Bottom-up design of spin-split and reshaped electronic band structures
in spin-orbit-coupling free antiferromagnets:
Procedure on the basis of augmented multipoles

The cluster multipole is defined to describe the onsite degree of freedom in the tight-binding Hamiltonian. As arbitrary onsite degrees of freedom are represented by a superposition of local potentials at each atomic site, all the anisotropic charge distributions in a cluster are systematically represented by using the spherical harmonics with the origin at the center of the cluster, which is related to the electric multipole degree of freedom Kusunose (2008); Kuramoto et al. (2009); Santini et al. (2009). Eventually, the anisotropic charge distributions on cluster sites are described as a cluster electric multipole $\tilde{Q}^{(c)}_{lm}$, which is given by

where $O_{lm}(\bm{R}_{i})=\sqrt{4\pi/(2l+1)}R_{i}^{l}Y^{*}_{lm}(\hat{\bm{R}}_{i})$, $\bm{R}_{i}=(X_{i},Y_{i},Z_{i})$ is the position vector of $i$th cluster site, $q^{(lm)}_{i}$ is the local electric charge of $i$th cluster site, and $N$ is the number of sites in a cluster. We omit electric charge unit $-e$ for notational simplicity. $Y_{lm}(\hat{\bm{R}}_{i})$ is the spherical harmonics as a function of angle $\hat{\bm{R}}_{i}=\bm{R}_{i}/|\bm{R}_{i}|$ with the azimuthal and magnetic quantum numbers, $l$ and $m$ ($-l\leq m\leq l$). In the following, we regard the symbol $O_{lm}(\bm{r})$ as the harmonics of the point group such as cubic and hexagonal ones instead of the spherical harmonics, which are real functions given by linear combinations of $O_{lm}$ and $O_{l-m}$ as shown in Appendix A Kusunose (2008). Through this expression, we define $q^{(lm)}_{i}$ for the specified electric multipole, and the corresponding matrix (operator) expression is given by $Q^{\rm(c)}_{lm}=\sum_{i}q^{(lm)}_{i}\ket{i}\bra{i}$ where $\ket{i}$ is the atomic site basis.

Such a cluster multipole can also describe magnetic ordering patterns in a cluster Suzuki et al. (2017, 2019). In the spin-orbital decoupled basis, it is useful to express the magnetic structure as a linear combination of direct products of $Q^{\rm(c)}_{lm}$ and the Pauli matrices of spin $\bm{\sigma}=(\sigma_{x},\sigma_{y},\sigma_{z})$. Then, any types of magnetic orderings coupled with the corresponding molecular fields are expressed by a linear combination of $Q^{\rm(c)}_{lm}\sigma_{\mu}$. Namely, the mean-field Hamiltonian of the AFM ordering is represented by

where the coefficient $m^{\mu}_{lm}$ is a conjugate field of an order parameter in the AFM state. As $Q^{\rm(c)}_{lm}$ is time-reversal-even and $\sigma_{\mu}$ is time-reversal-odd, $H_{m}$ is time-reversal-odd as it is a symmetry breaking term. From the multipole viewpoint, the ordering pattern is characterized by the type of the emergent multipole: The ferromagnetic structure corresponds to the isotropic electric monopole and the AFM structure corresponds to the anisotropic electric multipoles for $l\geq 1$.

In the case of the collinear AFM order, the mean-field Hamiltonian matrix is represented by the single component of $\sigma_{\mu}$ where $\mu$ denotes the ordered moment direction, although it is taken to be arbitrary in the absence of the SOC. We show an example of the staggered AFM ordering in a square cluster under the point group $D_{\rm 4h}$ in Fig. 3(a). By decomposing this magnetic structure into the alignment of point charges and spin as shown in Fig. 3(a), and then, evaluating $Q^{\rm(c)}_{lm}$ via Eq. (12), one can find that the corresponding multipole is $Q^{\rm(c)}_{xy}$.

Such a multipole description is also understood from a symmetry viewpoint. The mean-field matrix in a square cluster in Eq. (6) is represented by

where $\sigma=\pm 1$ for up and down spins. From Eq. (III), this matrix element except for $\sigma$ belongs to the irrep. ${B}_{2g}^{+}$ under the point group $D_{4{\rm h}}$, which is the same irrep. of the $Q^{\rm(c)}_{xy}$-type electric quadrupole (See also the correspondence between the irrep. under the point group and multipoles in Sec. IV.4) Hayami et al. (2018). More intuitively, the real-space point charge alignment in a square cluster clearly indicates the presence of $xy$-type electric quadrupole; the positive charges are in the $[110]$ direction, while the negative ones are in the $[\bar{1}10]$ direction.

In a similar manner, coplanar and noncoplanar magnetic structures are described by a linear combination of two and three components of $\sigma_{\mu}$, respectively. Figure 3(b) shows an example of the coplanar spin structure in a square cluster where each spin points to the $\langle 110\rangle$ radial direction. Also in this case, the plane including spins is taken to be arbitrary due to spin rotational symmetry. By using Eq. (12) for two spin components, one can find that the spin pattern in Fig. 3(b) is proportional to $Q^{\rm(c)}_{x}\sigma_{x}+Q^{\rm(c)}_{y}\sigma_{y}$. The mean-field matrix is given in the form of

It is apparent from Fig. 3(b) that $Q_{x}^{\rm(c)}$ and $Q_{y}^{\rm(c)}$ represent the $x$ and $y$ electric dipoles in the multipole language.

The bond multipole is introduced to describe the bond degree of freedom in the system, which corresponds to the off-diagonal hopping part in the tight-binding Hamiltonian Hayami et al. (2019b).

First, in order to get an intuitive insight into the relation between bond multipoles and hoppings, let us consider a two-site problem connected by the complex hopping $t=t^{\prime}+it^{\prime\prime}$. The two sites are denoted as A and B, which are separated by the distance $a$ in the $x$ direction. By using the molecular-orbital basis $\{\ket{\phi_{1}},\ket{\phi_{2}}\}$ instead of the atomic site basis $\{\ket{{\rm A}},\ket{{\rm B}}\}$, the real and imaginary hopping matrices are transformed as

where $\ket{\phi_{1}}=(1/\sqrt{2})(\ket{{\rm A}}+\ket{{\rm B}})$ and $\ket{\phi_{2}}=(1/\sqrt{2})(\ket{{\rm A}}-\ket{{\rm B}})$. As the anisotropy of the molecular orbitals $\{\ket{\phi_{1}},\ket{\phi_{2}}\}$ is the same as the $s$- and $p_{x}$-orbital wave functions, the ordinary atomic-scale multipole description in Ref. Hayami and Kusunose, 2018 can be applied. Then, by comparing the matrix elements in $s$-$p_{x}$ orbital basis, ${\rm Re}[H_{t}]$ corresponds to the electric monopole $Q_{0}$, while ${\rm Im}[H_{t}]$ corresponds to the magnetic toroidal dipole $T_{x}$ Hayami and Kusunose (2018). This result indicates that the real hopping is expressed as the electric monopole on the bond center, while the imaginary hopping is expressed as the magnetic toroidal dipole along the bond direction. This assignment of multipole moments on the bond center is reasonable from a symmetry viewpoint, since the real (imaginary) hopping is equivalent with the time-reversal-even scalar (time-reversal-odd polar vector), which corresponds to the electric monopole (magnetic toroidal dipole).

This result is generalized for arbitrary cluster systems. Any types of hoppings are represented by bond electric and magnetic toroidal multipoles, $\tilde{Q}^{\rm(b)}_{lm}$ and $\tilde{T}^{\rm(b)}_{lm}$, which are expressed as

where $N_{\rm bond}$ is the number of bonds in a cluster and $\bm{R}_{(ij)}$ is the position vector at the $i$-$j$ bond center. $q^{(lm)}_{(ij)}$ and $\bm{t}^{(lm)}_{(ij)}$ are the local electric charge and local magnetic toroidal dipole at $\bm{R}_{(ij)}$, respectively, where $\bm{t}^{(lm)}_{(ij)}=it^{\prime\prime(lm)}_{ij}\bm{n}_{ij}$ with $\bm{n}_{ij}$ being the unit vector connecting between $i$ and $j$ sites. The multipole assignment is independently performed per $n$th-neighbor bond. The corresponding matrices (operators) of bond electric and magnetic toroidal multipoles are given by $Q^{\rm(b)}_{lm}=\sum_{(ij)}q^{(lm)}_{(ij)}\ket{j}\bra{i}$ and $T^{\rm(b)}_{lm}=\sum_{(ij)}(\bm{t}^{(lm)}_{(ij)}\cdot\bm{n}_{ij})\ket{j}\bra{i}$.

By using these matrices, we express any bond modulations in terms of bond multipoles. In particular, the bond magnetic toroidal multipoles, $T^{\rm(b)}_{lm}$, represent the anisotropic current distribution including a loop-current distribution discussed in cuprates and iridates Shekhter_PhysRevB.80.214501; zhao2016evidence; murayama2020bond, as the imaginary hopping represents a local current along the bond.

Let us again consider an example in a square cluster under the point group $D_{4{\rm h}}$, whose Hamiltonian is shown in Eq. (6). There are six real and imaginary bond degrees of freedom, which are assigned as six electric and magnetic toroidal multipoles, respectively. By using Eqs. (33) and (34), the first-neighbor four real (imaginary) bonds correspond to electric monopole $Q^{\rm(b1)}_{0}$, electric quadrupole $Q^{\rm(b1)}_{v}$, and electric dipoles $(Q^{\rm(b1)}_{x},Q^{\rm(b1)}_{y})$ [magnetic toroidal hexadecapole $T^{\alpha{\rm(b1)}}_{4x}$, magnetic toroidal quadrupole $T^{\rm(b1)}_{xy}$, and magnetic toroidal dipoles $(T^{\rm(b1)}_{x},T^{\rm(b1)}_{y})$], while the second-neighbor two real (imaginary) bonds correspond to electric monopole $Q^{\rm(b2)}_{0}$ and electric quadrupole $Q^{\rm(b2)}_{xy}$ [magnetic toroidal dipoles $(T^{\rm(b2)}_{x},T^{\rm(b2)}_{y})$] where the integer $(n=1,2)$ in superscript represents the $n$th-neighbor bond. The specific examples of $Q^{\rm(b1)}_{v}$, $T^{\rm(b1)}_{x}$, $Q^{\rm(b1)}_{x}$, and $T^{\alpha{\rm(b1)}}_{4z}$ are shown in Fig. 4. It is noted that the magnetic toroidal hexadecapole $T_{4z}^{\rm\alpha(b1)}$ with the vortex-like alignment of $\bm{t}_{(ij)}$ in Fig. 4 is equivalent to the magnetic dipole along $z$ direction. Nevertheless, we use the higher-rank hexadecapole $T_{4z}^{\rm\alpha(b1)}$ since we use the convention in this paper that all bond degrees of freedom are described by the electric or magnetic toroidal multipoles.

Finally, we introduce momentum multipoles to represent the momentum dependence in crystals. In the $\bm{k}\to\bm{0}$ limit, the spherical harmonics as a function of $\hat{\bm{k}}=\bm{k}/|\bm{k}|$, $Y_{lm}(\hat{\bm{k}})$, gives the anisotropic momentum distribution. As $\bm{k}$ is a polar vector with time-reversal odd, the even-(odd-)rank component in $Y_{lm}(\hat{\bm{k}})$ corresponds to the electric (magnetic toroidal) multipoles, which are defined as

The explicit expressions are given by replacing $\bm{r}$ with $\bm{k}$ in Table 8 in Appendix A where the odd-rank multipoles in Table 8 should be replaced with $T_{lm}^{\rm(m)}(\bm{k})$.

In general, the momentum dependence in crystals has periodicity and is represented by a superposition of trigonometric functions of $\bm{k}$. In this case, the momentum form factor consists of the momentum multipoles up to the infinite rank belonging to the same irrep. For example, we consider the single-band tight-binding model on a simple square lattice under the point group $D_{4{\rm h}}$ with the lattice constant $a$. The momentum form factor for the nearest-neighbor bond $f(\bm{k})$ is given by a linear combination of momentum multipoles as

where $c_{i}$ ($i=1,2,\cdots$) are the expansion coefficients. $f(\bm{k})$ clearly consists of the multipoles belonging to the totally symmetric irrep. $A_{1g}$ under $D_{\rm 4h}$. To specify the type of multipole, we use the lowest-rank multipole in the superscript of $f(\bm{k})$ as the convention is often used in the field of superconductivity Nomoto et al. (2016); Sumita and Yanase (2020). In this case, $f(\bm{k})$ is expressed as $f^{Q_{0}}(\bm{k})$.

Similarly, the form factors belonging to the other irreps. are also described by the different set of multipoles. In the case of $D_{\rm 4h}$, when the sign of the hopping along the $y$ direction is opposite, the form factor is given by

which belongs to the irrep. $B_{1g}$. In this situation, we denote $f(\bm{k})$ as $f^{Q_{v}}(\bm{k})$.

Moreover, when the imaginary hopping appears only in the $x$ direction under $D_{\rm 4h}$, the form factor is given by

which belongs to the irrep. $E_{u}$. In this situation, we denote $f(\bm{k})$ as $f^{T_{x}}(\bm{k})$. Such form factors in Eqs. (40) and (43) can appear in the tight-binding Hamiltonian when the system has the sublattice degree of freedom, as the local site symmetry is lowered than the lattice symmetry.

In the crystal systems, a part of the rotational symmetry and/or inversion symmetry are lost due to the regular and discrete alignment of the ions. As a result, the multipole degrees of freedom belonging to the same irrep. are not distinguished from the symmetry viewpoint. In other words, the irrep. of the rotational group split into subgroups according to the point-group irrep. For example, some even-parity and odd-parity multipoles belong to the same irrep. in noncentrosymmetric crystals.

To avoid such confusion, we uniquely assign the multipoles of irrep. as in Sec. III by the following rules: Among the multipoles belonging to the same irrep., we adopt the lowest even-rank electric multipoles for time-reversal even quantities, whereas we adopt the lowest odd-rank magnetic toroidal multipoles for time-reversal odd quantities. In this convention, the momentum-type odd-rank electric multipoles and even-rank magnetic toroidal multipoles do not appear in the Hamiltonian, as will be clarified in Sec. V.

Following the above rules, the multipoles and the irrep. have a one-to-one correspondence. We summarize the corresponding multipole notations under 32 point groups in Appendix B, where we divide them into eleven Laue classes with the same number of the irreps., in Tables 9-19. The compatibility relation of multipole within the same Laue class is given in the same row in the table. On the other hand, for the compatibility relation of multipole between the different Laue classes, we use the group-subgroup compatibility relation and adopt the lower-rank multipole to assign the irrep. For example, by the relation between $T_{\rm d}$ and $T$, the electric monopole $Q_{0}$ belonging to $A_{1}$ and the electric hexadecapole $Q_{6t}$ belonging to $A_{2}$ under the point group $T_{\rm d}$ turn into the same irrep. $A$ under the point group $T$. In this case, the multipole belonging to $A$ under $T$ is denoted as $Q_{0}$.

Let us remark on the connection of the quantities introduced in the cluster and the lattice. Although we assign the multipoles to the electronic degrees of freedom by introducing the magnetic cluster, there is a situation where the lattice symmetry is higher than the cluster symmetry due to the additional operations combined with the translation. In such a situation, we replace the irreps. in a cluster with the corresponding ones in a lattice in accordance with the compatibility relation. Accordingly, the multipoles in a cluster are mapped onto those in a lattice. In Sec. VI, we exemplify this by considering the triangular and kagome lattices consisting of the triangle unit where the cluster and lattice symmetries are different with each other.

In the absence of the SOC, the single-orbital Hamiltonian consists of the hopping part without the spin dependence and the mean-field part gives rise to the symmetry breaking due to the magnetic ordering. The total tight-binding Hamiltonian is generally represented by

where $c^{\dagger}_{\bm{k}\gamma\sigma}$ ($c_{\bm{k}\gamma\sigma}$) is the creation (annihilation) operator at wave vector $\bm{k}$ and sublattice $\gamma$. $H_{t}^{Q}$ and $H_{t}^{T}$ stand for the real and imaginary hopping matrices, respectively, which are represented by a linear combination of the product between bond and momentum multipoles as

In Eq. (47), the electric monopole contribution $f^{Q_{0}}(\bm{k})Q^{\rm(b)}_{0}$ always appears in $H^{Q}_{t}$, while the higher-rank contribution depends on the lattice symmetry. On the other hand, $H^{T}_{t}$ in Eq. (48) exists only in the absence of the local inversion symmetry. It is noted that the cross terms $f^{T_{lm}}(\bm{k})Q^{\rm(b)}_{lm}$ and $f^{Q_{lm}}(\bm{k})T^{\rm(b)}_{lm}$ do not appear in the Hamiltonian due to the time-reversal symmetry. The mean-field term $H_{m}$ is represented by the cluster multipole as already introduced in Eq. (13).

Let us first discuss the essential points for the spin-split band structures in AFMs without the SOC. We consider three types of band deformations, the symmetric spin splitting, the antisymmetric spin splitting, and the antisymmetric band deformation, which are categorized into the different symmetry classes. Note that the time-reversal symmetry is always broken, as we focus on magnetic orderings.

The first category is the symmetric spin splitting with respect to $\bm{k}$ when there is the spatial inversion symmetry in the system. In this situation, the spin-dependent band dispersion is described by the product of the even function of $\bm{k}$ and spin $\sigma$. This means that the symmetric spin splitting arises through the effective coupling between the momentum electric multipole $Q^{\rm(m)}_{lm}(\bm{k})$ and $\sigma$. The lowest-order contribution is given by the rank-0 electric monopole, which merely corresponds to the momentum-independent Zeeman-like spin splitting in the band structure. In the following, we mainly focus on the higher-rank contribution for $l\geq 2$. The schematic example in the case $Q^{\rm(m)}_{xy}(\bm{k})\sigma\sim k_{x}k_{y}\sigma$ is shown in Fig. 5(a).

The second category is the antisymmetric spin splitting with respect to $\bm{k}$ in the absence of the spatial inversion symmetry and the product symmetry of time-reversal and spatial inversion operations in addition to the breaking of time-reversal symmetry. The functional form of the spin splittings is represented by the product of the odd function of $\bm{k}$ and spin $\sigma$. Thus, the antisymmetic spin splitting occurs when momentum magnetic toroidal multipole $T^{\rm(m)}_{lm}(\bm{k})$ is coupled with $\sigma$. The schematic example in the case of $T^{\rm(m)}_{x}(\bm{k})\sigma\sim k_{x}\sigma$ is shown in Fig. 5(b).

The third category is the antisymmetric band deformation with the spin degeneracy in the absence of spatial inversion symmetry, with preserving the product symmetry of time-reversal and spatial inversion operations. The band structure becomes asymmetric due to the contribution from the odd function of $\bm{k}$ in addition to the even function of $\bm{k}$. In terms of the multipole description, the antisymmetric part of the band deformation is described by the emergence of the momentum magnetic toroidal multipole $T^{\rm(m)}_{lm}(\bm{k})$ solely without spin dependence Dubovik and Tugushev (1990); Kopaev (2009); Yanase (2014); Hayami et al. (2014). The schematic example in the case of $T^{\rm(m)}_{x}(\bm{k})\sim k_{x}$ is shown in Fig. 5(c).

To clarify a necessary condition of the microscopic model parameters for the band deformations in AFM orderings beyond symmetry argument, one can need to know when and how the momentum multipoles, $Q^{\rm(m)}_{lm}(\bm{k})$ and $T^{\rm(m)}_{lm}(\bm{k})$, become active and are coupled with spin $\sigma$. To examine such conditions, we introduce the following quantity at wave vector $\bm{k}$ in the magnetic unit cell,

where $\mu=0,x,y,z$, $\mathcal{H}=\sum_{\bm{k}}\mathcal{H}_{\bm{k}}$ and $\beta$ is the inverse temperature. By means of a sort of high-temperature expansion, the $s$th order expansion coefficient of the $\mu$-component, $g_{s}^{\mu}(\bm{k})$, gives the corresponding effective multipole coupling as $g_{s}^{\mu}(\bm{k})\sigma_{\mu}/2$. As the Hamiltonian in Eq. (46) consists of the cluster and bond multipoles in the matrix form, the $s$th order expansion of $e^{-\beta\mathcal{H}_{\bm{k}}}$ can be described by the product of the $s$-tuple of the matrices, $Q^{\rm(b)}_{lm}$, $T^{\rm(b)}_{lm}$, and $Q^{\rm(c)}_{lm}$. It is noted that the $\bm{k}$ dependence arises from the momentum multipoles $f^{Q_{lm}}(\bm{k})$ and $f^{T_{lm}}(\bm{k})$, which are always coupled with the bond multipoles in the scalar form as Eqs. (47) and (48). This analysis can be applied to not only $\bm{Q}=\bm{0}$ orderings but also finite commensurate $\bm{Q}$ orderings by choosing the appropriate minimal magnetic unit cell.

We present microscopic conditions for the band deformations from a multipole viewpoint in the cases of symmetric spin splitting, antisymmetric spin splitting, and antisymmetric band deformation with spin degeneracy in Secs. V.2.1, V.2.2, and V.2.3, respectively.