q4

Eg+

Q4u

4

definition 1

Qα4x

Qα4z Qβ4x Qβ4y β

Q4z 6

+ A1g

Q6

+ A2g

Q6t

Eg+

Q6u Q6v

+ T1g

Qα6x Qα6y Qα6z

+ T2g

β1 Q6x β1

Q6y

β1

Q6z + T2g

β2 Q6x β2

Q6y

Qβ2 6z

3 (x 2 2

2

(20), (22) (21) , (21), (22) √ √ (4) ≡ 2√1 3 [ 5(44) + 7(40)] √ √ − 2√1 3 [ 7(44) − 5(40)]

+ y 4 + z4 − 35 r 4 ) √ 4 4 7 15 4 [z − x +y − 37 r 2 (3z2 − r 2 )] 6 2 √ 7 5 [x 4 − y 4 − 67 r 2 (x 2 − y 2 )] 4 √ 35 yz(y 2 − z2 ) 2 √ 35 zx(z2 − x 2 ) 2 √ 35 xy(x 2 − y 2 ) 2 √ 5 yz(7x 2 − r 2 ) 2 √ 5 zx(7y 2 − r 2 ) 2 √ 5 xy(7z2 − r 2 ) 2

Qα4y + T2g

2

√ 5 21 (x 4 12

Q4v + T1g

(00)

√

− r ), −y ) √ √ 3yz, 3zx, 3xy

1 (3z2 2

√

linear combination

−(42) √ − 2 2 [(43) + 7(41) ] √ − 2√1 2 [(43) − 7(41)] 1 √

−

√ 11 42 [x 6 8

x 6 +y 6 2

−

− y6 −

15 2 4 r (z 11 15 2 4 r (x 11

−

x 4 +y 4 2

)+

− y4 ) +

5 4 r (3z2 22

5 4 2 r (x 11

√ 3 7 yz(y 2 − z2 )(11x 2 − r 2 ) 4 √ 3 7 zx(z2 − x 2 )(11y 2 − r 2 ) 4 √ 3 7 xy(x 2 − y 2 )(11z2 − r 2 ) 4 √ 2 462 yz[y 4 + z4 − 58 (y 2 + z2 ) ] 2 √ 2 462 zx[z4 + x 4 − 58 (z2 + x 2 ) ] 2 √ 2 462 xy[x 4 + y 4 − 58 (x 2 + y 2 ) ] 2 √ 210 yz(33x 4 − 18x 2 r 2 + r 4 ) 16 √ 210 zx(33y 4 − 18y 2 r 2 + r 4 ) 16 √ 210 xy(33z4 − 18z2 r 2 + r 4 ) 16

taking into account the spin degree of freedom. In the oneelectron state, the total angular momentum J is represented by J = L − 1/2 or L + 1/2 and Jz = −J, −J + 1, . . . , J . Similar to the argument for the spinless basis, the active multipoles are uniquely identified for the spinful basis as well from the following conditions: (1) the even-rank E multipoles and the odd-rank M multipoles are active in nonhybrid orbitals, such as s-s and p-p orbitals with the same total angular momentum J . (2) The even(odd)-rank E and MT multipoles and odd(even)-rank M and ET multipoles are active in the even(odd)-parity hybrid orbitals with different J or L. (3) The rank of active multipole l is determined by |J1 − J2 | l J1 + J2 , where J1 and J2 are total angular momenta in the bra- and ket-basis functions. For example, in p-d orbitals with the same total angular momentum J = 3/2, the

(44)

7(43) − (41) ] √ − 2√1 2 [ 7(43) + (41)] 1 √ [ 2 2

(42) √ (6) ≡ 2√1 2 [− 7(64) + (60)] √ √ (6t) ≡ 41 [− 5(66) + 11(62)] √ 1 √ [(64) + 7(60)] 2 2

√ r2 r6 231 2 [x 2 y 2 z2 + 22 (x 4 + y 4 + z4 − 35 r 4 ) − 105 ] 8 √ 2310 [x 4 (y 2 − z2 ) + y 4 (z2 − x 2 ) + z4 (x 2 − y 2 )] 8

√ 11 14 6 [z 4

√

− r 2 )]

− y 2 )]

√ √ 11(66) + 5(62)] √ √ √ 1 [− 22(65) − 30(63) + 2 3(61) ] 8 √ √ √ 1 [ 22(65) − 30(63) − 2 3(61)] 8 1 [ 4

√

√

(64)

√ 3(65) + 55(63) + 3 22(61) ] √ √ √ 1 [ 3(65) − 55(63) + 3 22(61)] 16

1 [ 16

√

(66)

√ 165(65) − 9(63) + 10(61) ] √ √ 1 [ 165(65) + 9(63) + 10(61)] 16

1 [ 16

(62)

number of the independent active multipoles is 32 as indicated by the 11th row in Table III, which consist of M/ET monopole, E/MT dipole, M/ET quadrupole, and E/MT octupole without excess or deficiency. We summarize all the active multipoles in the spinful nonhybrid and hybrid orbitals in Table III, which will be useful to narrow down candidates of potential order parameters according to knowledge of relevant spin-orbital Hilbert space. In contrast to the cases for the spinless basis, ˆ are active in the spinful basis. ˆ 0 , and G Mˆ 0 , G The necessity of M and ET monopoles and ET dipole, which do not appear in the multipole expansion, implies that ˆ 0, the corresponding quantum-mechanical operators Mˆ 0 , G ˆ are definable in the spinful basis. To obtain explicit and G forms of these multipoles, we focus on the fact that the inner ˆ = m1 (r ) and the position product between the M dipole M

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TABLE II. Odd-parity cubic harmonics (E multipoles in unit of −e in the cubic Oh group). rank

irrep.

symbol

definition

linear combination

1

+ T1u

Qx , Qy , Qz

(11), (11) , (10)

3

+ A2u

Qxyz

x, y, z √ 15xyz

+ T1u

Qαx

1 x(5x 2 2

Qαz Qβx Qβy Qβz 5

Eu+

Q5u , Q5v

+ T1u

Qα1 5x Qα1 5y Qα1 5z

+ T1u

Qα2 5x Qα2 5y Qα2 5z

+ T2u

β Q5x β

Q5y Qβ5z

vector r has the same symmetry as the M monopole in the rotation group. Thus we define the M monopole as Mˆ 0 ≡ m1 (r j ) · r j = σ j · rj , (16) j

− 3r 2 )

1 y(5y 2 − 3r 2 ) 2 1 z(5z2 − 3r 2 ) 2 √ 15 x(y 2 − z2 ) 2 √ 15 y(z2 − x 2 ) 2 √ 15 z(x 2 − y 2 ) 2 √ √ 3 35 xyz(x 2 − y 2 ), − 105 xyz(3z2 − r 2 ) 2 2 2 x [8x 4 − 40x 2 (y 2 + z2 ) + 15(y 2 + z2 ) ] 8 2 y [8y 4 − 40y 2 (z2 + x 2 ) + 15(z2 + x 2 ) ] 8 2 z [8z4 − 40z2 (x 2 + y 2 ) + 15(x 2 + y 2 ) ] 8 √ 2 3 35 x[y 4 + z4 − 43 (y 2 + z2 ) ] 2 √ 2 3 35 y[z4 + x 4 − 43 (z2 + x 2 ) ] 2 √ 2 3 35 z[x 4 + y 4 − 43 (x 2 + y 2 ) ] 2 √ 105 x(y 2 − z2 )(3x 2 − r 2 ) 4 √ 105 y(z2 − x 2 )(3y 2 − r 2 ) 4 √ 105 z(x 2 − y 2 )(3z2 − r 2 ) 4

Qαy + T2u

√

j

(32)

√ 3(31)] √ 3(31) ]

1 √ [ 5(33) − 2 2 √ − 2√1 2 [ 5(33) +

(30)

√ [ 3(33) + 5(31)] √ √ 1 √ [− 3(33) + 5(31) ] 2 2 −2

1 √

√

2

(32)

(54) , −(52) √ √ √ 1 √ [3 7(55) − 35(53) + 30(51)] 8 2 √ √ √ 1 √ [3 7(55) + 35(53) + 30(51) ] 8 2 (50) √ √ √ 10(55) + 9 2(53) + 2 21(51)] √ √ √ 1 [ 10(55) − 9 2(53) + 2 21(51) ] 16 1 [ 16

(54) √ √ √ [− 15(55) − 3(53) + 14(51)] 4 2 √ √ √ 1 √ [ 15(55) − 3(53) − 14(51) ] 4 2 1 √

(52)

where we have used l · r = 0, and the prime represents that it does not appear in the multipole expansion. Mˆ 0 in Eq. (16) clearly exhibits a property of a time-reversal-odd pseudoscalar and is valid only for the spinful basis. Similarly, by using the fact that the MT dipole Tˆ = t 1 (r ) has the same spatial

TABLE III. Active multipoles in nonhybrid (intra) and hybrid (inter) orbitals with the spin degree of freedom. The number of independent multipoles within the relevant Hilbert space is indicated in parentheses. The columns of P and l represent the spatial parity and the rank of the active multipoles, respectively. The column of “orbital” indicates the atomic orbital that consists of the basis. basis 1/2-1/2 (4) 3/2-3/2 (16) 5/2-5/2 (36) 7/2-7/2 (64) 1/2-3/2 (16) 1/2-5/2 (24) 3/2-5/2 (48) 3/2-7/2 (64) 5/2-7/2 (96) 1/2-1/2 (8) 3/2-3/2 (32) 5/2-5/2 (72) 1/2-3/2 (16) 1/2-5/2 (24) 1/2-7/2 (32) 3/2-5/2 (48) 3/2-7/2 (64) 5/2-7/2 (96)

orbital

P

l = 0 (1)

1 (3)

2 (5)

3 (7)

4 (9)

5 (11)

6 (13)

7 (15)

s-s, p-p p-p, d-d d-d, f -f f -f s-d, p-p s-d p-f, d-d p-f f -f s-p p-d d-f s-p, p-d s-f, p-d s-f p-d, d-f d-f d-f

+

E E E E – – – – – M/ET M/ET M/ET – – – – – –

M M M M M/ET – M/ET – M/ET E/MT E/MT E/MT E/MT – – E/MT – E/MT

– E E E E/MT E/MT E/MT E/MT E/MT – M/ET M/ET M/ET M/ET – M/ET M/ET M/ET

– M M M – M/ET M/ET M/ET M/ET – E/MT E/MT – E/MT E/MT E/MT E/MT E/MT

– – E E – – E/MT E/MT E/MT – – M/ET – – M/ET M/ET M/ET M/ET

– – M M – – – M/ET M/ET – – E/MT – – – – E/MT E/MT

– – – E – – – – E/MT – – – – – – – – M/ET

– – – M – – – – – – – – – – – – – –

+

− −

165110-5



TABLE IV. Operator expressions of even-parity multipoles up to l = 4 in the cubic Oh group. For a noncommute product, a symmetrized αβ expression should be used, e.g., AB → (AB + B † A† )/2. ml , t l , and gl are defined in Eqs. (5)–(7). The multipole with the prime requires complementary definitions other than Eqs. (1)–(4). rank

type

0

irrep.

i (off-diagonal element)

Qu , Qv

+ T2g

Qyz , Qzx , Qxy

Eg−

Tu , Tv

− T2g

Tyz , Tzx , Txy

MT

ET

E

MT

y

mx1 , m1 , mz1

Mx , My , Mz Gx ,

− A2g

M

4

1

T0

E

M

3

Q0

Eg+

MT

2

definition

ET

E

1

symbol

+ A1g − A1g − T1g + T1g

Gy ,

Gz

xl · (r × σ ), yl · (r × σ ), zl · (r × σ )

Mxyz

− T1g

Mxα ,

− T2g

Mxβ , Myβ , Mzβ Gxyz

+ T1g

Gαx , Gαy , Gαz

+ T2g

Gβx , Gβy , Gβz

+ A1g

Q4

Eg+

Q4u , Q4v

+ T1g

Qα4x , Qα4y , Qα4z

+ T2g

Q4x , Q4y , Q4z

− A1g

T4

Eg−

y

3[ 21 (3x 2 − r 2 )mx3 − x(ym3 + zmz3 )], (cyclic) √ 1 2 y 15[ 2 (y − z2 )mx3 + x(ym3 − zmz3 )], (cyclic) √ yz xy 2 15(xg3 + yg3zx + zg3 ) αα xy zx xx 9xg3 − 6(yg3 + zg3 ) − 3x α g3 , (cyclic) √ xy yy 15[2(yg3 − zg3zx ) + x(g3 − g3zz )], (cyclic)

Myα , Mzα

+ A2g

β

β

√ 7 15 4 [z 6

√

15 [ 3

T4v − T1g

T4xα , T4yα , T4zα

− T2g

T4x , T4y , T4z

β

β

j

where we have used (r × l ) · l = (r × σ ) · σ = 0 and the ˆ 0 acts as an prefactor 6 for notational simplicity. Since G ˆ≡ elementary charge for the ET multipoles, the ET dipole G ˆ x, G ˆ y, G ˆ z ) is obtained by multiplying the position vector r (G ˆ to G0 as ˆ≡ ˆ 0 )j r j = G (G r j [l j · (r j × σ j )]. (18) j

j

ˆ are also valid only for the It is noteworthy that and G spinful basis. On the other hand, the MT monopole is identically zero even by taking an inner product between the MT dipole t 1 (r ) and the position vector r, since t 1 (r ) is perpendicular to r. This is consistent with the fact that the MT monopole is unnecessary to span an arbitrary matrix in both nonhybrid and hybrid orbitals as shown in Table III. It is interesting to note ˆ 0 G

−

√

α

x 4 +y 4 2

+ y 4 + z4 − 35 r 4 )

− 37 r 2 (3z2 − r 2 )], √

√ 7 5 [x 4 4

− y 4 − 67 r 2 (x 2 − y 2 )]

35 yz(y 2 − z2 ), (cyclic) 2 √ 5 yz(7x 2 − r 2 ), (cyclic) 2 √ y 5 21 [x 3 t4x + y 3 t4 + z3 t4z − 35 r 2 (r 3

α(5α − 3r 2

2

)t4α

− 3x(x − 3y 2

2

)t4x

· t 4 )] y

+ 3y(3x 2 − y 2 )t4 ]

y

5[x(x 2 − 3z2 )t4x − y(y 2 − 3z2 )t4 − 3z(x 2 − y 2 )t4z ]

√ y 35 [z(3y 2 − z2 )t4 + y(y 2 − 3z2 )t4z ], (cyclic) 2 √ √ √ y y 6 5xyzt4x + 25 (7x 2 − r 2 )(zt4 + yt4z ) − 5yz(yt4 + zt4z ),

inversion property as the E dipole, it is natural to define a ˆ 0 as time-reversal-even pseudoscalar G ˆ 0 ≡ 6 G m1 (r j ) · t 1 (r j ) = l j · (r j × σ j ), (17) j

√ 5 21 (x 4 12

β

T4u

β

√

− r 2 ), 23 (x 2 − y 2 ) √ √ √ 3yz, 3zx, 3xy √ y 3zt2z − r · t 2 , 3(xt2x − yt2 ) √ √ √ y y 3(yt2z + zt2 ), 3(zt2x + xt2z ), 3(xt2 + yt2x ) √ y z 15(yzmx3 + zxm3 + xym3 ) 1 (3z2 2

(cyclic)

that the MT monopole becomes active in hybrid orbitals with different additional quantum number within the same total angular momentum, e.g., 2p-4p and 3d-5d orbitals. From the correspondence between the E and MT multipoles, the expression of Tˆ0 should be given by n1 LM|Tˆ0 |n2 LM ≡ i

(n1 = n2 ),

(19)

where n1 and n2 are the quantum numbers other than L and M, such as principal quantum numbers. Equation (19) means that Tˆ0 represents the imaginary hybridization between the different orbitals with the same L and M. Finally, let us make a remark on the further extension of multipoles in the spinful basis. In the spinful basis, the scalar product of l and σ can also be regarded as the E monopole, i.e., ˆ 0 ≡ Q lj · σ j . (20) j

ˆ 0 with the Then, by combining this elementary charge Q polynomial Olm (r ), we obtain the higher-rank E multipoles including the spin degree of freedom. Similar extensions for

165110-6



TABLE V. Operator expressions of odd-parity multipoles up to l = 4 in the cubic Oh group. rank

type

0

M ET E MT M

1 2

ET 3

E

MT

4

M

ET

irrep.

symbol

definition

− A1u + A1u + T1u − T1u Eu− − T2u Eu+ + T2u + A2u + T1u

M0 G0 Qx , Qy , Qz Tx , Ty , Tz Mu , Mv Myz , Mzx , Mxy Gu , Gv Gyz , Gzx , Gxy Qxyz Qαx , Qαy , Qαz

r ·σ l · (r × σ ) x, y, z y t1x , t1 , t1z √ y z 3zm2 − r · m2 , 3(xmx2 − ym2 ) √ √ √ y y 3(ymz2 + zm2 ), 3(zmx2 + xmz2 ), 3(xm2 + ymx2 ) √ yy 3g2zz − α g2αα , 3(g2xx − g2 ) √ yz √ zx √ xy 2 3g2 , 2 3g2 , 2 3g2 √ 15xyz 1 2 x(5x − 3r 2 ), (cyclic) 2

+ T2u − A2u − T1u − T2u

Qβx , Qβy , Qβz Txyz Txα , Tyα , Tzα Txβ , Tyβ , Tzβ

15 x(y 2 − z2 ), (cyclic) √2 y 15(yzt3x + zxt3 + xyt3z ) y 1 2 2 x 3[ (3x − r )t3 − x(yt3 + zt3z )], (cyclic) √ 2 1 2 y 15[ 2 (y − z2 )t3x + x(yt3 − zt3z )], (cyclic)

− A1u

M4

− T1u

α α α M4x , M4y , M4z

+ T1u + T2u

β

y

+ y 3 m4 + z3 mz4 − 35 r 2 (r · m4 )] y 15 [ α(5α − 3r )mα4 − 3x(x 2 − 3y 2 )mx4 + 3y(3x 2 − y 2 )m4 ] 3 √α y 5[x(x 2 − 3z2 )mx4 − y(y 2 − 3z2 )m4 − 3z(x 2 − y 2 )mz4 ]

M4u M4v β

√ 5 21 [x 3 mx4 3 2 2

√

Eu−

− T2u + A1u Eu+

√

β

M4x , M4y , M4z G4 G4u G4v α G4x , Gα4y , Gα4z β β β G4x , G4y , G4z

√

y

35 [z(3y 2 − z2 )m4 + y(y 2 − 3z2 )mz4 ], (cyclic) 2√ √ √ y y 6 5xyzmx4 + 25 (7x 2 − r 2 )(zm4 + ymz4 ) − 5yz(ym4 + zmz4 ), (cyclic) √ √ xy yz zx −4 21(xyg4 + yzg4 + zxg4 ) + 21 α (3α 2 − r 2 )g4αα √ yy xy yz 2 15[(3y − r 2 )g4xx + (3x 2 − r 2 )g4 + (3z2 − r 2 )g4zz + 4(2xyg4 − xzg4xz − yzg4 )] √ yy yz zz xz 3 5[(x 2 − z2 )g4xx + (z2 − y 2 )g4 + (y 2 − x 2 )g4 − 4(xzg4 − yzg4 )] √ yy yz 3 35[yz(g4 − g4zz ) + (y 2 − z2 )g4 ], (cyclic) √ xy yy yz xz zz xx 3 5[4x(yg4 + zg4 ) + yz(2g4 − g4 − g4 ) + (3x 2 − r 2 )g4 ], (cyclic)

ˆ are possible by combining the elementary Mˆ lm , Tˆlm , and G lm ˆ 0 with Olm (r ). charges, Mˆ 0 , Tˆ0 , and G The proper expressions of the even- and odd-parity multipole operators up to l = 4 in the cubic Oh group are summarized in Tables IV and V. For the hexagonal D6h group, the proper expressions are also summarized in Tables XXIV and XXV in Appendix F. These expressions are sufficient to consider symmetry-classified multipoles in any point groups as they are subgroup of Oh or D6h groups.

(k × σ ) · ∇ k , which reverses its time-reversal property while keeping its parity and rank, the expressions of the odd-rank E multipole and the even-rank MT multipole are obtained. Thus, the expressions of E and MT multipole in momentum space are given by σ0 Olm (k) Qlm (k) ≡ (k × σ ) · ∇ k Olm (k) ⎧ ⎪ ⎨0 Tlm (k) ≡ (k × σ ) · ∇ k Olm (k) ⎪ ⎩σ O (k) 0 lm

B. Multipoles in momentum space

The classification of multipoles is also applicable in momentum space. We present the momentum-based multipoles in single-band systems, which are expressed in terms of (k, σ0 , σ ), where k is the wave vector of electrons, and σ0 and σ are the identity and Pauli matrices in the spin degree of freedom. The extension to multiband systems is not unique. We discuss a possible extension in Appendix B. To construct the multipoles in momentum space, we first introduce the harmonics Olm (k) by replacing the position vector r with the wave vector k in Olm (r ). Reflecting the polar vector of k, the harmonics σ0 Olm (k) represent the polar-type E and MT multipole degrees of freedom. As k is odd under time-reversal operation, the even-rank σ0 Olm (k) represents the E multipoles, while the odd-rank σ0 Olm (k) represents the MT multipoles. By multiplying Olm (k) by the operator

(l = 0, 2, 4, 6, . . . ) , (l = 1, 3, 5, . . . ) (21) (l = 0) (l = 2, 4, 6, . . . ) . (22) (l = 1, 3, 5, . . . )

Note that T0 vanishes owing to the derivative of Olm (k). Similarly, the axial-type M and ET multipoles in momentum space are expressed by multiplying σ0 Olm (k) by the operator σ · ∇ k , which reverses its parity while keeping its time-reversal property and rank. Therefore the oddrank M multipoles and the even-rank ET multipoles are given by σ · ∇ k Olm (k). On the other hand, it is impossible to construct the even-rank M multipoles and the oddrank ET multipoles within the single-band systems. They require the multiorbital or sublattice degree of freedom, as discussed in Appendix B. Thus the M and ET multipoles are

165110-7



TABLE VI. Even-parity multipoles in momentum space up to l = 4 in the cubic Oh group. The higher-order expression with respect to k is also shown for Q0 and (Mx , My , Mz ) in the bracket. The multipole with the prime becomes active in the multiorbital systems as discussed in Appendix B. rank

type

irrep.

symbol

definition

0

E MT M ET

+ A1g − A1g − T1g + T1g

Q0 T0 Mx , My , Mz Gx , Gy , Gz

σ0 [(kx2 + ky2 + kz2 )σ0 ] iσ0 σx , σy , σz [(k · σ )kx − 13 k 2 σx , (cyclic)] iσx , iσy , iσz

Eg+ + T2g Eg− − T2g − A2g − T1g − T2g + A2g + T1g + T2g

Qu , Qv Qyz , Qzx , Qxy Tu , Tv Tyz , Tzx , Txy Mxyz Mxα , Myα , Mzα Mxβ , Myβ , Mzβ Gxyz α Gx , Gαy , Gαz Gβx , Gβy , Gβz

+ A1g

Q4

1 2

E MT

3

M

ET

4

E

Eg+

MT

+ T1g + T2g − A1g Eg−

Qα4y , Qα4z β

β

Q4y , Q4z T4 T4u T4v

− T1g

T4xα , T4yα , T4zα

− T2g

T4x , T4y , T4z

β

β

β

0 σ · ∇ k Olm (k)

⎧ ⎨k · σ Glm (k) ≡ σ · ∇ k Olm (k) ⎩0

√ 5 21 (kx4 12

+ ky4 + kz4 − 35 k 4 )σ0 √ 3 2 − − 7 k (3kz2 − k 2 )]σ0 , 7 4 5 [kx4 − ky4 − 67 k 2 (kx2 − ky2 )]σ0 √ 35 ky kz (ky2 − kz2 )σ0 , (cyclic) √2 5 k k (7kx2 − k 2 )σ0 , (cyclic) 2 y z √ 5 21 [kx (kx2 − 35 k 2 )Qx + ky (ky2 − 35 k 2 )Qy + kz (kz2 − 35 k 2 )Qz ] 3 √ 15 [ k (5kα2 − 3k 2 )Qα − 3kx (kx2 − 3ky2 )Qx + 3ky (3kx2 − ky2 )Qy ] 3 √α α 5[kx (kx2 − 3kz2 )Qx − ky (ky2 − 3kz2 )Qy − 3kz (kx2 − ky2 )Qz ] √ 35 [ky (ky2 − 3kz2 )Qz − kz (kz2 − 3ky2 )Qy ], (cyclic) 2 √ 5 [12kx ky kz Qx + kz {(5kz2 − 3k 2 ) − 3(kz2 − 3kx2 )}Qy + ky {(5ky2 − 3k 2 ) − 3(ky2 − 3kx2 )}Qz ], 2 kx4 +ky4 2

given by Mlm (k) ≡

√

k 2 )σ0 , 23 (kx2 − ky2 )σ0 3ky kz σ0 , (cyclic) √ 3kz Qz − k · Q, 3(kx Qx − ky Qy ) √ 3(ky Qz + kz Qy ), (cyclic) √ 15(ky kz σx + kz kx σy + kx ky σz ) 3[ 1 (3k 2 − k 2 )σx − kx (ky σy + kz σz )], (cyclic) √ 2 1 x2 15[ 2 (ky − kz2 )σx + kx (ky σy − kz σz )], (cyclic) √ 15i(ky kz σx + kz kx σy + kx ky σz ) 3i[ 21 (3kx2 − k 2 )σx − kx (ky σy + kz σz )], (cyclic) √ 15i[ 21 (ky2 − kz2 )σx + kx (ky σy − kz σz )], (cyclic) √ 7 15 4 [kz 6

Q4u , Q4v Qα4x , β Q4x ,

1 (3kz2 − 2 √

(l = 0, 2, 4, 6, . . . ) , (l = 1, 3, 5, . . . )

(23)

(l = 0) (l = 2, 4, 6, . . . ) . (l = 1, 3, 5, . . . )

(24)

We complementarily define G0 (k) as k · σ from the symmetry consideration. The even- and odd-parity multipole operators in momentum space in the Oh group up to l = 4 are summarized in Tables VI and VII. The hexagonal version is summarized in Tables XXVI and XXVII in Appendix F.

the multipole moment affects the electronic band structures in Sec. III B. We also classify linear response tensors, such as magnetoelectric and magnetocurrent tensors, from the multipole point of view in Sec. III C.

A. Electromagnetic potential and electric and magnetic fields

By the definition of the multipole expansion of the scalar and vector potentials with the Coulomb gauge ∇ · A = 0 [21], we obtain

φ(r ) =

lm

A(r ) = i

III. PHYSICAL PROPERTIES IN THE PRESENCE OF MULTIPOLES

When a thermodynamic average of a multipole degree of freedom Xlm ≡ Xˆ lm , which we call a multipole moment, becomes nonzero, it generates anisotropic electric and/or magnetic fields via electromagnetic potentials. Moreover, it affects electronic states and leads to unconventional physical phenomena, such as the magnetocurrent effect through multipole-multipole interactions. In this section, we first give the formula for electric and magnetic fields in the presence of the multipole moment in Sec. III A. Then, we show how

(cyclic)

4π Ylm (ˆr ) Qlm l+1 , 2l + 1 r

4π (l + 1) Y l (ˆr ) Mlm lml+1 (2l + 1)l r lm Y l+1 (ˆr ) − 4π (l + 1)Tlm lml+2 , r

(25)

(26)

where Y llm (ˆr ) (l = l − 1, l, l + 1) is the vector spherical harmonics that transforms like Ylm (ˆr ) under spatial rotation and l is its orbital angular momentum [8,9]. By taking derivatives, we obtain the electric and magnetic fields in the presence of

165110-8



TABLE VII. Odd-parity multipoles in momentum space up to l = 4 in the cubic Oh group. rank

type

0

M ET E MT M

1 2

ET 3

E

MT

4

M

ET

irrep.

symbol

definition

− A1u + A1u + T1u − T1u Eu− − T2u Eu+ + T2u + A2u + T1u + T2u − A2u − T1u

M0 G0 Qx , Qy , Qz Tx , Ty , Tz Mu , Mv Myz , Mzx , Mxy Gu , Gv Gyz , Gzx , Gxy Qxyz Qαx , Qαy , Qαz Qβx , Qβy , Qβz Txyz Txα , Tyα , Tzα

ik · σ k·σ ky σz − kz σy , kz σx − kx σz , kx σy − ky σx kx σ0 , ky σ0 , kz σ0 √ i(3kz σz − k · σ ), 3i(kx σx − ky σy ) √ 3i(ky σz + kz σy ), (cyclic) √ 3kz σz − k · σ , 3(kx σx − ky σy ) √ 3(ky σz + kz σy ), (cyclic) √ 15(ky kz Qx + kz kx Qy + kx ky Qz ) 3[ 1 (3k 2 − k 2 )Qx − kx (ky Qy + kz Qz )], (cyclic) √ 2 1 x2 15[ 2 (ky − kz2 )Qx + kx (ky Qy − kz Qz )], (cyclic) √ 15kx ky kz σ0 1 2 k (5k − 3k 2 )σ0 , (cyclic) x x 2

− T2u

Txβ , Tyβ , Tzβ

15 kx (ky2 − kz2 )σ0 , (cyclic) 2 3 2 − 5 k )σx + ky (ky2 − 35 k 2 )σy + kz (kz2 − 35 k 2 )σz ] √ 15 i[ k (5kα2 − 3k 2 )σα − 3kx (kx2 − 3ky2 )σx + 3ky (3kx2 − ky2 )σy ] 3 √ α α 5i[kx (kx2 − 3kz2 )σx − ky (ky2 − 3kz2 )σy − 3kz (kx2 − ky2 )σz ] √ 35 i[ky (ky2 − 3kz2 )σz − kz (kz2 − 3ky2 )σy ], (cyclic) 2

− A1u Eu−

− T1u

α M4x ,

− T2u

M4x ,

β

M4 M4u M4v α M4y , β M4y ,

+ A1u

√

√ 5 21 i[kx (kx2 3

α M4z

β

M4z

√

+ kz {(5kz2 − 3k 2 ) − 3(kz2 − 3kx2 )}σy + ky {(5ky2 − 3k 2 ) − 3(ky2 − 3kx2 )}σz ], (cyclic)

5 i[12kx ky kz σx 2

G4

√

Eu+

G4u G4v

+ T1u

Gα4x , Gα4y , Gα4z

+ T2u

Gβ4x , Gβ4y , Gβ4z

√ 5 21 [kx (kx2 3

− 35 k 2 )σx + ky (ky2 − 35 k 2 )σy + kz (kz2 − 35 k 2 )σz ] 2 kα (5kα − 3k 2 )σα − 3kx (kx2 − 3ky2 )σx + 3ky (3kx2 − ky2 )σy ] 5[kx (kx2 − 3kz2 )σx − ky (ky2 − 3kz2 )σy − 3kz (kx2 − ky2 )σz ] √ 35 [ky (ky2 − 3kz2 )σz − kz (kz2 − 3ky2 )σy ], (cyclic) 2 kz {(5kz2 − 3k 2 ) − 3(kz2 − 3kx2 )}σy + ky {(5ky2 − 3k 2 ) − 3(ky2 − 3kx2 )}σz ],

15 [ 3 √α

√ 5 [12kx ky kz σx 2

+

the multipole moment as E(r ) = −∇φ = −

lm

B(r ) = ∇ × A = −

4π (l + 1)Qlm

lm

Y l+1 r) lm (ˆ , r l+2

(27)

Y l+1 r) lm (ˆ . r l+2

(28)

4π (l + 1)Mlm

Note that E(r ) and B(r ) have formally equivalent forms with Qlm and Mlm , and no toroidal moments Glm and Tlm appear in these expressions. Thus the toroidal moments Glm and Tlm affect physical quantities not through classical electromagnetic interactions but through the quantum-mechanical multipole-multipole interactions. The latter also affects the phase of electrons through the vector potential, e A. It is interesting to note that by using the dual nature of electric and magnetic quantities, we can introduce dual potentials with opposite space-time-parity counterparts as 4π Ylm (ˆr ) ∗ φ (r ) ≡ (29) Mlm l+1 , 2l + 1 r lm

4π (l + 1) Y l (ˆr ) ∗ i Qlm lml+1 A (r ) ≡ (2l + 1)l r lm Y l+1 r) lm (ˆ − 4π (l + 1)Glm l+2 . (30) r

(cyclic)

Then, the electric and magnetic fields are given by E = ∇ × A∗ and B = −∇φ ∗ . If a magnetic monopole charge e∗ were present, Glm could affect the phase of electrons via e∗ A∗ . The expression in Eq. (30) indicates why the electric toroidal multipoles Glm is the counterpart of magnetic toroidal multipoles Tlm , and represents a time-reversal-even axial degree of freedom in electrons. B. Electronic band structure

By using the expressions of multipoles in momentum space in Sec. II B, we find the effect of multipoles on the electronic band structure from the microscopic viewpoint. Conversely, we can deduce what type of multipoles are activated when the electronic band structure is determined from the firstprinciples band calculations or detected by the ARPES and de Haas-van Alphen measurements, and so on. A Hamiltonian in condensed matter physics must be totally symmetric and time-reversal even. Therefore the one-electron Hamiltonian must be in the scalar-product form as H=

Q,M,T ,G X

†

kσ σ

†

ext σ σ Xlm Xlm (k)ckσ ckσ ,

(31)

lm

where ckσ (ckσ ) is the creation (annihilation) operator of an electron with the wave vector k and spin σ . Note that

165110-9



TABLE VIII. Representative external fields. u represents a displacement vector field. multipole

P

T

external field

symbol

G ext ext Qext 0 , Q2m ext M Q ext T ext

+

+ + − + −

rotation strain magnetic field electric field electric current

ω = (∇ × u)/2 εij = (∂i uj + ∂j ui )/2 H E J, ∇ × H

−

we assume Xlm in real representation so that the Hermite conjugation is omitted in the scalar product. For example, the lower-rank contributions read from Tables VI and VII as h¯ 2 k2 H= σ0 + Q ext · (k × σ ) + M ext · σ 2m kσ σ † ext ext + T · k σ0 + G0 (k · σ ) + · · · ckσ ckσ , (32) σσ

where it has taken the form Qext ¯ 2 /2m. 0 = h ext Xlm (X = Q, M, T , and G) represent “symmetrybreaking” fields, which bring about symmetry-breaking for ext the c-electron systems. The microscopic origins of Xlm are the external fields applied to the systems, the crystalline electric field (CEF) from ligand ions, and molecular fields originating from multipole-multipole interactions by the spontaneous electronic orderings, and so on. For example, M ext arises from an external magnetic field H or molecular field of ferromagnetic ordering. The representative external fields are summarized in Table VIII. In order to examine the effect on the band structure, the classification according to their spatial-time inversion properties is useful. In the single-band systems, the Hamiltonian in Eq. (31) can be divided as H= εS (k)δσ σ + εA (k)δσ σ kσ σ

† + fσSσ (k) + fσAσ (k) ckσ ckσ ,

(33)

where ε(fσ σ ) is the charge(spin) sector and the superscript S(A) represents symmetric(antisymmetric) contribution with respect to k. As the even-rank Qlm (k), Tlm (k), and the oddrank Mlm (k) are even function of k, and other multipoles are odd function of k, each coefficient is identified as ε (k) = S

even

Qext lm Qlm (k),

ε (k) = A

ext Tlm Tlm (k),

(35)

lm

fσSσ (k) = fσAσ (k) =

odd lm even lm

ext σσ Mlm Mlm (k) +

σσ Gext lm Glm (k) +

even

lm odd lm

type

P

T

multipole

band structure

εS (k) fσSσ (k) fσAσ (k) εA (k)

+

+ − + −

Qext lm (even) ext ext Mlm (odd), Tlm (even) ext Gext (even), Q lm lm (odd) Tlm (odd)

S w/o SS S w/ SS A w/ SS A w/o SS

ext σ σ Tlm Tlm (k),

σσ Qext lm Qlm (k).

(36) (37)

−

When we consider the electronic band structure under symmetry-breaking fields, we apply symmetry operations only to the c-electron systems Xlm (k) without acting on ext Xlm . Thus, for single-band systems, the spatial inversion operation P corresponds to (k, σ0 , σ ) → (−k, σ0 , σ ), while the time-reversal operation T corresponds to (k, σ0 , σ ) → (−k, σ0 , −σ ). The effects of the multipole moments on each part of the electronic structure are summarized in Table IX. We discuss coefficients in Eqs. (34)–(37) one by one below. εS (k) in Eq. (34) represents the symmetric-type band dispersions without the spin splitting. This band structure is present when there are both the spatial inversion and time-reversal symmetries. For example, the kinetic energy of free electron, which is given by εS (k) = h¯ 2 k2 /2m, corresponds to the E monopole in terms of multipole terminology. The higher-rank quadrupole-type deformation in the band structure is attributed to the presence of Qext 2m = ext ext ext ext (Qext u , Qv , Qyz , Qzx , Qxy ). The band deformation caused by Qext xy is shown in Fig. 2, for instance. This quadrupole-type deformation corresponds to orbital (nematic) orderings, which have been discussed in iron-based superconductors [116–121] and Sr 3 Ru2 O7 [122–124]. fσSσ (k) in Eq. (36) represents the symmetric-type band dispersions with the spin splitting. This band structure appears once the time-reversal symmetry is broken while the spatial inversion symmetry is still present. The odd-rank M multipoles and even-rank MT multipoles are categorized in this symmetry group. The band structure shows the spin polarization owing to the time-reversal symmetry breaking. The typical example is the band structure in the ferromagnetic ordering; the M dipole Mz leads to the symmetric spin splitting with the dispersions εσ (k) =

(34)

lm odd

TABLE IX. Interplay between the multipole moments and spacetime classified part of electronic structure. P and T represent the spatial inversion and time-reversal operations corresponding to (k, σ0 , σ ) → (−k, σ0 , σ ) and (k, σ0 , σ ) → (−k, σ0 , −σ ), respectively. S(A) represents the symmetric (antisymmetric) contribution arising in the band structure, and SS is the abbreviation of the spin splitting.

h¯ 2 k2 + σ Hz , 2m

(38)

where Mzext = Hz and σ = ±1. The results indicate a uniform spin splitting in the band structure, as shown in Fig. 2. In the case of the higher-rank multipoles, e.g., the MT quadrupole Txy , the band dispersions are given by 2 h¯ 2 k2 ext + σ Txy kx2 − ky2 + kx2 + ky2 kz2 , (39) εσ (k) = 2m which indicates the quadrupole-type spin splitting, as shown in Fig. 2.

165110-10


Rank

Electric


Magnetic

Magnetic toroidal

Electric toroidal

FIG. 2. Examples of the typical electronic band modulations in the kx -ky plane at kz = 0 in the presence of the E, M, MT, and ET multipoles up to the rank l = 3 in the single-band picture. The blue (red) boxes stand for the band modulations by the even-parity (odd-parity) multipoles. The green arrows represent the in-plane spin moments at k, while the red (blue) curves show the up (down)-spin moments along the out-of-plane direction. The dashed circles in each box represent the energy contours in the absence of the multipole, which corresponds to the energy contour ext for Qext 0 . The superscript “ext” of Xlm is omitted for simplicity.

fσAσ (k) in Eq. (37) represents the asymmetric-type band dispersions (the antisymmetric contribution to the background symmetric band dispersions) with the spin splitting. The odd-rank E multipoles and even-rank ET multipoles contribute to fσAσ (k). This band structure is characterized by the breaking of the spatial inversion symmetry and is often realized in the presence of the spin-orbit coupling. For example, the Rashba-type spin-orbit coupling with the form of ky σx − kx σy corresponds to the E dipole Qz , while the Dresselhaus-type spin-orbit coupling kx (ky2 − kz2 )σx + ky (kz2 − kx2 )σy + kz (kx2 − ky2 )σz corresponds to E octupole Qxyz , as shown in Table VII. In other words, the Rashba-type (Dresselhaus-type) spin-orbit coupling appears when the corresponding odd-parity Q ext (Qext xyz ) is present due to the crystal structure. To be specific, the E dipole component in fσAσ (k) is induced by polar point groups (C4v , C4 , C2v , C2 , Cs , C6v , C6 , C3v , C3 , C1 ), while ET monopole, E dipole, or the ET quadrupole component in fσAσ (k) is induced by gyrotropic point

groups (O, T , D4 , D2d , C4v , C4 , S4 , D2 , C2v , C2 , Cs , D6 , C6v , C6 , D3 , C3v , C3 , C1 ), as discussed in Sec. IV. εA (k) in Eq. (35) represents the asymmetric-type band dispersions with the spin degeneracy. This band structure is obtained when the systems lack both the spatial inversion and time-reversal symmetries. The multipoles in this category are the odd-rank MT ones, which exhibit the odd-order k dispersions. For example, the MT dipole Tz order leads to the k-linear dispersions, ε(k) =

h¯ 2 k2 + Tzext kz , 2m

(40)

whereas the MT octupole leads to the third-order modulations with respect to k; kx (5kx2 − 3k2 ) for Txα ext , which is schematically shown in Fig. 2. Note that asymmetric band modulations are also obtained by combining the multipoles belonging to fσSσ (k) and fσAσ (k). For example, the noncentrosymmetric polar ferromagnets where the M dipole moment lies in the x direction

165110-11



TABLE X. Relation between the components of the linear-response rank-2 tensors χij(2) = χij(J) + χij(E) and relevant multipoles in the presence () or absence (×) of (P, T , PT ), where the superscript J (E) represents the current driven dissipative (electric-field driven nondissipative) part. The superscripts S and A represent the symmetric and antisymmetric components of the linear-response tensor, respectively. The symbol “=” represents that there are no additional contributions from that in the presence of both spatial inversion and time-reversal symmetries. Note that we restrict our discussion only to the electronic contributions in the thermal conductivity although phonons usually contribute to it as well. The Seebeck coefficient corresponds to the symmetric components of βij . tensor

type

electric conductivity thermoelectric conductivity [127–129] thermal conductivity [127–130] magnetoelectric(current) tensor symmetric (S) components antisymmetric (A) components

(P, T , PT ) = (, , )

polar polar polar axial

(Mxext ) with the amplitude Hx and the E dipole moment lies in the z direction (Qext z ) with the amplitude of Ez . In this case, the Hamiltonian is written as h¯ 2 k2 −E (k + ik ) + H z y x x 2m H= , (41) h¯ 2 k2 −Ez (ky − ikx ) + Hx 2m where the eigenvalues are easily obtained as h¯ 2 k2 2 2 ± Ez kx + ky2 + Hx2 − 2Ez Hx ky . (42) εσ (k) = 2m The obtained band dispersions clearly show a shift of the band bottom along the ky direction when Ez Hx = 0, which indicates the emergence of the MT dipole along the y direction T ext ≡ Q ext × M ext = Ez Hx ey , where ey is the unit vector along the y direction. Note that there is no spin degeneracy in this situation because the symmetry of the product PT is broken. This result is also confirmed by the multipole-operator aspect in momentum space, since the direct product of Mx ∝ σx and Qz ∝ kx σy − ky σx , which are given in Tables VI and VII respectively, leads to the functional form of the MT dipole as {σx , kx σy − ky σx } = −2ky σ0 ∝ Ty . Similar discussions can be straightforwardly extended to the antiferromagnets accompanied by spatial-inversion symmetry breaking. For example, the staggered-type antiferromagnetic orderings on the one-dimensional zigzag and two-dimensional honeycomb structures are regarded as the MT dipole and octupole orders, respectively [35,39,48,93]. C. Linear-response tensors

Next, we discuss the cross-correlated phenomena in terms of multipole degrees of freedom. We examine what types of linear responses are expected in the presence of multipoles on the basis of the Kubo formula. In the linear response Aˆ i = j χij Fj , where Fj is an external field along the j direction (i, j = x, y, z), the coupling with the external perturbation is given by Hext = −Cˆ j Fj , and Bˆ j = d Cˆ j /dt, the linear response tensor is generally written by χij(2) = −

i fnk − fmk ψnk |Aˆ i |ψmk ψmk |Bˆ j |ψnk , V εnk − εmk εnk − εmk + iγ k,n,m

(43)

(, ×, ×)

(×, , ×)

(×, ×, )

(×, ×, ×)

σij(J,S) βij(J) κij(J)

+σij(E,A) +βij(E) +κij(E)

– Q0 , Qij Gi

– T0 , Tij Mi

= = = αij(J) G0 , Gij Qi

= = = αij(E) M0 , Mij Ti

both both both both

where V is the system volume, γ is a broadening factor, fmk = f (εmk ) is the Fermi distribution function, and εmk and ψmk are the eigenvalue and eigenstate of the Hamiltonian. m and n are the band indices. This expression does not take account of the orbital motion due to the Lorentz force in the presence of the external magnetic field, which could be important for some response functions such as the normal Hall conductivity [125,126]. Aˆ i and Bˆ j are rank-1 tensor and Hermite operators and χij(2) is rank-2 tensor [the superscript (l) in χij represents the rank-l tensor], although the following discussion is straightforwardly extended to higher-rank operators, such as the E quadrupole representing the elastic property, as discussed in Appendix D. We decompose χij(2) in Eq. (43) into the energy degenerate (J: εnk = εmk ) and nondegenerate (E: εnk = εmk ) parts as χij(2) = χij(J) + χij(E) , χij(J)

= mn Anm γ fnk − fmk ik Bj k =− , V εnk − εmk (εnk − εmk )2 + γ 2 knm

χij(E) = −

= nm mn i (fnk − fmk )Aik Bj k , V (εnk − εmk )2 + γ 2 knm

(44)

(45)

mn ˆ ˆ where Anm ik = ψnk |Ai |ψmk and Bj k = ψmk |Bj |ψnk , and

(fnk − fmk )/(εnk − εmk ) → ∂f/∂εnk . χij(J) is the dissipative (current driven) part, while χij(E) is the nondissipative (electricfield driven) part. As discussed in Appendix C [42], χij(J) and χij(E) have different transformation properties under the timereversal operation, i.e., the latter (former) changes the sign when the time-reversal properties of Aˆ i and Bˆ j are the same (different). On the other hand, the spatial inversion changes the sign of χij(J,E) when the parities of Aˆ i and Bˆ j are different. These facts indicate that different multipoles contribute to χij(J) and χij(E) . The relation between the 2nd-rank linearresponse tensors and relevant multipoles is summarized in Table X. The linear-response (rank-2) tensor χij(2) has nine independent components when Aˆ i and Bˆ j are rank-1 tensors. As will be discussed in Appendix D, each component is related with the multipole degree of freedom. χij(2) is spanned by the

165110-12



multipoles in the following form:

χˆ (2)

⎛ X0 − Xu + Xv ⎜ = ⎝ Xxy − Yz Xzx + Yy

Xxy + Yz X0 − Xu − Xv Xyz − Yx

⎞ Xzx − Yy ⎟ Xyz + Yx ⎠, X0 + 2Xu

Magneto-current effect (Magneto-gyrotropic effect) Magneto-electric effect

Magneto-elastic effect

(46) where X = Q or T (G or M) and Y = G or M (Q or T ) for the polar (axial) 2nd-rank tensor depending on their timereversal property. Among 9 multipoles, rank-0 (X0 ), rank-1 (Yx , Yy , Yz ), and rank-2 (Xu , Xv , Xyz , Xzx , Xxy ) multipoles represent the isotropic component, and antisymmetric and symmetric traceless components, respectively. Since χij(J) and χij(E) have definite parities under spatial inversion and time-reversal operations (χij(J) and χij(E) have opposite time-reversal property), the type of multipoles is determined so as to share the same parities. When the linearresponse tensors χij(J) or χij(E) are both the time-reversal and spatial inversion even, i.e., time-reversal-even polar tensor, the relevant multipoles are the E monopole Q0 , ET dipole Gi , and E quadrupole Qij . When the time-reversal (spatial inversion) operation changes the sign of the linear-response tensors, i.e., time-reversal-odd polar tensor (time-reversal-even axial tensor), they are characterized by the MT monopole T0 , M dipole Mi , and MT quadrupole Tij (ET monopole G0 , E dipole Qi , and ET quadrupole Gij ), which become finite in the absence of the time-reversal (spatial inversion) symmetry. Moreover, the M monopole M0 , MT dipole Ti , and M quadrupole Mij contribute to the time-reversal-odd axial tensors, and it can be finite in the case where there are no time-reversal and spatial inversion symmetries while their product is preserved. In the following, we discuss two fundamental examples of the linear-response tensors: one is the electrical conductivity tensor and the other is the magnetoelectric (current) tensor. We also discuss the other linear-response tensors, such as the spin conductivity tensor and piezoelectric (current) tensor in Appendix D.

Piezo-electric effect

FIG. 3. Schematic picture of a revised Heckmann diagram showing the relation among electric, magnetic, and mechanical properties in solids. E, H, J, and ζkl = εkl + m klm ωm are the external electric field, magnetic field, electric current, and strain-rotation field, while P, M, and σij represent the electric polarization, magnetization, and stress tensor, respectively. The relevant multipoles in the linear-response tensor for each cross-correlated coupling are also shown.

for σˆ (J,S) and σˆ (E,A) are identified as ⎛ Q0 − Qu + Qv Qxy Qxy Q0 − Qu − Qv σˆ (J,S) = ⎝ Qzx Qyz ⎛ ⎞ 0 Mz −My 0 Mx ⎠. σˆ (E,A) = ⎝−Mz My −Mx 0

⎞ Qzx Qyz ⎠, Q0 + 2Qu (48)

The result shows that the anomalous Hall conductivity corresponding to the antisymmetric nondissipative part becomes nonzero in the presence of the M dipole moment, and arises from the interband contribution. 2. Magnetoelectric(current) tensor

Let us consider the magnetoelectric (current) effect where the magnetization is induced by the electric field (current),

1. Electric conductivity tensor

M = αˆ E = (αˆ (J) + αˆ (E) )E.

First, we consider the polar second-rank electric conductivity,

The axial second-rank magnetoelectric (current) tensor αˆ is obtained by using Aˆ i = Mˆ i and Bˆ j = Jˆj in Eq. (43). As Mˆ i is time-reversal odd and spatial inversion even, the dissipative part αˆ (J) becomes nonzero in the absence of only the spatial inversion symmetry. On the other hand, the nondissipative αˆ (E) becomes nonzero only in the absence of both the timereversal and spatial inversion symmetries. Thus αˆ (J) and αˆ (E) are identified as ⎛ ⎞ G0 − Gu + Gv Gxy + Qz Gzx − Qy G0 − Gu − Gv Gyz + Qx ⎠, αˆ (J) = ⎝ Gxy − Qz Gzx + Qy Gyz − Qx G0 + 2Gu ⎛ ⎞ M0 − Mu + Mv Mxy + Tz Mzx − Ty M 0 − Mu − Mv Myz + Tx ⎠. αˆ (E) = ⎝ Mxy − Tz Mzx + Ty Myz − Tx M0 + 2Mu (50)

J = σˆ E = (σˆ (J) + σˆ (E) )E,

(47)

namely, we adopt Aˆ i = Bˆ i = Jî in the general formula (43). As the electric current operator Jî is both time-reversal and spatial-inversion odd, the polar electric conductivity tensor σˆ (E) (electric-field driven nondissipative part) becomes nonzero only when the time-reversal symmetry is broken, while there is no symmetry restriction for the current driven dissipative part σˆ (J) . Meanwhile, from Eqs. (45) and (44), (E,A) and σˆ (E) is the antisymmetric tensor σij(E) = −σj(E) i ≡ σij (J) (J) (J,S) σˆ (J) is the symmetric tensor σij = σj i ≡ σij . According to these facts, the corresponding multipole degrees of freedom

165110-13

(49)



TABLE XI. Finite CEF parameters in terms of the tesseral harmonics under the point group except for triclinic crystals (C1 , Ci ), where (c) (s) , Olm , and Ol (see Table I for the expressions of Ol ). The checkmark in the columns N, P, C, (lm), (lm) , and (l) means the presence of Olm and G represents the noncentrosymmetric (no inversion operation), polar (presence of rank-1 CEFs, see also rank-1 multipoles in Table XVI), chiral (no improper rotations, PCn ), and gyrotropic (presence of less than rank-2 odd-parity multipoles in Table XVI, i.e., presence of χijA ) point groups, respectively. We take the x axis as the C2 rotation axis. For C3v , we take the zx plane as the σv mirror plane. For monoclinic crystals, the standard orientation is taken. crystal system point group cubic

tetragonal

orthorhombic

monoclinic

hexagonal

trigonal

Oh O Td T Th D4h D4 D2d C4v C4h C4 S4 D2h D2 C2v C2h

N

P

C

G

C2

Cs

D6h D6 D3h C6v C6h C6 C3h D3d D3 C3v C3i C3

even-parity CEF (4), (6) (4), (6) (4), (6) (4), (6), (6t ) (4), (6), (6t ) (20), (40), (44), (60), (64) (20), (40), (44), (60), (64) (20), (40), (44), (60), (64) (20), (40), (44), (60), (64) (20), (40), (44), (60), (64), (44) , (64) (20), (40), (44), (60), (64), (44) , (64) (20), (40), (44), (60), (64), (44) , (64) (20), (22), (40), (42), (44), (60), (62), (64), (20), (22), (40), (42), (44), (60), (62), (64), (20), (22), (40), (42), (44), (60), (62), (64), (20), (22), (40), (42), (44), (60), (62), (64), (22) , (42) , (44) , (62) , (64) , (66) (20), (22), (40), (42), (44), (60), (62), (64), (22) , (42) , (44) , (62) , (64) , (66) (20), (22), (40), (42), (44), (60), (62), (64), (22) , (42) , (44) , (62) , (64) , (66) (20), (40), (60), (66) (20), (40), (60), (66) (20), (40), (60), (66) (20), (40), (60), (66) (20), (40), (60), (66), (66) (20), (40), (60), (66), (66) (20), (40), (60), (66), (66) (20), (40), (60), (66), (43) , (63) (20), (40), (60), (66), (43) , (63) (20), (40), (43), (60), (63), (66) (20), (40), (43), (60), (63), (66), (43) , (63) , (66) (20), (40), (43), (60), (63), (66), (43) , (63) , (66)

The isotropic longitudinal magnetoelectric (current) response is realized in the presence of the M (ET) monopole, the antisymmetric transverse response in the presence of the MT (E) dipole, and the symmetric transverse and traceless longitudinal responses in the presence of the M (ET) quadrupoles. The nondissipative αˆ (E) originates from the nondegenerate (interband) contributions, while the dissipative αˆ (J) mainly arises from the intraband contributions. In other words, αˆ (E) plays an important role in insulating systems, while αˆ (J) becomes important for metallic systems. Especially, in the presence of PT symmetry, the latter contribution is forbidden. From the above considerations, the former tensor is called magnetoelectric tensor and the latter tensor is called magne-

odd-parity CEF

(66) (66) (66) (66)

– – (32) (32) – – (54) (32) , (52) (10), (30), (50), (54) – (10), (30), (50), (54), (54) (32), (52), (32) , (52) – (32) , (52) , (54) (10), (30), (32), (50), (52), (54) –

(66)

(10), (30), (32), (50), (52), (54) (32) , (52) , (54) (66) (11), (31), (33), (51), (53), (55) (11) , (31) , (33) , (51) , (53) , (55) – – (33), (53) (10), (30), (50) – (10), (30), (50) (33), (53), (33) , (53) – (33), (53) (10), (30), (33), (50), (53) – (10), (30), (33), (50), (53), (33) , (53)

tocurrent (magnetogyrotropic) tensor, which has essentially the same origin as the so-call Edelstein effect [131,132]. The relations between the cross-correlated responses including the piezoelectric effect and the relevant multipoles are summarized in Fig. 3. IV. POINT-GROUP IRREDUCIBLE REPRESENTATIONS

In the crystal systems, the rotational symmetry and inversion symmetry in some cases, are lost. In such cases, the components of the same rank (the irreducible representation of the rotational group) split into subgroups according to the point-group irreducible representation. In this section, we

165110-14



TABLE XII. Multipoles under cubic, tetragonal, orthorhombic, and monoclinic crystals. The upper and lower columns represent evenparity and odd-parity multipoles, respectively. We take the x axis as the C2 rotation axis. E Q0 , Q4 – Qu , Q4u Qv , Q4v Qα4x Qα4y Qα4z β Qyz , Q4x β Qzx , Q4y β Qxy , Q4z – Qxyz – – Qx , Qαx Qy , Qαy Qz , Qαz Qβx Qβy Qβz

ET

M

MT

– – T0 , T4 Gxyz Mxyz – – – Tu , T4u – – Tv , T4v Gx , Gαx Mx , Mxα T4xα α α Gy , Gy My , My T4yα α α Gz , Gz Mz , Mz T4zα β Gβx Mxβ Tyz , T4x β Gβy Myβ Tzx , T4y β β β Gz Mz Txy , T4z G0 , G4 M0 , M4 – – – Txyz Gu , G4u Mu , M4u – Gv , G4v Mv , M4v – α Gα4x M4x Tx , Txα α Gα4y M4y Ty , Tyα α Gα4z M4z Tz , Tzα β β Gyz , G4x Myz , M4x Txβ β β Gzx , G4y Mzx , M4y Tyβ β β Gxy , G4z Mxy , M4z Tzβ

Oh

O

Td

Th

A1g A2g Eg

A1 A1 Ag A2 A2 Ag E E Eg

T1g

T1

T1

T2g

T2

T2

T

D4h D4 C4h D2d C4v C4 S4 D2h D2 C2v C2h C2

Tg

A A1g A B1g E A1g B1g T Eg

A1 B1 A1 B1 E

Ag Bg Ag Bg Eg

A1 B1 A1 B1 E

A1 B1 A1 B1 E

A B A B E

A B A B E

Tg

T

A2g Eg

A2 E

Ag Eg

A2 E

A2 E

A E

A E

B2g A1u A1 A2 Au A A1u A2u A2 A1 Au A B1u Eu E E Eu E A1u B1u T1u T1 T2 Tu T Eu

B2 A1 B1 A1 B1 E

Bg Au Bu Au Bu Eu

B2 B1 A1 B1 A1 E

B2 A2 B2 A2 B2 E

B A B A B E

B B A B A E

A2u Eu

A2 E

Au Eu

B2 E

A1 E

A E

B E

B2u

B2

Bu

A2

B1

B

A

T2u

T2

T1

Tu

T

Ag Ag Ag Ag B3g B2g B1g B3g B2g B1g Au Au Au Au B3u B2u B1u B3u B2u B1u

A A A A B3 B2 B1 B3 B2 B1 A A A A B3 B2 B1 B3 B2 B1

A1 A1 A1 A1 B2 B1 A2 B2 B1 A2 A2 A2 A2 A2 B1 B2 A1 B1 B2 A1

Ag Ag Ag Ag Bg Bg Ag Bg Bg Ag Au Au Au Au Bu Bu Au Bu Bu Au

A A A A B B A B B A A A A A B B A B B A

Cs A A A A A A A A A A A A A A A A A A A A

discuss such a reduction for multipoles. First, we discuss the CEF potential under all the point groups in Sec. IV A, which is nothing but the sum of the E multipoles in the totally + representation. Then, we show how to classify symmetric A1g multipoles under point-group irreducible representations in Sec. IV B. Such analyses offer a microscopic investigation of potential active multipole degrees of freedom in solids. We show several examples by considering specific basis functions in the tetragonal D2d group. Similar analysis can be applied to any point-group symmetry. For such purpose, we provide various tables for the parent cubic Oh group in the main text and the hexagonal D6h group in Appendix F, and the compatible relations between these parent groups and subgroups in Tables XII and XXVIII.

parity hybridizations for s-p, p-d, d-f , and s-f orbitals at the same site. Note that the odd-parity CEF is present only in the lack of inversion center at lattice sites. The nonzero CEF parameters under each point group except for triclinic crystals (C1 , Ci ) are summarized in Table XI, which is constructed by reading the E multipoles belonging to + A1g representation in the reduction rules in Tables XII and XXVIII in Appendix F. The relevant harmonics are given in Tables I, II, XXII, and XXIII. Now, let us consider a typical example by considering the system with the p-d hybridized orbitals under the D2d group. From Table XI, the atomic CEF Hamiltonian is represented by

A. CEF potential

Under the point-group symmetry, some components of + the E multipoles reduce to the totally symmetric A1g representation. These components constitute the CEF potentials. Therefore the CEF potential is represented by + ∈A1g

V (r ) =

clm Olm (r ),

(51)

lm

where clm is the CEF parameter. Since we consider the largest orbital-angular momentum L = 3 corresponding to the f orbital in the one-electron state, going up to the rank l = 2L = 6 is sufficient in the summation. The even-parity (even-rank) CEF leads to even-parity hybridizations such as s-d and p-f orbitals, while the odd-parity (odd-rank) CEF leads to odd-

even odd HCEF = HCEF + HCEF + H ,

(52)

(c) (c) (c) even = c20 O20 + c40 O40 + c44 O44 , HCEF

(53)

(s) odd = c32 O32 . HCEF

(54)

c52

Note that c60 , c64 , and become zero for the p-d hybridized systems, as the maximum rank is given by 2 max(L1 = even represents the even-parity CEF Hamil1, L2 = 2) = 4. HCEF tonian, which splits three p orbitals (φx , φy , φz ) into two orbitals (φx , φy ) with the irreducible representation E and an orbital φz with the representation B2 ; the five d orbitals (φu , φv , φyz , φzx , φxy ) split into three single orbitals φu with A1 , φv with B1 , and φxy with B2 , and two orbitals (φyz , φzx ) odd with E. Meanwhile, HCEF leads to the odd-parity hybridization between p and d orbitals belonging to the same irreducible representation. In the present case, φz hybridizes with φxy , and (φx , φy ) hybridize with (φyz , φzx ). H represents the atomic energy level for p and d orbitals where the energy difference is taken as = εp − εd .

165110-15


p

PHYSICAL REVIEW B 98, 165110 (2018) B. Active multipoles under point groups

d

(a)

Energy

0.4 0.0 -0.4 -0.8 (b) 0.8

Energy

0.4 0.0 -0.4 0.0

0.2

0.4

0.6

0.8

1.0

FIG. 4. Odd-parity CEF dependencies of the atomic energy level for p-d hybrid orbitals under the tetragonal D2d group. The colors show the weights for p and d orbitals. The difference of the atomic levels for p and d orbitals is taken as (a) = −0.7 and (b) = 0.2, and the CEF parameters are c20 = 1, c40 = 0.5, and c44 = 0.4.

The CEF levels in Eq. (52) are shown in Fig. 4, which largely depends on the model parameters. Strong odd-parity hybridization between φz and φxy is expected for = −0.7 in Fig. 4(a), while that between (φx , φy ) and (φyz , φzx ) occurs for = 0.2 in Fig. 4(b).

When a CEF splitting is large and one of CEF multiplets dominates the low-energy physics, the point-group irreducible representation is suitable to classify multipole moments. The classification is done by using the reduction rules, which are summarized in Tables XII and XXVIII in Appendix F for 32 point groups in seven crystal systems. Tables XII and XXVIII are useful to investigate what type of multipoles are activated in the specific crystal structures, which stimulate microscopic understanding of physical phenomena induced by multipoles, as discussed in Sec. III. For example, the magnetocurrent effect, where the uniform magnetization is induced by electric current as was discussed in Sec. III C, can occur in the point groups (O, T , D4 , C4 , D6 , C6 , D3 , C3 , D2 , D2d , C4v , S4 , C2 , Cs , C6v , C3v , C2v , C1 ), which is the so-called gyrotropic point groups. This is because any of the relevant multipoles for the magnetocurrent effect [G0 , (Qx , Qy , Qz ), and (Gu , Gv , Gyz , Gzx , Gxy )] belong to the totally symmetric representation in the gyrotropic point groups. Moreover, Tables XII and XXVIII show what multipoles are potential order parameters in the systems. Note that as the E (M) and ET (MT) multipoles belong to different irreducible representations in some point groups, the distinction of them should be essential in such point groups. Let us demonstrate how to identify active multipole degrees of freedom in the low-energy multiplets by using the p-d hybridized CEF states in the tetragonal D2d group as was discussed in the previous subsection. First, we consider the basis functions (φyz , φzx ) in the representation E under the D4h group, which is the parent group of D2d and does not show orbital hybridization with different parities (c32 = 0). By taking the direct product of the basis functions, the active multipole degrees of freedom is obtained as + − + ⊕ A2g ⊕ B+ Eg ⊗ Eg = A1g 1g ⊕ B2g ,

(55)

TABLE XIII. Multipoles under the tetragonal crystal system (D4h and related groups) in the spinless basis. We introduce the abbreviation ± ∓ ± ⊕ A2g,u ⊕ B± (g, u)± = A1g,u 1g,u ⊕ B2g,u . The subscripts g, u are omitted for the D4 group. The subscripts 1,2 are omitted for the C4h group. A1u ↔ B1u and A2u ↔ B2u , and then, the subscripts g, u are omitted for the D2d group. A1u ↔ B1u and A2u ↔ B2u , and then, the subscripts g, u and 1,2 are omitted for the S4 group. A1u ↔ A2u and B1u ↔ B2u , and then, the subscripts g, u are omitted for the C4v group. A1u ↔ A2u and B1u ↔ B2u , and then, the subscripts g, u and 1,2 are omitted for the C4 group.

(s) A1g (p) A2u Eu (d) A1g B1g B2g Eg (f) B1u A2u Eu B2u Eu

(s) A1g

(p) A2u

Eu

(d) A1g

B1g

B2g

Eg

(f) B1u

A2u

Eu

B2u

Eu

+ A1g

± A2u + A1g

Eu± Eg± (g)+

± A1g ± A2u Eu± + A1g

B± 1g B± 2u Eu± B± 1g + A1g

B± 2g B± 1u Eu± B± 2g ± A2g + A1g

Eg± Eu± (u)± Eg± Eg± Eg± (g)+

B± 1u B± 2g Eg± B± 1u ± A1u ± A2u Eu± + A1g

± A2u ± A1g Eg± ± A2u ± B2u B± 1u Eu± B± 2g + A1g

Eu± Eg± (g)± Eu± Eu± Eu± (u)± Eg± Eg± (g)+

B± 2u B± 1g Eg± B± 2u ± A2u ± A1u Eu± ± A2g ± B1g Eg± + A1g

Eu± Eg± (g)± Eu± Eu± Eu± (u)± Eg± Eg± (g)± Eg± (g)+

165110-16



TABLE XIV. Multipoles under the tetragonal crystal system (D4h and related groups) in the spinful basis. We introduce the abbreviation ± ∓ ± ∓ ± ⊕ A2g,u ⊕ Eg,u , and g, u± = B± (g, u)± = A1g,u 1g,u ⊕ B2g,u ⊕ Eg,u . The subscripts g, u are omitted for the D4 group. The subscripts 1,2 are omitted for the C4h group. A1u ↔ B1u , A2u ↔ B2u , and E1/2u ↔ E3/2u , and then, the subscripts g, u are omitted for the D2d group. A1u ↔ B1u , A2u ↔ B2u , and E1/2u ↔ E3/2u , and then, the subscripts g, u and 1,2 are omitted for the S4 group. A1u ↔ A2u and B1u ↔ B2u , and then, the subscripts g, u are omitted for the C4v group. A1u ↔ A2u and B1u ↔ B2u , and then, the subscripts g, u and 1,2 are omitted for the C4 group. (s) E1/2g (p) E1/2u E1/2u E3/2u (d) E1/2g E3/2g E3/2g E1/2g E3/2g (f) E3/2u E1/2u E1/2u E3/2u E3/2u E1/2u E3/2u (s) A1g ⊗ E1/2g → E1/2g (p) A2u ⊗ E1/2g → E1/2u Eu ⊗ E1/2g → E1/2u E3/2u (d) A1g ⊗ E1/2g → E1/2g B1g ⊗ E1/2g → E3/2g B2g ⊗ E1/2g → E3/2g Eg ⊗ E1/2g → E1/2g E3/2g (f) B1u ⊗ E1/2g → E3/2u A2u ⊗ E1/2g → E1/2u Eu ⊗ E1/2g → E1/2u E3/2u B2u ⊗ E1/2g → E3/2u Eu ⊗ E1/2g → E1/2u E3/2u

(g)+

(u)± (g)+

(u)± u± (g)± g± (g)+ g± (g)+

(g)± (u)± (u)± u± (g)+

where the superscripts + (−) represent time-reversal even (odd), and all the multipoles are even-parity (the subscript g). Note that the symmetrized (antisymmetrized) product of bases corresponds to time-reversal even(odd) operators when the basis functions are spinless (the angular momentum is an integer). On the other hand, the symmetrized (antisymmetrized) product of the basis functions correspond to time-reversal odd(even) operators when the basis functions are spinful (the angular momentum is a half integer). The above irreducible decomposition for the spinless basis in the tetragonal crystal system is summarized in Table XIII. The corresponding multipoles are given by + : A1g

ρˆ0 ↔ Q0 ,

(56)

− A2g :

ρˆy ↔ Mz ,

(57)

B+ 1g :

ρˆz ↔ Qv ,

(58)

B+ 2g :

ρˆx ↔ Qxy ,

(59)

where ρˆ0 and ρˆ are 2 × 2 unit and Pauli matrices for doubly degenerate orbitals (φyz , φzx ), respectively. Obviously, only even-parity E and M multipoles are active in these CEF multiplets. Next, let us consider the basis functions (φz , φxy ) belonging to the representation B2 under the D2d group. The multipole degrees of freedom are obtained by decomposing the direct product of the basis functions (using Table XIII) as

g± u± u± (u)± g± (g)+

g± u± u± (u)± g± (g)± (g)+

(g)± (u)± (u)± u± (g)± g± g± (g)+

g± u± u± (u)± g± (g)± (g)± g± (g)+

u± g± g± (g)± u± (u)± (u)± u± (u)± (g)+

(u)± (g)± (g)± g± (u)± u± u± (u)± u± g± (g)+

(u)± (g)± (g)± g± (u)± u± u± (u)± u± g± (g)± (g)+

u± g± g± (g)± u± (u)± (u)± u± (u)± (g)± g± g± (g)+

u± g± g± (g)± u± (u)± (u)± u± (u)± (g)± g± g± (g)± (g)+

(u)± (g)± (g)± g± (u)± u± u± (u)± u± g± (g)± (g)± g± g± (g)+

u± g± g± (g)± u± (u)± (u)± u± (u)± (g)± g± g± (g)± (g)± g± (g)+

where (. . . )intra and (. . . )inter represent the multipoles activated in nonhybrid and hybrid orbitals, respectively. A set of independent operators in the basis (φz , φxy ) is given by A1+ :

{τˆ0 , τˆz } ↔ {Q0 , Qu }, A1− :

τˆx ↔ Gv ,

τˆy ↔ Mv ,

(61) (62)

where τˆ0 and τˆ are respectively a 2 × 2 unit matrix and Pauli matrices in the basis (φz , φxy ), and {. . . } represent the appropriate linear combination. τ0 and τz represent the intraorbital degree of freedom, while τx and τy represent the interorbital degree of freedom. The expected linear responses are uniquely found once the multipole degrees of freedom are identified. In this case, the magnetocurrent effect is induced by the emergence of the ET quadrupole Gv ∝ kx σx − ky σy ; the induced magnetization Mx by the electric current in the x direction and My by the electric current in the y direction should have opposite sign and the same magnitude. The result implies that a larger magnetocurrent response is expected for larger ET quadrupole Gv . In a similar way, the magnetoelectric effect is anticipated once the time-reversal symmetry breaking occurs, as the antisymmetrized representation is the M quadrupole Mv . In the spinful basis, possible multipoles are obtained by changing the irreducible representation of the basis function as B2 ⊗ E1/2 → E3/2 , and the decomposition according to Table XIV gives

(B2 ⊕ B2 ) ⊗ (B2 ⊕ B2 ) = (2A1+ )intra ⊕ (A1+ ⊕ A1− )inter , (60) 165110-17

(E3/2 ⊕ E3/2 ) ⊗ (E3/2 ⊕ E3/2 ) = (2A1+ ⊕ 2A2− ⊕ 2E− )intra ⊕ (A1+ ⊕ A2+ ⊕ E+ ⊕ A1− ⊕ A2− ⊕ E− )inter .

(63)



TABLE XV. Even-parity multipoles belonging to a totally symmetric representation in the point groups except for the triclinic ones (C1 , Ci ). χijP and χijAk represent rank-2 polar and rank-3 axial tensors, respectively. χijP (χij(2) ) has 9 components and χijAk has 27 components at most. The components of χijAk are decomposed into a symmetric monopole response denoted by 1 with 3 components (χi(1) ), antisymmetric dipole response denoted by 2 with nine components (χij(2) ), and symmetric quadrupole response denoted by 3 with 15 components (χij(3)k ), respectively. The independent number of components in χijP and χijAk is shown for each point group. Responses in the absence of time-reversal symmetry, such as the magnetoelectric effect and Nernst effect, are obtained by replacing (Qlm , Glm ) with (Tlm , Mlm ). χijP

totally symmetric rep. crystal system point group cubic

Oh O Td T Th D4h D4 D2d C4v C4h C4 S4 D2h D2 C2v C2h C2 Cs D6h D6 D3h C6v C6h C6 C3h D3d D3 C3v C3i C3

tetragonal

orthorhombic

monoclinic

hexagonal

trigonal

N

P

C

G

rank 0 rank 1

Q0 Q0 Q0 Q0 Q0 Q0 Q0 Q0 Q0 Q0 Q0 Q0 Q0 Q0 Q0 Q0 Q0 Q0 Q0 Q0 Q0 Q0 Q0 Q0 Q0 Q0 Q0 Q0 Q0 Q0

rank 2

Gxyz Gxyz

Gz Gz Gz

Gz Gz Gz

Gz Gz Gz

Gz Gz

Qu Qu Qu Qu Qu Qu Qu Qu , Qv Qu , Qv Qu , Qv Qu , Qv , Qxy Qu , Qv , Qxy Qu , Qv , Qxy Qu Qu Qu Qu Qu Qu Qu Qu Qu Qu Qu Qu

A set of independent operators in the basis (φz , φxy ) is given by A1+ : {τˆ0 σˆ 0 , τˆz σˆ 0 } ↔ {Q0 , Qu },

E+ :

E− :

τˆx σˆ 0 ↔ Gv ,

(64)

A2+ : τˆy σˆ z ↔ Gxy ,

(65)

(τˆy σˆ x , τˆy σˆ y ) ↔ (Qx , Qy ),

(66)

A1− :

(67)

τˆy σˆ 0 ↔ Mv ,

A2− : {τˆ0 σˆ z , τˆz σˆ z } ↔ {Mz , Mzα },

τˆx σˆ z ↔ Mxy ,

(68)

{(τˆ0 σˆ x , τˆ0 σˆ y ), (τˆz σˆ x , τˆz σˆ y )} ↔ {(Mx , My ), (Mxα , Myα )}, (τˆx σˆ x , τˆx σˆ y ) ↔ (Tx , Ty ),

rank 3

(69)

Gαz Gαz Gαz Gxyz Gxyz Gxyz Gxyz , Gαz , Gβz Gxyz , Gαz , Gβz Gxyz , Gαz , Gβz

Gαz Gαz Gαz G3a G3a G3b Gαz , G3a , G3b Gαz , G3a , G3b

χijAk

rank 4

2

1 2

3

Q4 Q4 Q4 Q4 Q4 Q4 , Q4u Q4 , Q4u Q4 , Q4u Q4 , Q4u Q4 , Q4u , Qα4z Q4 , Q4u , Qα4z Q4 , Q4u , Qα4z Q4 , Q4u , Q4v Q4 , Q4u , Q4v Q4 , Q4u , Q4v Q4 , Q4u , Q4v , Qα4z , Qβ4z β Q4 , Q4u , Q4v , Qα4z , Q4z β Q4 , Q4u , Q4v , Qα4z , Q4z Q40 Q40 Q40 Q40 Q40 Q40 Q40 Q40 , Q4a Q40 , Q4a Q40 , Q4b Q40 , Q4a , Q4b Q40 , Q4a , Q4b

1 1 1 1 1 2 2 2 2 3 3 3 3 3 3 5 5 5 2 2 2 2 3 3 3 2 2 2 3 3

1 1 1 1 1 2 2 2 2 3 3 3 3 3 3 5 5 5 2 2 2 2 3 3 3 2 2 2 3 3

1 1 1 1 1 1 3 3 3 3 3 3 7 7 7 1 1 1 1 3 3 3 2 2 2 5 5

1 1 1

1 1 1

1 1 1

1 1

In the case of the spinful basis functions, several oddparity multipole degrees of freedom are potentially activated in addition to Mv and Gv in the spinless basis functions. For example, the MT dipole (Tx , Ty ), which is the origin of the magnetoelectric effect, can be a primary order parameter when the thermodynamic averages of (τˆx σˆ x , τˆx σˆ y ) become nonzero. Interestingly, it is also possible to activate the MT dipole (Tx , Ty ) by an external magnetic field or spontaneous magnetic ordering in the xy plane, since the M dipoles (Mx , My ) belong to the same irreducible representation as (Tx , Ty ). This fact indicates that E dipole Q is not necessary for the emergence of the MT dipole T , which has never been clarified in previous interpretations of the MT dipole. Similarly, the M quadrupole Mxy becomes active under an external magnetic field along the z direction.

165110-18



TABLE XVI. Odd-parity multipoles belonging to a totally symmetric representation in the point groups except for the triclinic ones (C1 , Ci ) and other centrosymmetric point groups. χijA and χijP k represent rank-2 axial and rank-3 polar tensors, respectively. χijA has nine components and χijP k has 27 components at most. Note that the polar point groups (P) have rank-1 E multipoles ( Q), the chiral point groups (C) have rank-0 ET multipoles (G0 ), and the gyrotropic point groups (G) have less than or equal to rank-2 E/ET multipoles (G0 , Q, G2m ) yielding nonzero χijA . The gyrotropic point groups with only rank-1 E dipoles ( Q) in (C4v , C6v , C3v ) are called the weak gyrotropic point groups, which do not show a natural optical rotation. χijA

χijP k

rank 4

2

1 2 3

G4

1

1

G4 G4 , G4u G4v Gα4z G4 , G4u , Gα4z β G4v , G4z G4 , G4u , G4v β Gα4z , G4z G4 , G4u , G4v , Gα4z , Gβ4z β β Gα4x , Gα4y , G4x , G4y G40 G4a

1 2 1 1 3 2 3 2 5 4 2

1 2 1 1 3 2 3 2 5 4 2

totally symmetric rep. crystal system point group N cubic

O Td T D4 D2d C4v C4 S4 D2 C2v C2 Cs D6 D3h C6v C6 C3h D3 C3v C3

tetragonal

orthorhombic monoclinic hexagonal

trigonal

P

C G rank 0

rank 1

rank 2

rank 3

G0

Qxyz Qxyz

G0 G0

G0

Gu Gv Qz Qz

G0 G0

Qz Qz Qx , Q y

G0

G0

Gu Gv , Gxy Gu , Gv Gxy Gu , Gv , Gxy Gyz , Gzx Gu

Qz Qz

Gu

G0 G0

Gu Qz Qz

Gu

The final example is the basis functions (φx , φy , φyz , φzx ) belonging to the E representation under the D2d group. The direct product of the basis functions is given by

Qxyz Qαz Qαz Qxyz , Qβz Qxyz Qαz , Qβz Qxyz , Qαz , Qβz Qαx , Qαy , Qβx , Qβy Q3a Qαz Qαz Q3a , Q3b Q3a Qαz , Q3a Qαz , Q3a , Q3b

G40 G4a , G4b G40 , G4a G4a G40 , G4a , G4b

1 1

1 1 2

1 3

1 1 1 3

2 1 3

2 1 1 1 3

1 1 1 2 2 3 4 3 4 7 8 1 1 2 3 2 2 3 5

(φx , φy , φyz , φzx ), there are several odd-parity multipole degrees of freedom even in the spinless basis functions, such as the E dipole Qz and MT dipole Tz .

(E ⊕ E) ⊗ (E ⊕ E) + + + = (2A1+ ⊕ 2A2− ⊕ 2B+ 1 ⊕ 2B2 )intra ⊕ (A1 ⊕ A2

⊕

B+ 1

⊕

B+ 2

⊕

A1−

⊕

A2−

⊕

B− 1

⊕

B− 2 )inter .

C. Linear response tensors

(70)

This gives us a set of independent operators as A1+ :

{ρˆ0 τˆ0 , ρˆ0 τˆz } ↔ {Q0 , Qu }, A2+ :

ρˆ0 τˆx ↔ Gv ,

ρˆy τˆy ↔ Gxy ,

ρˆz τˆx ↔ Gu , B+ 1 : {ρˆz τˆ0 , ρˆz τˆz } ↔ {Qv , Q4v }, β ρˆx τˆx ↔ Qz , B+ 2 : {ρˆx τˆ0 , ρˆx τˆz } ↔ Qxy , Q4z , A1− :

A2− :

ρˆ0 τˆy ↔ Mv , {ρˆy τˆ0 , ρˆy τˆz } ↔ Mz , Mzα , ρˆy τˆx ↔ Mxy ,

(71) (72) (73) (74) (75) (76)

B− 1 :

ρˆz τˆy ↔ Mu ,

(77)

B− 2 :

ρˆx τˆy ↔ Tz ,

(78)

where τˆ0 and τˆ are the unit and Pauli matrices acting on the different parity orbitals. In the case of the basis functions

Finally, we show nonzero rank-2 and rank-3 tensors in each point group in terms of totally symmetric even-parity multipoles in Table XV and odd-parity multipoles in Table XVI. As described in Sec. III C, the rank-2 tensor χij(2) has nine independent components, which are characterized by the rank-0, rank-1, and rank-2 multipoles. Similarly, as discussed in Appendix D, the rank-2 tensor response by the rank-1 external field is described by the rank-3 tensor χij k with 27 components, which are decomposed into a symmetric monopole response tensor with 3 components χi(1) , antisymmetric dipole response tensor with 9 components χij(2) , and symmetric quadrupole response tensor with 15 components χij(3)k . Moreover, the components of χˆ (1,2,3) are characterized by multipoles. For example, in the case of D2d group, one component in the rank-2 axial tensor and three components of the rank-3 polar tensor become nonzero, as odd-parity multipoles Gv and Qxyz belong to the totally symmetric representation in D2d group as shown in Table XVI. Nonzero χijA and χijP k imply the emergence of the magnetoelectric effect,

165110-19



TABLE XVII. Representative polar (P) and axial (A) tensors in condensed matter physics. The odd-parity tensors in the lower rows can be finite only when the spatial inversion symmetry is broken. tensor χiA χijP χijAk χijP kl χiP χijA χijP k χijAkl

P

rank

+ magnetocaloric coefficient (thermo)electric/thermal conductivity spin conductivity, Nernst coefficient elastic stiffness tensor − electrocaloric coefficient linear magnetoelectric tensor piezoelectric tensor third-order magnetoelectric tensor

1 2 3 4 1 2 3 4

piezoelectric tensor, and so on. We summarize the representative tensors in condensed matter physics in Table XVII. V. SUMMARY

In summary, we have investigated a general description of multipoles from the microscopic viewpoint. We have presented a definition of four multipoles in both real and momentum spaces, and how to apply to 32 point groups in seven crystal systems. We have demonstrated which multipole degrees of freedom become active in the tetragonal D2d group as an example. We showed that physical properties in electron systems, such as the electromagnetic fields, band structures, and linear responses, are closely related with the multipoles, and hence, the multipole formulation gives a comprehensive and systematic understanding of physical phenomena in condensed matter physics at the microscopic level. Such a comprehensive investigation of multipoles could stimulate an identification of unknown order parameters, such as MT and ET multipoles, and a further exploration of cross-correlated couplings through multipole-multipole interactions, since we present more than 30 tables as a useful reference in order to cover most cases in all point groups.

for three p orbitals, √ 5 1 3z2 − r 2 5 3 x2 − y2 , φv = , φu = 4π 2 r2 4π 2 r2 5 √ yz 5 √ zx 3 2 , φzx = 3 2, (A3) φyz = 4π r 4π r 5 √ xy 3 2, φxy = 4π r for five d orbitals, and 7 √ xyz 7 1 x(5x 2 − 3r 2 ) φxyz = 15 3 , φxα = , 4π r 4π 2 r3 7 1 y(5y 2 − 3r 2 ) 7 1 z(5z2 − 3r 2 ) α α φy = , φ = , z 4π 2 r3 4π 2 r3 √ √ 7 15 x(y 2 − z2 ) 7 15 y(z2 − x 2 ) β β φx = , φy = , 4π 2 r3 4π 2 r3 √ 7 15 z(x 2 − y 2 ) φzβ = . (A4) 4π 2 r3 for seven f orbitals. They are proportional to E multipoles in the cubic Oh representation in Tables I and II. APPENDIX B: MOMENTUM-SPACE MULTIPOLES IN MULTIORBITAL SYSTEMS

As mentioned in Sec. II B, the extension of the momentumspace multipoles to multiorbital systems is not unique. Here, we discuss one possible extension. As was discussed, in the single-band systems, the odd-rank M and even-rank ET multipoles are given by σ · ∇ k Olm (k) in Eqs. (23) and (24). By applying the time-reversal-conversion operator (σ × k) · ∇ k , we obtain the expressions for the oddrank ET and even-rank M multipoles in multiorbital systems as

ACKNOWLEDGMENTS

We thank H. Harima and Y. Motome for fruitful discussions. This research was supported by JSPS KAKENHI Grants No. JP15K05176, No. JP15H05885, No. JP18H04296 (J-Physics), No. JP16H06590, and No. JP18K13488.

φy =

3 y , 4π r

Mlm = (σ × k) · ∇ k [σ · ∇ k Olm (k)]

(even).

Mlm

ik · σ = ik · ∇ k [σ · ∇ k Olm (k)]

Glm = ik · ∇ k [σ · ∇ k Olm (k)]

We show the atomic spinless basis functions for s, p, d, and f orbitals as a function of angles rˆ /r, which are used in the main text [133]. The basis functions are given by

for an s orbital, 3 x , φx = 4π r

(odd),

These expressions are simplified by simple algebra as

APPENDIX A: ATOMIC BASIS WAVE FUNCTIONS

1 φ0 = √ , 4π

Glm = (σ × k) · ∇ k [σ · ∇ k Olm (k)]

(l = 0) , (l = 2, 4, . . . )

(B1)

(l = 1, 3, . . . ),

(B2)

where we have used ∇ 2 Olm = 0. The specific form of the even-rank M and odd-rank ET multipoles up to l = 4 and the MT monopole in momentum space is shown in Tables VI and VII.

(A1) APPENDIX C: SPACE-TIME INVERSION OPERATIONS OF LINEAR RESPONSE TENSOR

φz =

3 z , 4π r

(A2)

We discuss the space-time inversion properties of linear response tensors, which is also discussed in Ref. [42]. As discussed in Sec. III C, the linear response tensors in Eq. (43)

165110-20



are generally decomposed into =−

χ = χ (J) + χ (E) , χ (J) = −

= mn Anm γ fnk − fmk k Bk , V εnk − εmk (εnk − εmk )2 + γ 2 knm

(C1)

= mn i (fnk − fmk )Anm k Bk =− , V (εnk − εmk )2 + γ 2 knm

(C2)

χ (E)

ˆ = t A, ˆ T (A)

ˆ P (Bˆ ) = s B,

ˆ T (Bˆ ) = t B,

(s, t = ±1),

(s , t = ±1).

(C4)

= mn sAnm γ fn−k − fm−k k s Bk V εn−k − εm−k (εn−k − εm−k )2 + γ 2 knm

Due to the above symmetry properties, in the case of s = s a breaking of the spatial inversion is necessary to obtain a finite χ (J,E) . Similarly, in the case of t = t (t = t ), a breaking of time-reversal symmetry is necessary to obtain a finite χ (J) (χ (E) ). APPENDIX D: OTHER LINEAR RESPONSE TENSORS

In this section, we show a natural extension of linearresponse tensors as discussed in Sec. III C. In the main text, we discuss the rank-2 linear-response tensors χij(2) with nine independent components when Aˆ i and Bˆ j are both rank-1 tensors. The extension of the linear response tensors when Aˆ i and Bˆ j are higher-rank tensors is straightforward. Let us consider the rank-2 response to the rank-1 external field, namely, Aij = χij k Bk . (D1)

l

where we have used εn−k = εnk . A similar discussion holds for χ (E) , and we obtain χ (E) = ss χ (E) .

(C5)

On the other hand, in the presence of the time-reversal symmetry, we have = ¯ n¯ n¯ m ¯ tAm γ fnk − fmk −k t B−k 2 V εnk − εmk (εnk − εmk ) + γ 2 knm

where ij l is the totally antisymmetric tensor (Levi-Civita D symbol), and AQ ij is symmetric and traceless. Note that Ai is polar (axial) when others are axial (polar) since ij k and δij are axial and polar tensors, respectively, and AM , AD i , and AQ eventually have the same parity. For each contribution, the ij linear-response tensor is introduced as (1) 1 Aii = χk B k , 3 i k (2) 1 ij k Aj k = χik Bk AD i = 2 jk k (3) χij k Bk , AQ ij = AM =

= mn tAnm − fn−k γ fm−k ¯ ¯ k t Bk =− V εm−k − εn−k (εm−k )2 + γ 2 − εn−k ¯ ¯ ¯ ¯ knm

=−

= mn Anm γ fmk − fnk k Bk tt V εmk − εnk (εmk − εnk )2 + γ 2 knm

= tt χ (J) , where we have used εn−k = εnk . Similarly, ¯ χ (E) = −

=−

(C6)

We can decompose Aij into the monopole, dipole, and quadrupole contributions as Q Aij = δij AM + ij l AD (D2) l + Aij ,

= ss χ (J) ,

χ (J) = −

χ (E) = −tt χ (E) .

k

= mn Anm γ fnk − fmk k Bk =− ss V εnk − εmk (εnk − εmk )2 + γ 2 knm

χ (J) = ss χ (J) ,

χ (J) = tt χ (J) ,

(C3)

Moreover, for the Bloch states, P|nk = |n − k and T |nk = |n¯ − k, where n¯ is the time-reversal partner of n state. As T is antiunitary, these relations lead to Anm k = ¯ n¯ ¯ nm nm mn m nm n¯ m sAnm , A = tA , B = s B , and B = t B . −k k k −k k −k −k Thus, in the presence of the spatial inversion symmetry, we show that = mn sAnm γ fnk − fmk −k s B−k (J) χ =− V εnk − εmk (εnk − εmk )2 + γ 2 knm =−

= −tt χ (E) . Thus we obtain

where Aˆ and Bˆ are Hermite operators and these equations correspond to Eqs. (44) and (45), respectively. We assume that Aˆ and Bˆ have definite parties with respect to the spatial inversion and time-reversal operations as ˆ = s A, ˆ P (A)

= mn i (fmk − fnk )Anm k Bk tt 2 2 V (ε ) − ε + γ mk nk knm

= ¯ n¯ n¯ m ¯ i (fnk − fmk )tAm −k t B−k 2 V (εnk − εmk ) + γ 2 knm

= mn − fn−k )tAnm i (fm−k ¯ ¯ k t Bk V (εm−k )2 + γ 2 − εn−k ¯ ¯ knm

(D3) (D4) (D5)

k

where χij(3)k is symmetric and traceless with respect to (i, j ), (3) (2) i.e., χij(3)k = χj(3) i χiik = 0. Note that χij corresponds ik and to Eq. (46) in the main text. By inserting these expressions into the decomposition of Aij , we obtain the relation to χij k as (1) (2) (3) ij l χlk + χij k Bk . (D6) δij χk + Aij = k

165110-21

l



where χijQ is symmetric and traceless, and χijOk is totally symmetric with respect to any permutations of indices and O D i χiik = −3χk . Since χ M , χkD , χijQ , χijOk are characterized by Y0 (monopole), Xi (dipole), Yij (quadrupole), and Xij k (octupole), respectively, where X is polar (axial) and Y is axial (polar) when χij(3)k is polar (axial), we explicitly express the linearresponse functions in terms of multipoles as

Each linear-response function can be decomposed as

χij(2)

χk(1) = χkD , = δij χ M + ij k χkD + χijQ ,

χij(3)k = δij χkD +

(D7) (D8)

k

Q (ikl δj m + j kl δim )χlm + χijOk ,

(D9)

lm

χ (1) = (Xx

⎛

⎛

χ (3)

Y0 − Yu + Yv χ (2) = ⎝ Yxy − Xz Yzx + Xy β

Xy

Xz ),

(D10) ⎞

Yxy + Xz Y0 − Yu − Yv Yyz − Xx

Yzx − Xy Yyz + Xx ⎠, Y0 + 2Yu β

Xx + Yyz − Xxα − Xx

⎜ ⎜ −3Xx + Yyz + 3Xα − Xxβ x ⎜ ⎜ ⎜ =⎜ −3Yu − Yv + Xxyz ⎜ ⎜−3Xz + Yxy − 2Xα + 2Xzβ z ⎝ β −3Xy − Yzx − 2Xyα − 2Xy

Xy − Yzx − Xyα + Xy

−2Xz + 2Xzα

β

⎞

⎟ ⎟ ⎟ ⎟ β⎟ α −3Xy + Yzx − 2Xy + 2Xy ⎟, ⎟ β −3Xx − Yyz − 2Xxα − 2Xx ⎟ ⎠ 2Yv + Xxyz β

3Xy + Yzx − 3Xyα − Xy

−2Yxy + 2Xz

β

−3Xz − Yxy − 2Xzα − 2Xz 3Yu − Yv + Xxyz

β

−3Xx + Yyz − 2Xxα + 2Xx

(D11)

(D12)

where we have put χ M = Y0 , χiD = χiD = Xi , χijQ = χijQ = Yij , χijOk = Xij k , and χiD = 5Xi for notational simplicity. We have also used the relations 1 1 Yii = 0, Yu = (2Yzz − Yxx − Yyy ), Yv = (Yxx − Yyy ), (D13) 6 2 i for quadrupoles and

Xiik = −15Xk

(k = x, y, z),

i

Xxα = Xyα = Xzα = Xxβ = Xyβ = Xzβ =

1 (2Xxxx − 3Xyyx − 3Xzzx ), 20 1 (2Xyyy − 3Xzzy − 3Xxxy ), 20 1 (2Xzzz − 3Xxxz − 3Xyyz ), 20 1 (Xyyx − Xzzx ), 4 1 (Xzzy − Xxxy ), 4 1 (Xxxz − Xyyz ), 4

(D14)

for octupoles. The prefactors in the right-hand sides are chosen in order to simplify the resultant expressions in the linear-response (3) (3) (3) (3) = (2χzzk − χxxk − χyyk )/6 tensors. In the expression of χ (3) , the rows correspond to (i, j ) = (u, v, yz, zx, xy), e.g., χuk according to Eq. (D13). In order to complete the comprehensive lists, we also give the expression of χ (3) in the hexagonal D6h group as ⎞ ⎛ Xx + Yyz + 2X3u Xy − Yzx + 2X3v −2Xz + 2Xzα ⎟ ⎜ β ⎟ ⎜ −3Xx + Yyz + X3a − X3u 3Xy + Yzx + X3b + X3v −2Yxy + 2Xz ⎟ ⎜ ⎟ ⎜ β (3) α α (D15) χ =⎜ −3Yu − Yv + Xxyz −3Xz − Yxy − 2Xz − 2Xz −3Xy + Yzx − 2Xy + 4X3v ⎟, ⎟ ⎜ ⎟ ⎜ β 3Yu − Yv + Xxyz −3Xx − Yyz − 2Xxα + 4X3u ⎠ ⎝−3Xz + Yxy − 2Xzα + 2Xz −3Xy − Yzx + X3b − X3v

−3Xx + Yyz − X3a − X3u 165110-22

2Yv + Xxyz


where we have used the relations


X3u = − 21 Xxα + Xxβ , X3v = − 21 Xyα − Xyβ , X3a = 21 5Xxα − 3Xxβ , X3b = − 21 5Xyα + 3Xyβ .

(D16)

We present two examples in the following: spin conductivity and piezoelectric tensors. 1. Spin conductivity tensor

We here consider the axial third-rank spin conductivity tensor, which is defined as σijs k Ek , Jijs =

(D17)

k

where Aˆ ij = Jîjs and Bˆ j = Jˆj in the general formula in Eq. (43). Note that there are 9 independent components for the spin current tensor, as it is characterized by the product of rank-1 tensors, Jî and σˆ j (σˆ is the spin degree of freedom). From the symmetry point of view, the spin conductivity tensor is decomposed as ij k Jks,D + Jijs,Q , (D18) Jijs ≡ Ji σj = δij J s,M + k

where

J s,M =

1 J · σ, 3

J s,D =

1 ( J × σ ), 2

Jˆs,Q

⎞ ⎛ [2Jz σz − Jx σx − Jy σy ]/3 Jx σx − Jy σy ⎟ 1⎜ ⎟ ⎜ Jy σz + Jz σy = ⎜ ⎟. ⎠ 2⎝ Jz σx + Jx σz Jx σy + Jy σx

(D19)

J s,M , J s,D , and Jˆs,Q possess the same symmetry as ET monopole, E dipole, and ET quadrupole, respectively. The corresponding tensors σˆ s(1) , σˆ s(2) , and σˆ s(3) are expressed as σˆ s(1,J) = (Mx My Mz ), σˆ s(1,E) = (Gx Gy Gz ), ⎛ ⎞ T0 − Tu + Tv Txy + Mz Tzx − My T0 − Tu − Tv Tyz + Mx ⎠, σˆ s(2,J) = ⎝ Txy − Mz Tzx + My Tyz − Mx T0 + 2Tu ⎛ ⎞ Q0 − Qu + Qv Qxy + Gz Qzx − Gy Q0 − Qu − Qv Qyz + Gx ⎠, σˆ s(2,E) = ⎝ Qxy − Gz Qzx + Gy Qyz − Gx Q0 + 2Qu and

⎛

β

Mx + Tyz − Mxα − Mx ⎜ −3M + T + 3M α − M β x x yz ⎜ x ⎜ σˆ s(3,J) = ⎜ −3Tu − Tv + Mxyz ⎜ ⎝−3Mz + Txy − 2Mzα + 2Mzβ β −3My − Tzx − 2Myα − 2My ⎛ β Gx + Qyz − Gαx − Gx ⎜ −3G + Q + 3Gα − Gβ x x yz ⎜ x ⎜ σˆ s(3,E) = ⎜ −3Qu − Qv + Gxyz ⎜ ⎝−3Gz + Qxy − 2Gαz + 2Gβz β −3Gy − Qzx − 2Gαy − 2Gy

β

My − Tzx − Myα + My β 3My + Tzx − 3Myα − My β α −3Mz − Txy − 2Mz − 2Mz 3Tu − Tv + Mxyz β −3Mx + Tyz − 2Mxα + 2Mx β

Gy − Qzx − Gαy + Gy β 3Gy + Qzx − 3Gαy − Gy β −3Gz − Qxy − 2Gαz − 2Gz 3Qu − Qv + Gxyz β −3Gx + Qyz − 2Gαx + 2Gx

⎞ −2Mz + 2Mzα β ⎟ −2Txy + 2Mz ⎟ β⎟ α −3My + Tzx − 2My + 2My ⎟, β⎟ −3Mx − Tyz − 2Mxα − 2Mx ⎠ 2Tv + Mxyz ⎞ −2Gz + 2Gαz β ⎟ −2Qxy + 2Gz ⎟ β⎟ −3Gy + Qzx − 2Gαy + 2Gy ⎟. β⎟ −3Gx − Qyz − 2Gαx − 2Gx ⎠ 2Qv + Gxyz

(D20)

(D21)

(D22)

2. Piezoelectric tensor

Next, we consider a piezoelectric effect where the strain and rotation is induced by the electric field (or current). The polar third-rank piezoelectric tensor dij k is given by ζij = dij k Ek . (D23) k

165110-23



ζij ≡ ∂ui /∂xj , where u is a displacement vector field. It can be decomposed as ζij = δij ε0 + ij k ωk + ε˜ ij ,

(D24)

k

where ε0 = (∇ · u)/3, ω = (∇ × u)/2, and εij = (ζij + ζj i )/2 = δij ε0 + ε˜ ij represent the bulk modulus, rotation, and symmetric strain, respectively. Since ε0 , (ωx , ωy , ωz ), and (εu , εv , εyz , εzx , εxy ) are the same symmetry as E monopole, ET dipole, and E quadrupole, respectively, the corresponding tensors dˆ (1) , dˆ (2) , and dˆ (3) are given by dˆ (1,J) = (Tx ˆ (1,E)

d ⎛

= (Qx

M0 − Mu + Mv dˆ (2,J) = ⎝ Mxy − Tz Mzx + Ty ⎛ G0 − Gu + Gv dˆ (2,E) = ⎝ Gxy − Qz Gzx + Qy and

Ty Qy

Tz ), Qz ),

Mxy + Tz M 0 − Mu − Mv Myz − Tx Gxy + Qz G0 − Gu − Gv Gyz − Qx

(D25) ⎞ Mzx − Ty Myz + Tx ⎠, M0 + 2Mu ⎞ Gzx − Qy Gyz + Qx ⎠, G0 + 2Gu

(D26)

⎛

⎞ β β Tx + Myz − Txα − Tx Ty − Mzx − Tyα + Ty −2Tz + 2Tzα β β ⎜ −3T + M + 3T α − T β ⎟ 3Ty + Mzx − 3Tyα − Ty −2Mxy + 2Tz x x yz ⎜ ⎟ x ⎜ ⎟ β β (3,J) dˆ = ⎜ −3Mu − Mv + Txyz −3Tz − Mxy − 2Tzα − 2Tz −3Ty + Mzx − 2Tyα + 2Ty ⎟, ⎜ β⎟ ⎝−3Tz + Mxy − 2Tzα + 2Tzβ 3Mu − Mv + Txyz −3Tx − Myz − 2Txα − 2Tx ⎠ β β −3Ty − Mzx − 2Tyα − 2Ty −3Tx + Myz − 2Txα + 2Tx 2Mv + Txyz ⎞ ⎛ β β Qx + Gyz − Qαx − Qx Qy − Gzx − Qαy + Qy −2Qz + 2Qαz β β ⎟ ⎜ −3Q + G + 3Qα − Qβ 3Qy + Gzx − 3Qαy − Qy −2Gxy + 2Qz x x yz ⎟ ⎜ x ⎜ β β⎟ dˆ (3,E) = ⎜ −3Gu − Gv + Qxyz −3Qz − Gxy − 2Qαz − 2Qz −3Qy + Gzx − 2Qαy + 2Qy ⎟. ⎜ β⎟ ⎝−3Qz + Gxy − 2Qαz + 2Qβz 3Gu − Gv + Qxyz −3Qx − Gyz − 2Qαx − 2Qx ⎠ β β −3Qy − Gzx − 2Qαy − 2Qy −3Qx + Gyz − 2Qαx + 2Qx 2Gv + Qxyz

(D27)

For example, the rotation by the electric field, i.e., nonzero dˆ (2,E) , occurs for the gyrotropic point groups, and the symmetric strain by the electric field, i.e., nonzero dˆ (3,E) , occurs for the noncentrosymetric point groups, as shown in Table XVI.

APPENDIX E: Oh AND ITS SUBGROUPS

We summarize the active multipoles for the basis for s, p, d, and f orbitals with/without the spin degree of freedom for the Oh and its subgroups. The active multipoles for cubic and orthorhombic crystals in the spinless (spinful) basis are shown in Table XVIII (XIX) and XX (XXI), respectively. The active multipoles for tetragonal crystals in the spinless (spinful) basis are shown in Table XIII (XIV) in the main text. ± ∓ ± ± TABLE XVIII. Multipoles under the cubic crystal system (Oh ) in the spinless basis. (g, u)± = A1g,u ⊕ Eg,u ⊕ T1g,u ⊕ T2g,u ; g, u± = ± ± ± ± ⊕ Eg,u ⊕ T1g,u ⊕ T2g,u . The superscript ± includes both time-reversal-even (+ ) and time-reversal-odd (− ) operators. The subscripts g, u A2g,u are omitted for the O group. A1u ↔ A2u and T1u ↔ T2u for the Td group. The subscripts 1,2 are omitted for the Th group. The subscripts g, u, 1, 2 are omitted for the T group.

(s) A1g (p) T1u (d) Eg T2g (f) A2u T1u T2u

(s) A1g

(p) T1u

(d) Eg

T2g

(f) A2u

T1u

T2u

+ A1g

± T1u (g)+

Eg± ± ± T1u , T2u + − A1g , A2g , Eg+

± T2g u± ± ± T1g , T2g + (g)

± A2u ± T2g Eu± ± T1u + A1g

± T1u (g)± ± ± T1u , T2u ± u ± T2g (g)+

± T2u g± ± ± T1u , T2u ± (u) ± T1g g± (g)+

165110-24



± ∓ ∓ ∓ ± TABLE XIX. Multipoles under the cubic crystal system (Oh ) in the spinful basis. (g, u)± = A± 1g,u ⊕ Eg,u ⊕ T2g,u ⊕ A2g,u ⊕ 2T1g,u ⊕ T2g,u ; ± ± ± ± g, u = Eg,u ⊕ T1g,u ⊕ T2g,u . The subscripts g, u are omitted for the O group. A1u ↔ A2u , T1u ↔ T2u , and E1/2u ↔ E5/2u for the Td group. The subscripts 1,2 are omitted for the Th group. The subscripts g, u and 1,2 are omitted for the T group.

(s) A1g ⊗ E1/2g → E1/2g (p) T1u ⊗ E1/2g → E1/2u

(s) E1/2g

(p) E1/2u

G3/2u

(d) G3/2g

E5/2g

G3/2g

(f) E5/2u

E1/2u

G3/2u

E5/2u

G3/2u

+ − A1g , T1g

± ± A1u , T1u + − A1g , T1g

u± g± (g)+

g± u± (u)± (g)+

± ± A2g , T2g ± ± A2u , T2u ± u g± + − A1g , T1g

g± u± (u)± (g)± g± (g)+

± ± A2u , T2u ± ± A2g , T2g ± g u± ± ± A1u , T1u ± u + − A1g , T1g

± ± A1u , T1u ± ± A1g , T1g ± g u± ± ± A2u , T2u ± u ± ± A2g , T2g + − A1g , T1g

u± g± (g)± (u)± u± (u)± g± g± (g)+

± ± A2u , T2u ± ± A2g , T2g ± g u± ± ± A1u , T1u ± u ± ± A1g , T1g ± ± A2g , T2g ± g + − A1g , T1g

u± g± (g)± (u)± u± (u)± g± g± (g)± g± (g)+

G3/2u (d) Eg ⊗ E1/2g → G3/2g T2g ⊗ E1/2g → E5/2g G3/2g (f) A2u ⊗ E1/2g → E5/2u T1u ⊗ E1/2g → E1/2u G3/2u T2u ⊗ E1/2g → E5/2u G3/2u

APPENDIX F: D6h AND ITS SUBGROUPS

Here, we summarize various tables for the hexagonal D6h and its subgroups. Tables XXII and XXIII represent even- and odd-parity hexagonal harmonics. Tables XXIV and XXV represent the operator expressions of even- and odd-parity multipoles in real space. Tables XXVI and XXVII represent the even- and odd-parity multipoles in momentum space. Table XXVIII represents the relation of multipoles under the D6h group and its subgroups. Tables XXIX (XXX) and XXXI (XXXII) represent the active multipoles under the hexagonal and trigonal crystals in the spinless (spinful) basis, respectively.

TABLE XX. Multipoles under the orthorhombic crystal system (D2h ) in the spinless basis. The subscripts g, u are omitted for the D2 group. Ag , B1u → A1 , Au , B1g → A2 , B3g,u → B1 , and B2g,u → B2 for the C2v group.

(s) Ag (p) B3u B2u B1u (d) Ag Ag B3g B2g B1g (f) Au B3u B2u B1u B3u B2u B1u

(s) Ag

(p) B3u

B2u

B1u

(d) Ag

Ag

B3g

B2g

B1g

(f) Au

B3u

B2u

B1u

B3u

B2u

B1u

Ag+

B± 3u Ag+

B± 2u B± 1g Ag+

B± 1u B± 2g B± 3g Ag+

Ag± B± 3u B± 2u B± 1u Ag+

Ag± B± 3u B± 2u B± 1u Ag± Ag+

B± 3g Au± B± 1u B± 2u B± 3g B± 3g Ag+

B± 2g B± 1u Au± B± 3u B± 2g B± 2g B± 1g Ag+

B± 1g B± 2u B± 3u Au± B± 1g B± 1g B± 2g B± 3g Ag+

Au± B± 3g B± 2g B± 1g Au± Au± B± 3u B± 2u B± 1u Ag+

B± 3u Ag± B± 1g B± 2g B± 3u B± 3u Au± B± 1u B± 2u B± 3g Ag+

B± 2u B± 1g Ag± B± 3g B± 2u B± 2u B± 1u Au± B± 3u B± 2g B± 1g Ag+

B± 1u B± 2g B± 3g Ag± B± 1u B± 1u B± 2u B± 3u Au± B± 1g B± 2g B± 3g Ag+

B± 3u Ag± B± 1g B± 2g B± 3u B± 3u Au± B± 1u B± 2u B± 3g Ag± B± 1g B± 2g Ag+

B± 2u B± 1g Ag± B± 3g B± 2u B± 2u B± 1u Au± B± 3u B± 2g B± 1g Ag± B± 3g B± 1g Ag+

B± 1u B± 2g B± 3g Ag± B± 1u B± 1u B± 2u B± 3u Au± B± 1g B± 2g B± 3g Ag± B± 2g B± 3g Ag+

165110-25



∓ ∓ ± TABLE XXI. Multipoles under the orthorhombic crystal system (D2h ) in the spinful basis. (g, u)± = Ag,u ⊕ B∓ 1g,u ⊕ B2g,u ⊕ B3g,u . The subscripts g, u are omitted for the D2 group. Ag , B1u → A1 , Au , B1g → A2 , B3g,u → B1 , and B2g,u → B2 for the C2v group.

(s) E1/2g (p) E1/2u E1/2u E1/2u (d) E1/2g E1/2g E1/2g E1/2g E1/2g (f) E1/2u E1/2u E1/2u E1/2u E1/2u E1/2u E1/2u (s) Ag ⊗ E1/2g → (p) B3u ⊗ E1/2g → B2u ⊗ E1/2g → B1u ⊗ E1/2g → (d) Ag ⊗ E1/2g → Ag ⊗ E1/2g → B3g ⊗ E1/2g → B2g ⊗ E1/2g → B1g ⊗ E1/2g → (f) Au ⊗ E1/2g → B3u ⊗ E1/2g → B2u ⊗ E1/2g → B1u ⊗ E1/2g → B3u ⊗ E1/2g → B2u ⊗ E1/2g → B1u ⊗ E1/2g →

E1/2g E1/2u E1/2u E1/2u E1/2g E1/2g E1/2g E1/2g E1/2g E1/2u E1/2u E1/2u E1/2u E1/2u E1/2u E1/2u

(g)+

(u)± (g)+

(u)± (u)± (g)± (g)± (g)+ (g)± (g)+

(g)± (u)± (u)± (u)± (g)+

(g)± (u)± (u)± (u)± (g)± (g)+

(g)± (u)± (u)± (u)± (g)± (g)± (g)+

(g)± (u)± (u)± (u)± (g)± (g)± (g)± (g)+

(g)± (u)± (u)± (u)± (g)± (g)± (g)± (g)± (g)+

(u)± (g)± (g)± (g)± (u)± (u)± (u)± (u)± (u)± (g)+

(u)± (g)± (g)± (g)± (u)± (u)± (u)± (u)± (u)± (g)± (g)+

(u)± (g)± (g)± (g)± (u)± (u)± (u)± (u)± (u)± (g)± (g)± (g)+

(u)± (g)± (g)± (g)± (u)± (u)± (u)± (u)± (u)± (g)± (g)± (g)± (g)+

(u)± (g)± (g)± (g)± (u)± (u)± (u)± (u)± (u)± (g)± (g)± (g)± (g)± (g)+

(u)± (g)± (g)± (g)± (u)± (u)± (u)± (u)± (u)± (g)± (g)± (g)± (g)± (g)± (g)+

(u)± (g)± (g)± (g)± (u)± (u)± (u)± (u)± (u)± (g)± (g)± (g)± (g)± (g)± (g)± (g)+

TABLE XXII. Even-parity hexagonal harmonics (E multipoles in unit of −e in the hexagonal D6h group) up to l = 6. The correspondence to the tesseral harmonics is shown. A1 , A2 , B1 , B2 , E1 , and E2 correspond to 1 , 2 , 3 , 4 , 5 , and 6 , respectively, in the Bethe notation. β1 β2 β2 β β1 β1 β2 β2 Note that Q4v = Qα4z , Q4u = −Q4v , Q4v = Q4z , Q6s = Q6z , Q6v = Qα6z , and Q6v = Q6z in Table I. rank

irrep.

symbol

definition

correspondence

0

+ A1g

Q0

1

(00)

2

+ A1g

Qu

(20)

+ E1g

Qzx , Qyz

+ E2g

Qv , Qxy

− r2) √ √ 3zx, 3yz √ √ 3 (x 2 − y 2 ), 3xy 2

+ A1g

Q40

B+ 1g

Q4a

4

B+ 2g + E1g

+ E2g + A1g

Q60

+ A1g

Q6c

+ A2g

Q6s

B+ 1g

Q6a

B+ 2g

Q6b

+ E1g

α1 Qα1 6u , Q6v

+ E1g

α2 Qα2 6u , Q6v β1 β1 Q6u , Q6v β2 β2 Q6u , Q6v

+ E2g

6

Q4b Qα4u , Qα4v β1 Qβ1 4u , Q4v β2 Qβ2 4u , Q4v

+ E2g + E2g

1 (3z2 2

1 (35z4 − 30z2 r 2 + 3r 4 ) 8 √ 70 yz(3x 2 − y 2 ) 4 √ 70 zx(x 2 − 3y 2 ) 4 √ √ 10 2 zx(7z − 3r 2 ), 410 yz(7z2 − 3r 2 ) 4 √ √ 35 (x 4 − 6x 2 y 2 + y 4 ), 235 xy(x 2 − y 2 ) 8 √ √ 5 (x 2 − y 2 )(7z2 − r 2 ), 25 xy(7z2 − r 2 ) 4 1 (231z6 − 315z4 r 2 + 105z2 r 4 − 5r 6 ) 16 √ 462 [x 6 − 15x 2 y 2 (x 2 − y 2 ) − y 6 ] 32 √ 462 xy(3x 4 − 10x 2 y 2 + 3y 4 ) 16 √ 210 yz(3x 2 − y 2 )(11z2 − 3r 2 ) 16 √ 210 zx(x 2 − 3y 2 )(11z2 − 3r 2 ) 16 √ √ 3 154 zx(x 4 − 10x 2 y 2 + 5y 4 ), 3 16154 yz(5x 4 − 10x 2 y 2 + y 4 ) 16 √ √ 21 zx[5r 4 + 3z2 (11z2 − 10r 2 )], 821 yz[5r 4 + 3z2 (11z2 − 10r 2 )] 8 √ √ 3 7 (x 4 − 6x 2 y 2 + y 4 )(11z2 − r 2 ), 3 4 7 xy(x 2 − y 2 )(11z2 − r 2 ) 16 √ √ 210 210 (x 2 − y 2 )[r 4 + 3z2 (11z2 − 6r 2 )], 16 xy[r 4 + 3z2 (11z2 − 6r 2 )] 32

165110-26

(21), (21) (22), (22) (40) (43) (43) (41), (41) (44), (44) (42), (42) (60) (66) (66) (63) (63) (65), (65) (61), (61) (64), (64) (62), (62)



TABLE XXIII. Odd-parity hexagonal harmonics (E multipoles in unit of −e in the hexagonal D6h group) up to l = 6. Note that Q50 = β1 β1 β2 β β2 α2 Qα1 5z , Q5u = Q5z , Q5v = Q5u , Q5u = Q5z , and Q5v = −Q5v in Table II. rank

irrep.

1

3

symbol

correspondence

+ A2u

Qz

z

(10)

+ E1u

Qx , Qy

x, y

(11), (11)

+ A2u

Qαz Q3a

B+ 2u

Q3b

+ E1u

Q3u , Q3v

1 z(5z2 − 3r 2 ) 2 √ 10 x(x 2 − 3y 2 ) 4 √ 10 y(3x 2 − y 2 ) 4 √ √ 6 x(5z2 − r 2 ), 46 y(5z2 − r 2 ) 4 √ √ 15 z(x 2 − y 2 ), 15xyz 2 1 z(63z4 − 70z2 r 2 + 15r 4 ) 8 √ 70 x(x 2 − 3y 2 )(9z2 − r 2 ) 16 √ 70 y(3x 2 − y 2 )(9z2 − r 2 ) 16 √ √ 3 14 4 2 2 x(x − 10x y + 5y 4 ), 3 1614 y(5x 4 − 10x 2 y 2 + y 4 ) 16 √ √ 15 x[r 4 + 7z2 (3z2 − 2r 2 )], 815 y[r 4 + 7z2 (3z2 − 2r 2 )] 8 √ √ 3 35 z(x 4 − 6x 2 y 2 + y 4 ), 3 235 xyz(x 2 − y 2 ) 8 √ √ 105 z(x 2 − y 2 )(3z2 − r 2 ), 105 xyz(3z2 − r 2 ) 4 2

(30)

B+ 1u

+ E2u + A2u

5

definition

β Qz ,

Qxyz

Q50

B+ 1u

Q5a

B+ 2u

Q5b

+ E1u

α1 Qα1 5u , Q5v

+ E1u

α2 Qα2 5u , Q5v β1 β1 Q5u , Q5v β2 β2 Q5u , Q5v

+ E2u + E2u

(33) (33) (31), (31) (32), (32) (50) (53) (53) (55), (55) (51), (51) (54), (54) (52), (52)

TABLE XXIV. Operator expressions of even-parity multipoles up to l = 4 in the hexagonal D6h group. rank 0

1

type

irrep.

symbol

E

+ A1g

Q0

1

MT

− A1g

T0

i (off-diagonal element)

M

− A2g

Mz

mz1

− E1g

Mx , My

mx1 , m1

ET

2

E

MT

3

M

ET

4

E

+ A2g + E1g + A1g + E1g + E2g − A1g − E1g − E2g − A2g

y

Gz

zl · (r × σ )

Gx , Gy

xl · (r × σ ), yl · (r × σ )

Qu

1 (3z2 − r 2 ) 2 √ √ 3zx, 3yz √ √ 3 (x 2 − y 2 ), 3xy 2

Qzx , Qyz Qv , Qxy

Tv , Txy

3zt2z − r · t 2 √ √ y 3(zt2x + xt2z ), 3(yt2z + zt2 ) √ √ y y 3(xt2x − yt2 ), 3(xt2 + yt2x )

Mzα

3[ 12 (3z2 − r 2 )mz3 − z(xmx3 + ym3 )]

Tu Tzx , Tyz

B− 1g

M3a

B− 2g

M3b

− E1g − E2g + A2g

definition

y

√

M3u , M3v β Mz ,

Mxyz

Gαz

B+ 1g

G3a

B+ 2g

G3b

+ E1g

G3u , G3v

+ E2g

Gz , Gxyz

β

+ A1g

Q40

+ E2g

β2 β2 Q4u , Q4v

B+ 1g

Q4a

B+ 2g

Q4b

+ E1g

Qα4u , Qα4v

+ E2g

Q4u , Q4v

β1

√ y 3 410 [(x 2 − y 2 )mx3 − 2xym3 ] √ y 3 410 [2xymx3 + (x 2 − y 2 )m3 ] √ y 2x(5zmz3 − r · m3 )], 46 [(5z2 − r 2 )m3

− r 2 )mx3 + + 2y(5zmz3 − r · m3 )] √ √ 1 2 y y z x x 2 15[ 2 (x − y )m3 + z(xm3 − ym3 )], 15(yzm3 + zxm3 + xymz3 ) αα yz zz zx 9zg3 − 6(xg3 + yg3 ) − 3z α g3 yy xy 3 25 [x(g3xx − g3 ) − 2yg3 ] xy yy 3 25 [2xg3 + y(g3xx − g3 )] yz 3 [x(5g3zz − α g3αα ) + 2(5zg3xz − α αg xα )], 23 [y(5g3zz − α g3αα ) + 2(5zg3 − α αg yα )] 2 √ √ yz yy yz xy 15[2(xg3zx − yg3 ) + z(g3xx − g3 )], 2 15(xg3 + yg3zx + zg3 ) 6 [(5z2 4

√

1 (35z4 8

− 30z2 r 2 + 3r 4 )

β1

165110-27

− y )(7z − r 2 ),

√

5 xy(7z2 − r 2 ) 2 70 2 2 yz(3x − y ) 4 √ 70 zx(x 2 − 3y 2 ) 4 √ √ 10 2 2 zx(7z − 3r ), 410 yz(7z2 − 3r 2 ) 4 √ √ 35 (x 4 − 6x 2 y 2 + y 4 ), 235 xy(x 2 − y 2 ) 8 5 (x 2 4

2

2

√



TABLE XXIV. (Continued.) rank

type

irrep.

symbol

MT

− A1g

T40

B− 1g

T4a

1 2

T4b

1 2

B− 2g − E1g

definition

− E2g

1 [3(5z2 2 35 [6xyzt4x 2 35 [3z(x 2 2

− r 2 )(5zt4z − r · t 4 ) − 40zt4z ] y

+ 3z(x 2 − y 2 )t4 + y(3x 2 − y 2 )t4z ] y

− y 2 )t4x − 6xyzt4 + x(x 2 − 3y 2 )t4z ]

t 4 · ∇Qα4u , t 4 · ∇Qα4v √ √ y y 35 [x(x 2 − 3y 2 )t4x − y(3x 2 − y 2 )t4 ], 235 [y(3x 2 − y 2 )t4x + x(x 2 − 3y 2 )t4 ] 2 √ y β2 5[x(3z2 − x 2 )t4x + y(y 2 − 3z2 )t4 + 3z(x 2 − y 2 )t4z ], t 4 · ∇Q4v

α α T4u , T4v β1 β1 T4u , T4v β2 β2 T4u , T4v

− E2g

TABLE XXV. Operator expressions of odd-parity multipoles up to l = 4 in the hexagonal D6h group. rank

type

0

M ET E

1

MT 2

M

ET

3

E

MT

4

M

ET

irrep.

symbol

definition

− A1u + A1u + A2u + E1u − A2u − E1u − A1u

M0 G0 Qz Qx , Qy Tz Tx , Ty Mu

− E1u

Mzx , Myz

r ·σ l · (r × σ ) z x, y t1z y x t 1 , t1 z 3zm2 − r · m2 √ √ y 3(zmx2 + xmz2 ), 3(ymz2 + zm2 ) √ √ y y 3(xmx2 − ym2 ), 3(xm2 + ymx2 ) αα zz 3g2 − α g2 √ √ yz 2 3g zx , 2 3g2 √ xx 2 yy √ xy 3(g2 − g2 ), 2 3g2 1 z(5z2 − 3r 2 ) 2

− E2u + A1u + E1u + E2u + A2u B+ 1u B+ 2u + E1u + E2u − A2u B− 1u B− 2u − E1u − E2u − A1u

Mv , Mxy Gu Gzx , Gyz Gv , Gxy Qαz

√

10 x(x 2 − 3y 2 ) √4 10 y(3x 2 − y 2 ) 4 √ √ 6 2 x(5z − r 2 ), 46 y(5z2 − r 2 ) 4 √ √ 15 z(x 2 − y 2 ), 15xyz 2 y 3[ 21 (3z2 − r 2 )t3z − z(xt3x + yt3 )] √ y 3 410 [(x 2 − y 2 )t3x − 2xyt3 ] √ y 3 410 [2xyt3x + (x 2 − y 2 )t3 ] √ √ y 6 [(5z2 − r 2 )t3x + 2x(5zt3z − r · t 3 )], 46 [(5z2 − r 2 )t3 + 2y(5zt3z − r 4 √ √ 1 2 y y x x 2 z 15[ 2 (x − y )t3 + z(xt3 − yt3 )], 15(yzt3 + zxt3 + xyt3z ) z z 1 [3(5z2 − r 2 )(5zm4 − r · m4 ) − 40zm4 ] 2

Q3a Q3b Q3u , Q3v Qβz , Qxyz Tzα T3a T3b T3u , T3v Tzβ , Txyz M40

B− 1u

M4a

1 2

B− 2u

M4b

1 2

− E1u

α α M4u , M4v

− E2u

β1 β1 M4u , M4v β2 β2 M4u , M4v

− E2u + A1u

B+ 1u

G40 G4a

B+ 2u

G4b

+ E1u + E2u + E2u

Gα4u , Gα4v β1 β1 G4u , G4v β2 Gβ2 4u , G4v

√

+ y2 −

35 [6xyzmx4 2 35 [3z(x 2 2

y

+ 3z(x 2 − y 2 )m4 + y(3x 2 − y 2 )mz4 ] y

− y 2 )mx4 − 6xyzm4 + x(x 2 − 3y 2 )mz4 ] m4 · ∇Qα4u , m4 · ∇Qα4v y

√

y 35 [y(3x 2 − y 2 )mx4 + x(x 2 − 3y 2 )m4 ] 2 y β2 5[x(3z2 − x 2 )mx4 + y(y 2 − 3z2 )m4 + 3z(x 2 − y 2 )mz4 ], m4 · ∇Q4v xy 2 xx 2 2 2 yy 2 2 zz 4z )g4 + (x + 3y − 4z )g4 + 4(3z − r )g4 + 4xyg4 − 16z(xg4xz +

35 [x(x 2 2 √

3 [(3x 2 2

· t 3 )]

− 3y 2 )mx4 − y(3x 2 − y 2 )m4 ],

yz

yg4 )] yy xy xz 35 xx 2 2 yz 3 2 [yz(g4 − g4 ) + 2x(zg4 + yg4 ) + (x − y )g4 ] yy yz xy 3 35 [zx(g4xx − g4 ) − 2y(xg4 + zg4 ) + (x 2 − y 2 )g4xz ] 2 αβ αβ α α αβ g4 ∇α ∇β Q4u , αβ g4 ∇α ∇β Q4v √ √ yy xy yy xy 3 235 [(x 2 − y 2 )(g4xx − g4 ) − 4xyg4 ], 3 35[xy(g4xx − g4 ) + (x 2 − y 2 )g4 ] √ yy yz 3 5[(z2 − x 2 )g4xx + (y 2 − z2 )g4 + (x 2 − y 2 )g4zz + 4z(xg4zx − yg4 )], αβ g4αβ ∇α ∇β Qβ2 4v

165110-28



TABLE XXVI. Even-parity multipoles in momentum space up to l = 4 in the hexagonal D6h group. The higher-order representation with respect to k is also shown for Q0 and (Mx , My , Mz ) in the bracket. rank 0

type

irrep.

symbol

definition

E

+ A1g

Q0

σ0 [(kx2 + ky2 + kz2 )σ0 ]

T0

iσ0

Mz

σz [(k · σ )kz − 13 k 2 σz ]

Mx , My

σx , σy [(k · σ )kx − 13 k 2 σx , (k · σ )ky − 13 k 2 σy ]

Gz Gx , Gy

iσz

MT 1

M ET

2

E

MT

3

M

ET

4

E

− A1g − A2g − E1g + A2g + E1g + A1g + E1g + E2g − A1g − E1g − E2g − A2g

MT

Qv , Qxy

1 (3kz2 2

Tv , Txy

3kz Qz − k · Q √ 3(kz Qx + kx Qz ), 3(ky Qz + kz Qy ) √ √ 3(kx Qx − ky Qy ), 3(kx Qy + ky Qx )

Mzα

3[ 21 (3kz2 − k 2 )σz − kz (kx σx + ky σy )]

Tu

√

Tzx , Tyz

M3a

B− 2g

M3b

√

√

6 [(5kz2 4

M3u , M3v Mzβ ,

√

Mxyz

− k 2 )σx +

15[ 21 (kx2

−

ky2 )σz

Gαz

G3a

B+ 2g

G3b

+ E1g + E2g + A1g

G3u , G3v Gβz ,

√

6 i[(5kz2 4

√

Gxyz

− k 2 )σx +

15i[ 21 (kx2

−

Q4a

B+ 2g

Q4b

+ E1g

Qα4u , Qα4v

+ E2g

β1 Qβ1 4u , Q4v

+ E2g

Q4u , Q4v

− A1g

T40

B− 1g

T4a

1 2

B− 2g

T4b

1 2

− E1g

α T4u , T4vα

− E2g

T4u , T4v

− E2g

T4u , T4v

β1

β2

β2

β1 β2

+ kz (kx σx − ky σy )],

√

+ 2ky (5kz σz − k · σ )]

15(ky kz σx + kz kx σy + kx ky σz )

3i[ 21 (3kz2 − k 2 )σz − kz (kx σx + ky σy )] √ 3i 410 [(kx2 − ky2 )σx − 2kx ky σy ] √ 3i 410 [2kx ky σx + (kx2 − ky2 )σy ] √ 2kx (5kz σz − k · σ )], 46 i[(5kz2 − k 2 )σy +

ky2 )σz

+ kz (kx σx − ky σy )],

√

2ky (5kz σz − k · σ )]

15i(ky kz σx + kz kx σy + kx ky σz )

1 (35kz4 − 30kz2 k 2 + 3k 4 )σ0 8 √ 70 ky kz (3kx2 − ky2 )σ0 4 √ 70 kz kx (kx2 − 3ky2 )σ0 4 √ √ 10 kz kx (7kz2 − 3k 2 )σ0 , 410 ky kz (7kz2 − 3k 2 )σ0 4 √ √ 35 (kx4 − 6kx2 ky2 + ky4 )σ0 , 235 kx ky (kx2 − ky2 )σ0 8 √ √ 5 (kx2 − ky2 )(7kz2 − k 2 )σ0 , 25 kx ky (7kz2 − k 2 )σ0 4

Q40

β2

10 [(kx2 − ky2 )σx − 2kx ky σy ] 4 √ 3 410 [2kx ky σx + (kx2 − ky2 )σy ] √ 2kx (5kz σz − k · σ )], 46 [(5kz2 − k 2 )σy

3

B+ 1g

B+ 1g

− k 2 )σ0 √ 3kz kx σ0 , 3ky kz σ0 √ √ 3 (kx2 − ky2 )σ0 , 3kx ky σ0 2 √

Qzx , Qyz

B− 1g − E1g − E2g + A2g

iσx , iσy

Qu

1 [3(5kz2 2

− k 2 )(5kz Qz − k · Q) − 40kz Qz ]

35 [6kx ky kz Qx 2 35 [3kz (kx2 2

+ 3kz (kx2 − ky2 )Qy + ky (3kx2 − ky2 )Qz ]

− ky2 )Qx − 6kx ky kz Qy + kx (kx2 − 3ky2 )Qz ] Q · ∇Qα4u , Q · ∇Qα4v

√

35 [kx (kx2 2

√

− 3ky2 )Qx − ky (3kx2 − ky2 )Qy ],

√

35 [ky (3kx2 2

− ky2 )Qx + kx (kx2 − 3ky2 )Qy ] β2

5[kx (3kz2 − kx2 )Qx + ky (ky2 − 3kz2 )Qy + 3kz (kx2 − ky2 )Qz ], Q · ∇Q4v

165110-29



TABLE XXVII. Odd-parity multipoles in momentum space up to l = 4 in the hexagonal D6h group. The higher-order representation with respect to k is also shown for Q0 and (Mx , My , Mz ) in the bracket. rank

type

irrep.

symbol

definition

0

M ET

− A1u + A1u

M0 G0

ik · σ k·σ

1

E

+ A2u

Qz

kx σy − ky σx

Qx , Qy

ky σz − kz σy , kz σx − kx σz

Tz

kz σ0

MT 2

M

ET

3

E

MT

4

M

ET

+ E1u − A2u − E1u − A1u − E1u − E2u + A1u + E1u + E2u + A2u

Tx , Ty

kx σ0 , ky σ0

Mu

3ikz σz − ik · σ √ √ 3i(kz σx + kx σz ), 3i(ky σz + kz σy ) √ √ 3i(kx σx − ky σy ), 3i(kx σy + ky σx )

Mzx , Myz Mv , Mxy

Gv , Gxy

3kz σz − k · σ √ 3(kz σx + kx σz ), 3(ky σz + kz σy ) √ √ 3(kx σx − ky σy ), 3(kx σy + ky σx )

Qαz

3[ 21 (3kz2 − k 2 )Qz − kz (kx Qx + ky Qy )]

Gu Gzx , Gyz

B+ 1u

Q3a

B+ 2u

Q3b

+ E1u + E2u − A2u

√

√

√

Q3u , Q3v Qβz ,

6 [(5kz2 4

√

Qxyz

− k 2 )Qx +

15[ 21 (kx2 − ky2 )Qz + kz (kx Qx − ky Qy )],

T3a

B− 2u

T3b

− E1u

T3u , T3v

− E2u

Tzβ , Txyz

− A1u

M40

B− 1u

M4a

1 2

B− 2u

M4b

1 2

− E1u

α α M4u , M4v

− E2u

M4u , M4v

− E2u

M4u , M4v

+ A1u

G40

B+ 1u

G4a

1 2

B+ 2u

G4b

1 2

+ E1u

Gα4u , Gα4v

+ E2u

G4u , G4v

+ E2u

β2 β2 G4u , G4v

β1

β2

β1

1 i[3(5kz2 2

35 i[3kz (kx2 2

+ 3kz (kx2 − ky2 )σy + ky (3kx2 − ky2 )σz ]

− ky2 )σx − 6kx ky kz σy + kx (kx2 − 3ky2 )σz ] iσ · ∇Qα4u , iσ · ∇Qα4v

√

35 i[kx (kx2 2

√

β2

β1

+ 2ky (5kz Qz − k · Q)]

15(ky kz Qx + kz kx Qy + kx ky Qz )

− k 2 )(5kz σz − k · σ ) − 40kz σz ]

35 i[6kx ky kz σx 2

β1

√

1 k (5kz2 − 3k 2 )σ0 2 z √ 10 kx (kx2 − 3ky2 )σ0 4 √ 10 ky (3kx2 − ky2 )σ0 4 √ √ 6 k (5kz2 − k 2 )σ0 , 46 ky (5kz2 − k 2 )σ0 4 x √ √ 15 kz (kx2 − ky2 )σ0 , 15kx ky kz σ0 2

Tzα

B− 1u

10 [(kx2 − ky2 )Qx − 2kx ky Qy ] 4 √ 3 410 [2kx ky Qx + (kx2 − ky2 )Qy ] √ 2kx (5kz Qz − k · Q)], 46 [(5kz2 − k 2 )Qy

3

− 3ky2 )σx − kx (3kx2 − ky2 )σy ],

√

35 i[ky (3kx2 2

− ky2 )σx + kx (kx2 − 3ky2 )σy ] β2

5i[kx (3kz2 − kx2 )σx + ky (ky2 − 3kz2 )σy + 3kz (kx2 − ky2 )σz ], iσ · ∇Q4v

1 [3(5kz2 2

− k 2 )(5kz σz − k · σ ) − 40kz σz ]

35 [6kx ky kz σx 2 35 [3kz (kx2 2

+ 3kz (kx2 − ky2 )σy + ky (3kx2 − ky2 )σz ]

− ky2 )σx − 6kx ky kz σy + kx (kx2 − 3ky2 )σz ] σ · ∇Qα4u , σ · ∇Qα4v

√

35 [kx (kx2 2

√

− 3ky2 )σx − ky (3kx2 − ky2 )σy ],

√

35 [ky (3kx2 2

− ky2 )σx + kx (kx2 − 3ky2 )σy ] β2

5[kx (3kz2 − kx2 )σx + ky (ky2 − 3kz2 )σy + 3kz (kx2 − ky2 )σz ], σ · ∇Q4v

165110-30



TABLE XXVIII. Multipoles under hexagonal and trigonal crystals. The upper and lower columns represent even-parity (g) and odd-parity (u) multipoles. We take the x axis as the C2 rotation axis. For C3v , we take the zx plane as the σv mirror plane. C3i = S6 . The basis functions are taken as (x, y ) → x ± iy for C6h , C6 , C3h , and C3i . E Q0 , Qu , Q40 – Q4a Q4b Qzx , Qα4u Qyz , Qα4v β1 β2 Qv , Q4u , Q4u β1 β2 Qxy , Q4v , Q4v – Qz , Qαz Q3a Q3b Qx , Q3u Qy , Q3v Qxyz Qβz

ET

M

MT

D6h

D6

C6h

C6v

C6

D3h

C3h

D3d

D3

C3v

C3i

C3

– Gz , Gαz G3a G3b Gx , G3u Gy , G3v Gxyz Gβz G0 , Gu , G40 – G4a G4b Gzx , Gα4u Gyz , Gα4v β1 β2 Gv , G4u , G4u β1 β2 Gxy , G4v , G4v

– Mz , Mzα M3a M3b Mx , M3u My , M3v Mxyz Mzβ M0 , Mu , M40 – M4a M4b α Mzx , M4u α Myz , M4v β1 β2 Mv , M4u , M4u β1 β2 Mxy , M4v , M4v

T0 , Tu , T40 – T4a T4b α Tzx , T4u Tyz , T4vα β1 β2 Tv , T4u , T4u β2 β2 Txy , T4v , T4v – Tz , Tzα T3a T3b Tx , T3u Ty , T3v Txyz Tzβ

A1g A2g B1g B2g E1g

A1 A2 B1 B2 E1

Ag Ag Bg Bg E1g

A1 A2 B2 B1 E1

A A B B E1

A1 A2 A1 A2 E

A A A A E

A1g A2g A1g A2g Eg

A1 A2 A1 A2 E

A1 A2 A2 A1 E

Ag Ag Ag Ag Eg

A A A A E

E2g

E2

E2g

E2

E2

E

E

Eg

E

E

Eg

E

A1u A2u B1u B2u E1u

A1 A2 B1 B2 E1

Au Au Bu Bu E1u

A2 A1 B1 B2 E1

A A B B E1

A1 A2 A1 A2 E

A A A A E

A1u A2u A1u A2u Eu

A1 A2 A1 A2 E

A2 A1 A1 A2 E

Au Au Au Au Eu

A A A A E

E2u

E2

E2u

E2

E2

E

E

Eu

E

E

Eu

E

± ∓ ± TABLE XXIX. Multipoles under the hexagonal crystal system (D6h ) in the spinless basis. (g, u)± = A1g,u ⊕ A2g,u ⊕ E2g,u and g, u± = ± ± ⊕ B2g,u ⊕ E1g,u . The subscripts g, u are omitted for the D6 group. The subscripts 1,2 in A and B are omitted for the C6h group. A1u ↔ A2u and B1g ↔ B2g , and then, the subscripts g, u are omitted for the C6v group. The subscripts g, u and 1,2 in A and B are omitted for the C6 group. A1g , B1u → A1 , A2g , B2u → A2 , A1u , B1g → A1 , A2u , B2g → A2 , E2u , E1g → E , E2g , and E1u → E for the D3h group. A1g , A2g , B1u , B2u → A , A1u , A2u , B1g , B2g → A , E2u , E1g → E , E2g , and E1u → E for the C3h group.

B± 1g,u

(s) A1g (p) A2u E1u (d) A1g E1g E2g (f) A2u B1u B2u E1u E2u

(s) A1g

(p) A2u

E1u

(d) A1g

E1g

E2g

(f) A2u

B1u

B2u

E1u

E2u

+ A1g

± A2u + A1g

± E1u ± E1g +

± A1g ± A2u ± E1u + A1g

± E1g ± E1u ±

± E2g ± E2u ±

± A2u ± A1g ± E1g ± A2u ± E1u ± E2u + A1g

B± 1u B± 2g ± E2g B± 1u ± E2u ± E1u B± 2g + A1g

B± 2u B± 1g ± E2g B± 2u ± E2u ± E1u B± 1g ± A2g + A1g

± E1u ± E1g ±

± E2u ± E2g g± ± E2u u± (u)± ± E2g ± E1g ± E1g g± (g)+

(g)

(u) ± E1g (g)+

165110-31

u ± E2g g± (g)+

(g) ± E1u (u)± u± ± E1g ± E2g ± E2g (g)+



± ∓ ∓ TABLE XXX. Multipoles under the hexagonal crystal system (D6h ) in the spinful basis. (g, u)± = A1g,u ⊕ A2g,u ⊕ E1g,u , (g , u )± = ∓ ∓ ∓ ± ± ± ± ⊕ A2g,u ⊕ B1g,u ⊕ B2g,u , and g, u = B1g,u ⊕ B2g,u ⊕ E2g,u . The subscripts g, u are omitted for the D6 group. The subscripts 1,2 in A and B are omitted for the C6h group. A1u ↔ A2u and B1g ↔ B2g , and then, the subscripts g, u are omitted for the C6v group. The subscripts g, u and 1,2 in A and B are omitted for the C6 group. A1g , B1u → A1 , A2g , B2u → A2 , A1u , B1g → A1 , A2u , B2g → A2 , E2u , E1g → E , E2g , E1u → E , and E1/2u ↔ E3/2u for the D3h group. A1g , A2g , B1u , B2u → A , A1u , A2u , B1g , B2g → A , E2u , E1g → E , E2g , E1u → E , and E1/2u ↔ E3/2u for the C3h group. ± A1g,u

(s) E1/2g (p) E1/2u E1/2u (s) A1g ⊗ E1/2g → E1/2g (p) A2u ⊗ E1/2g → E1/2u E1u ⊗ E1/2g →

+

(g)

±

(u)

(g)+

±

(u)

(g)± (g)+

E1/2u

E5/2u ± E1u , ± E1g , ± E1g ,

(g )+

E5/2u

(d) E1/2g

E1/2g

±

(g)

± E1g , ± E1u , ± E1u ,

(g)

(u)±

(u)±

(u)±

± ± ± ± E1u , E2u E1u , E2u

E1/2g E5/2g

E2g ⊗ E1/2g →

E5/2g

±

(u)±

(g)+

(d) A1g ⊗ E1/2g → E1/2g E1g ⊗ E1/2g →

± E2u ± E2g ± E2g

± E2g ± E2u ± E2u

E3/2g g

E5/2g ± E1g , ± E1u , ± E1u ,

±

±

u

u±

(u )±

± ± E1u , E2u

(u )±

(g)±

± ± E1g , E2g

g±

± ± E1g , E2g

(g)+

± ± E1g , E2g

g±

± ± E1g , E2g

(g )+

± ± E1g , E2g

(g )±

(g)+

± ± E1g , E2g

E3/2g

(g )+

E5/2g

B2u ⊗ E1/2g → E1u ⊗ E1/2g →

(f) E1/2u ±

(u)

E3/2u

E3/2u

E1/2u

±

±

±

u

±

(g)±

g

(g)±

g±

u

(u)

±

(g)±

g±

(g)±

g

± ± ± ± ± ± ± ± E1g , E2g E1g , E2g E1g , E2g E1g , E2g

± E1u , ± E1g , ± E1g ,

E3/2u

± E2u ± E2g ± E2g

±

u

g

±

g±

E5/2u ± E1u , ± E1g , ± E1g ,

± E2u ± E2g ± E2g

(g )±

± ± E1g , E2g

(g )±

u±

u±

(u)±

± ± E1u , E2u

u±

± ± E1u , E2u

(u)±

u±

u±

(u)±

± ± E1u , E2u

u±

± ± E1u , E2u

(u )±

± ± E1u , E2u

(u )±

± ± E1u , E2u

(u)±

± ± E1u , E2u

(u )±

± ± E1u , E2u

(u )±

± ± ± ± ± ± ± ± E1u , E2u E1u , E2u E1u , E2u E1u , E2u

u±

(u)±

(u)±

u±

± ± ± ± ± ± ± ± E1u , E2u E1u , E2u E1u , E2u E1u , E2u

(g)

g

± +

E3/2u

(g)

g

± ±

(g)

+

E3/2u

(g)

±

(g) g

±

g

± +

E1/2u

(g)

± E1g , ± E1g , ± E1g , ± E1g ,

± E2g ± E2g ± E2g ± E2g

+

E5/2u E2u ⊗ E1/2g →

E5/2u

(u)±

+

(f) A2u ⊗ E1/2g → E1/2u B1u ⊗ E1/2g →

± E2g ± E2u ± E2u

(g )

g

±

± ± E1g , E2g

±

± ± E1g , E2g

±

± ± E1g , E2g

±

± ± E1g , E2g

(g) (g) g ± E1g ,

± E2g +

E3/2u

(g)

(g )± ± ± E1g , E2g

(g )+

E5/2u

± ∓ ± TABLE XXXI. Multipoles under the trigonal crystal system (D3d ) in the spinless basis. (g, u)± = A1g,u ⊕ A2g,u ⊕ Eg,u . The subscripts g, u are omitted for the D3 group. The subscripts g, u are omitted, and Q3a ↔ Q3b for the C3v group. The subscripts 1,2 are omitted for the C3i group. The subscripts g, u and 1,2 are omitted for the C3 group.

(s) A1g (p) A2u Eu (d) A1g Eg Eg (f) A2u A1u A2u Eu Eu

(s) A1g

(p) A2u

Eu

(d) A1g

Eg

Eg

(f) A2u

A1u

A2u

Eu

Eu

+ A1g

± A2u + A1g

Eu± Eg± +

± A1g ± A2u Eu± + A1g

Eg± Eu± ±

Eg± Eu± ±

± A2u ± A1g Eg± ± A2u Eu± Eu± + A1g

± A1u ± A2g Eg± ± A1u Eu± Eu± ± A2g + A1g

± A2u ± A1g Eg± ± A2u Eu± Eu± ± A1g ± A2g + A1g

Eu± Eg± ±

Eu± Eg± (g)± Eu± (u)± (u)± Eg± Eg± Eg± (g)± (g)+

(g)

(u) Eg± (g)+

165110-32

(u) Eg± (g)± (g)+

(g) Eu± (u)± (u)± Eg± Eg± Eg± (g)+



± ∓ ∓ TABLE XXXII. Multipoles under the trigonal crystal system (D3d ) in the spinful basis. (g, u)± = A1g,u ⊕ A2g,u ⊕ Eg,u and (g , u )∓ = ± ± ⊕ A1g,u ⊕ 2A2g,u . The subscripts g, u are omitted for the D3 group. The subscripts g, u are omitted, and Q3a ↔ Q3b for the C3v group. The subscripts 1,2 are omitted for the C3i group. The subscripts g, u and 1,2 are omitted for the C3 group. ∓ A1g,u

(s) E1/2g (p) E1/2u E1/2u E3/2u (d) E1/2g E1/2g E3/2g E1/2g E3/2g (f) E1/2u E1/2u E1/2u E1/2u E3/2u E1/2u E3/2u (s) A1g ⊗ E1/2g → E1/2g (p) A2u ⊗ E1/2g → E1/2u Eu ⊗ E1/2g → E1/2u E3/2u (d) A1g ⊗ E1/2g → E1/2g Eg ⊗ E1/2g → E1/2g E3/2g Eg ⊗ E1/2g → E1/2g E3/2g (f) A2u ⊗ E1/2g → E1/2u A1u ⊗ E1/2g → E1/2u A2u ⊗ E1/2g → E1/2u Eu ⊗ E1/2g → E1/2u E3/2u Eu ⊗ E1/2g → E1/2u E3/2u

(g)+

(u)± (g)+

(u)± 2Eu± (g)± 2Eg± (g)+ 2Eg± (g )+

(g)± (u)± (u)± 2Eu± (g)+

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(g)± (u)± (u)± 2Eu± (g)± (g)+

2Eg± 2Eu± 2Eu± (u )± 2Eg± 2Eg± (g )+

(g)± (u)± (u)± 2Eu± (g)± (g)± 2Eg± (g)+

2Eg± 2Eu± 2Eu± (u )± 2Eg± 2Eg± (g )± 2Eg± (g )+

(u)± (g)± (g)± 2Eg± (u)± (u)± 2Eu± (u)± 2Eu± (g)+

(u)± (g)± (g)± 2Eg± (u)± (u)± 2Eu± (u)± 2Eu± (g)± (g)+

(u)± (g)± (g)± 2Eg± (u)± (u)± 2Eu± (u)± 2Eu± (g)± (g)± (g)+

(u)± (g)± (g)± 2Eg± (u)± (u)± 2Eu± (u)± 2Eu± (g)± (g)± (g)± (g)+

2Eu± 2Eg± 2Eg± (g )± 2Eu± 2Eu± (u )± 2Eu± (u )± 2Eg± 2Eg± 2Eg± 2Eg± (g )+

(u)± (g)± (g)± 2Eg± (u)± (u)± 2Eu± (u)± 2Eu± (g)± (g)± (g)± (g)± 2Eg± (g)+

2Eu± 2Eg± 2Eg± (g )± 2Eu± 2Eu± (u )± 2Eu± (u )± 2Eg± 2Eg± 2Eg± 2Eg± (g )± 2Eg± (g )+

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[114] [115] [116] [117]

[118] [119] [120] [121] [122]

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165110-35

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Classification of atomic-scale multipoles under

Classification of atomic-scale multipoles under

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