Essential model parameters for nonreciprocal magnons in multisublattice systems

Conductive phenomena in solids have long been studied in various fields of condensed matter physics, such as the giant magnetoresistance [1, 2, 3, 4, 5] and the anomalous Hall effect [6, 7, 8, 9, 10, 11, 12, 13, 14]. For these physical phenomena, the electronic band structures play an important role. The flat band structures give rise to magnetism, superconductivity, and the fractional quantum Hall effect [15, 16, 17, 18, 19, 20], while the linear band dispersions around the Dirac/Weyl points lead to unconventional topological properties [21, 22, 23, 24, 25]. Besides, the spin splittings in the band structure bring about fascinating physical phenomena, such as the Edelstein effect in noncentrosymmetric systems [26, 27, 28, 29], spin current generation in antiferromagnetic systems without the relativistic spin-orbit coupling [30, 31, 32, 33], and the spin-orbit-momentum locking in magnetic quadrupole systems [34].

Under space-time inversion symmetry, the electronic band structures are categorized into four groups: the $\bm{k}$-symmetric band dispersion with the spin degeneracy in the presence of both spatial inversion ($\mathcal{P}$) and time-reversal ($\mathcal{T}$) symmetries, the $\bm{k}$-(anti)symmetric spin-split band dispersion without $\mathcal{T}$ ($\mathcal{P}$) while keeping $\mathcal{P}$ ($\mathcal{T}$), and the $\bm{k}$-antisymmetric band dispersion without both $\mathcal{P}$ and $\mathcal{T}$, where $\bm{k}$ is the wave vector of electrons. In particular, the $\bm{k}$-antisymmetric band dispersion has been extensively studied in recent years, since it becomes a source of nonreciprocal conductive phenomena owing to the inequivalence between $\bm{k}$ and $-\bm{k}$ [35]. The nonreciprocal nonlinear optical effect is a typical example [36, 37, 38, 39]. The microscopic origin of the $\bm{k}$-antisymmetric band dispersion is accounted for by the active magnetic toroidal moment, which corresponds to a polar tensor with time-reversal odd [40, 41, 42, 43, 44, 45, 46, 47].

The nonreciprocal phenomena have also been discussed in magnetic insulators [48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 35, 62, 63, 64, 65, 66, 67, 68, 69]. In spite of the absence of carriers, the collective excitaions of magnons lead to directional-dependent dynamical properties, where we refer it to the nonreciprocal (asymmetric) magnons [35, 62]. Similar to the electron band dispersion, an appearance of nonreciprocal magnons is attributed to the active magnetic toroidal moment [70]. Although they were mainly studied for ferromagnetic slabs [48, 49] and for magnetic orderings in the noncentrosymmetric crystals [50, 51, 59, 71, 72], where the magnetic dipolar interaction and/or the Dzyaloshinskii-Moriya (DM) interaction are important [73, 74], it was shown that they occur even via other mechanisms, such as frustrated exchange interactions [75, 76] and bond-dependent symmetric exchange interactions [77, 78]. The nonreciprocal magnons have a potential to exhibit further intriguing nonreciprocal phenomena, such as the magneto-optical effect [79, 80, 81] and spin Seebeck effect [82], which avoid Joule heating.

Engineering asymmetric band deformations in the systems without $\mathcal{P}$ and $\mathcal{T}$ symmetries is important for nonreciprocal conductive phenomena irrespective of electrons and magnons. Meanwhile, the microscopic conditions have not been fully clarified yet, although active magnetic toroidal multipoles are necessary from the symmetry aspect [83, 47, 84, 85]. Recently, a useful framework to extract essential model parameters for the asymmetric band structure in the electron systems has been proposed on the basis of augmented multipoles [86]. Similar approach has also been performed in the magnon systems by introducing the bond-type magnetic toroidal dipole degree of freedom, which is only applied to the mechanism induced by the DM interaction [72]. It is desired to have a simple formula to investigate which model parameters contribute to the asymmetric band deformations in magnon systems with arbitrary spin interactions.

In the present study, we investigate the microscopic conditions for emergent nonreciprocal magnons in multisublattice systems in an analytical way. We show that the product of the Bogoliubov Hamiltonian after the linear spin wave approximation provides two important information for nonreciprocal magnons without the cumbersome Bogoliubov transformation. One is the momentum-space functional form and the other is the essential model parameters to cause the antisymmetric band deformations. We demonstrate that our scheme ubiquitously accounts for the microscopic key ingredients irrespective of the mechanisms by analyzing a spin Hamiltonian with the general exchange interactions in the one- to four-sublattice systems. We discuss the important magnon-hopping processes that arise from the exchange interactions in real space. We also test our scheme for both ferromagnetic and antiferromagnetic orderings with the DM interaction and the symmetric anisotropic interaction. Our results will be useful to extract the significant model parameters in inducing the nonreciprocal magnons under complicated noncollinear magnetic orderings.

The remaining of the paper is organized as follows. In Sec. II, we present a general method of extracting the essential model parameters from the Bogoliubov Hamiltonian. We present a general expression contributing to nonreciprocal magnons on the basis of the spin Hamiltonian with both symmetric and antisymmetric exchange interactions in the one- to four-sublattice systems in Sec. III. We apply the method for the ferromagnetic ordering in the breathing kagome lattice structure and the collinear/noncollinear antiferromagnetic orderings in the honeycomb and breathing kagome lattice structures in Sec. IV. Section V is devoted to a summary of the present paper. Appendix A provides lengthy expressions in terms of momentum-space functions in the three- and four-sublattice cases.

Let us start a general spin Hamiltonian, which is given by

$\displaystyle H=\sum_{ll^{\prime}}\sum_{\alpha\beta}S^{\alpha}_{l}\mathcal{J}^{\alpha\beta}_{ll^{\prime}}S^{\beta}_{l^{\prime}},$

(1)

with

$\displaystyle\mathcal{J}_{ll^{\prime}}=\left(\begin{array}[]{ccc}J_{ll^{\prime}}^{\perp}+J_{ll^{\prime}}^{v}&J_{ll^{\prime}}^{xy}+D_{ll^{\prime}}^{z}&J_{ll^{\prime}}^{zx}-D_{ll^{\prime}}^{y}\\ J_{ll^{\prime}}^{xy}-D_{ll^{\prime}}^{z}&J_{ll^{\prime}}^{\perp}-J_{ll^{\prime}}^{v}&J_{ll^{\prime}}^{yz}+D_{ll^{\prime}}^{x}\\ J_{ll^{\prime}}^{zx}+D_{ll^{\prime}}^{y}&J_{ll^{\prime}}^{yz}-D_{ll^{\prime}}^{x}&J_{ll^{\prime}}^{z}\\ \end{array}\right),$

(5)

where $S^{\alpha}_{l}$ is an $\alpha$ ($=x$, $y$, and $z$) component of classical spin at site $l$. $J^{\perp}_{ll^{\prime}}$, $J^{z}_{ll^{\prime}}$, $J^{v}_{ll^{\prime}}$, $J^{xy}_{ll^{\prime}}$, $J^{yz}_{ll^{\prime}}$, and $J^{zx}_{ll^{\prime}}$ are the symmetric exchange interactions, while $D^{x}_{ll^{\prime}}$, $D^{y}_{ll^{\prime}}$, and $D^{z}_{ll^{\prime}}$ are the antisymmetric exchange interactions. The latter corresponds to the DM interaction. The nonzero components of $\mathcal{J}_{ll^{\prime}}$ are determined by point group symmetry of the bond. For later convenience, the spin is rotated so as to align the local axis along the $z$ direction:

$\displaystyle(S_{l}^{x},S_{l}^{y},S_{l}^{z})^{\rm T}=R_{z}(\phi_{l})R_{y}(\theta_{l})(\tilde{S}_{l}^{x},\tilde{S}_{l}^{y},\tilde{S}_{l}^{z})^{\rm T},$

(6)

where $R_{z}(\phi_{l})$ and $R_{y}(\theta_{l})$ are the rotation matrices around the $z$ and $y$ axes, respectively, and T is the transpose of the vector. Then, the Hamiltonian in Eq. (1) is rewritten as

	$\displaystyle H$	$\displaystyle=\sum_{ll^{\prime}}\sum_{\alpha\beta}\tilde{S}^{\alpha}_{l}\mathcal{\tilde{J}}_{ll^{\prime}}^{\alpha\beta}\tilde{S}^{\beta}_{l^{\prime}}$
		$\displaystyle=\sum_{ll^{\prime}}\left(H^{\perp}_{ll^{\prime}}+H^{\rm{DM}}_{ll^{\prime}}+H^{v}_{ll^{\prime}}+H^{xy}_{ll^{\prime}}+H^{z}_{ll^{\prime}}+H^{yz/zx}_{ll^{\prime}}\right),$		(7)

where

	$\displaystyle H^{\perp}_{ll^{\prime}}$	$\displaystyle=\frac{\tilde{J}_{ll^{\prime}}^{\perp}}{2}(\tilde{S}^{+}_{l}\tilde{S}^{-}_{l^{\prime}}+\tilde{S}^{-}_{l}\tilde{S}^{+}_{l^{\prime}}),$		(8)
	$\displaystyle H^{\rm{DM}}_{ll^{\prime}}$	$\displaystyle=\frac{i\tilde{D}_{ll^{\prime}}}{2}(\tilde{S}^{+}_{l}\tilde{S}^{-}_{l^{\prime}}-\tilde{S}^{-}_{l}\tilde{S}^{+}_{l^{\prime}}),$		(9)
	$\displaystyle H^{v}_{ll^{\prime}}$	$\displaystyle=\frac{\tilde{J}_{ll^{\prime}}^{v}}{2}(\tilde{S}^{+}_{l}\tilde{S}^{+}_{l^{\prime}}+\tilde{S}^{-}_{l}\tilde{S}^{-}_{l^{\prime}}),$		(10)
	$\displaystyle H^{xy}_{ll^{\prime}}$	$\displaystyle=-\frac{i\tilde{J}_{ll^{\prime}}^{xy}}{2}(\tilde{S}^{+}_{l}\tilde{S}^{+}_{l^{\prime}}-\tilde{S}^{-}_{l}\tilde{S}^{-}_{l^{\prime}}),$		(11)
	$\displaystyle H^{z}_{ll^{\prime}}$	$\displaystyle=\tilde{J}^{z}_{ll^{\prime}}\tilde{S}^{z}_{l}\tilde{S}^{z}_{l^{\prime}}.$		(12)

$H^{zx}_{ll^{\prime}}$ and $H^{yz}_{ll^{\prime}}$ consist of the product of $\tilde{S}^{x}\tilde{S}^{z}$ and $\tilde{S}^{y}\tilde{S}^{z}$, respectively. The interaction tensor $\mathcal{\tilde{J}}_{ll^{\prime}}$ is represented by rotating $\mathcal{J}_{ll^{\prime}}$.

We investigate magnon spectra within a linear spin wave approximation. By applying the Holstein-Primakov transformation, which is given by $\tilde{S}^{+}_{i\eta}=\sqrt{2S}a_{i\eta}$, $\tilde{S}^{-}_{i\eta}=\sqrt{2S}a_{i\eta}^{\dagger}$, and $\tilde{S}^{z}_{i\eta}=S-a_{i\eta}^{\dagger}a_{i\eta}$ (the subscripts $i$ and $\eta$ denote the indices for a unit cell and a sublattice, respectively, and $a_{i\eta}$ is the boson operator for sublattice $\eta$), to the spin Hamiltonian in Eq. (II), the Bogoliubov Hamiltonian is derived. By performing the Fourier transformation as $a_{i\eta}\to a_{\bm{q}\eta}$, the resultant Bogoliubov Hamiltonian in the $n$-sublattice system is given by

	$\displaystyle H^{\rm B}$	$\displaystyle=\frac{S}{2}\sum_{\bm{q}}\Psi^{\dagger}_{\bm{q}}H^{\rm B}_{\bm{q}}\Psi_{\bm{q}},$		(13)
	$\displaystyle H^{\rm B}_{\bm{q}}$	$\displaystyle=\left(\begin{array}[]{cc}\mathcal{X}_{\bm{q}}&\mathcal{Y}_{\bm{q}}\\ \mathcal{Y}^{\dagger}_{\bm{q}}&\mathcal{X}^{*}_{-\bm{q}}\end{array}\right),$		(16)

where $\Psi^{\dagger}_{\bm{q}}=(a^{\dagger}_{\bm{q}1},a^{\dagger}_{\bm{q}2},\cdots,a^{\dagger}_{\bm{q}n},a_{-\bm{q}1},a_{-\bm{q}2},\cdots,a_{-\bm{q}n})$ and $\mathcal{X}_{\bm{q}}$ and $\mathcal{Y}_{\bm{q}}$ are the $n\times n$ matrices.

In Eq. (II), $H^{z}_{ll^{\prime}}$ corresponds to the diagonal elements of $\mathcal{X}_{\bm{q}}$, while $H^{\perp}_{ll^{\prime}}$, $H^{\rm{DM}}_{ll^{\prime}}$, $H^{v}_{ll^{\prime}}$, and $H^{xy}_{ll^{\prime}}$ correspond to the off-diagonal elements $\mathcal{X}_{\bm{q}}$ and $\mathcal{Y}_{\bm{q}}$. In other words, only the spin components perpendicular to $\tilde{S}^{z}_{l}$ contribute to a magnon hopping process. Meanwhile, $H^{yz/zx}_{ll^{\prime}}$ does not appear in Eq. (13), since it consists of the odd number of boson operators.

When $H^{\rm B}_{\bm{q}}$ is a positive-definite matrix, the Cholesky decomposition is possible as $H^{\rm B}_{\bm{q}}=K^{\dagger}_{\bm{q}}K_{\bm{q}}$, where $K_{\bm{q}}$ is the upper triangular matrix. Then, $H^{\rm B}_{\bm{q}}$ is transformed into the Hermitian matrix $H_{\bm{q}}$ as

$\displaystyle H_{\bm{q}}=K_{\bm{q}}gK^{\dagger}_{\bm{q}},$

(17)

where the $2n\times 2n$ matrix $g$ satisfies $(g)_{\eta\eta^{\prime}}=[\Psi_{\bm{q}\eta},\Psi^{\dagger}_{\bm{q}\eta^{\prime}}]$. The eigenvalues $\omega_{\bm{q}m}$ ($m$ is the band index) in Eq. (16) are obtained by diagonalizing $H_{\bm{q}}$.

Nonreciprocal magnon excitations mean that the eigenvalues have an antisymmetric component with respect to $\bm{q}$, i.e., $\omega_{\bm{q}m}\neq\omega_{-\bm{q}m}$. To investigate important model parameters for the nonreciprocal magnons in a systematic way, we introduce a following quantity as

	$\displaystyle E^{(s)}_{\bm{q}}$	$\displaystyle={\rm Tr}[\underbrace{H_{\bm{q}}H_{\bm{q}}\cdots H_{\bm{q}}}_{s}],$		(18)
		$\displaystyle={\rm Tr}[\underbrace{(H^{\rm B}_{\bm{q}}g)(H^{\rm B}_{\bm{q}}g)\cdots(H^{\rm B}_{\bm{q}}g}_{s})],$		(19)

which is related to the eigenenergy. A similar quantity has been discussed in the antisymmetric band modulation and spin splittings in the electron system [87, 86, 88]. The antisymmetric component is extracted by

$\displaystyle F^{(s)}_{\bm{q}}$

$\displaystyle=\frac{1}{2}(E^{(s)}_{\bm{q}}-E^{(s)}_{-\bm{q}}).$

(20)

Thus nonzero $F^{(s)}_{\bm{q}}$ signals the appearance of nonreciprocal magnons.

From the expression of Eq. (19), one can deduce the essential model parameters inducing nonreciprocal magnons, as detailed in Sec. III. In Eqs. (8)-(12), there are four types of magnon hoppings and one onsite potential in the real space Bogoliubov Hamiltonian, which are expressed as

	$\displaystyle H^{\perp}_{ll^{\prime}}$	$\displaystyle=S\tilde{J}_{ll^{\prime}}^{\perp}(a_{l}a^{\dagger}_{l^{\prime}}+a_{l}^{\dagger}a_{l^{\prime}}),$		(21)
	$\displaystyle H^{\rm{DM}}_{ll^{\prime}}$	$\displaystyle=iS\tilde{D}_{ll^{\prime}}(a_{l}a^{\dagger}_{l^{\prime}}-a_{l}^{\dagger}a_{l^{\prime}}),$		(22)
	$\displaystyle H^{v}_{ll^{\prime}}$	$\displaystyle=S\tilde{J}_{ll^{\prime}}^{v}(a_{l}a_{l^{\prime}}+a_{l}^{\dagger}a_{l^{\prime}}^{\dagger}),$		(23)
	$\displaystyle H^{xy}_{ll^{\prime}}$	$\displaystyle=-iS\tilde{J}_{ll^{\prime}}^{xy}(a_{l}a_{l^{\prime}}-a^{\dagger}_{l}a^{\dagger}_{l^{\prime}}),$		(24)
	$\displaystyle H^{z}_{ll^{\prime}}$	$\displaystyle=S\tilde{J}^{z}_{ll^{\prime}}(S-a_{l}^{\dagger}a_{l}-a_{l^{\prime}}^{\dagger}a_{l^{\prime}}).$		(25)

From theses expressions, one finds that the real (imaginary) part of the standard hopping $a^{\dagger}_{i\eta}a_{j\eta^{\prime}}$ is related to $H^{\perp}_{ll^{\prime}}$ ($H^{\rm{DM}}_{ll^{\prime}}$), which corresponds to the off-diagonal part of $\mathcal{X}_{\bm{q}}$, while the real (imaginary) part of the anomalous hopping $a_{i\eta}^{\dagger}a^{\dagger}_{j\eta^{\prime}}$ is related to $H^{v}_{ll^{\prime}}$ ($H^{xy}_{ll^{\prime}}$), which corresponds to the off-diagonal part of $\mathcal{Y}_{\bm{q}}$. As only the hopping processes to satisfy the magnon-number conservation are important, one can find that an even order of $\tilde{J}^{v}_{ll^{\prime}}$ and $\tilde{J}^{xy}_{ll^{\prime}}$ can contribute to nonreciprocal magnon excitations. In addition, when taking into account the fact that an odd order of imaginary hopping can also contribute to nonreciprocal magnon excitations, we expect that the antisymmetric magnon band structure is related to the odd order of an effective antisymmetric DM interaction or the even order of an effective symmetric anisotropic interaction. This indicates that the antisymmetric magnon band structure can be reversed regarding $\bm{q}$ by the sign of $\tilde{D}_{ll^{\prime}}$, while that is not by the sign of $\tilde{J}^{v}_{ll^{\prime}}$ and $\tilde{J}^{xy}_{ll^{\prime}}$. As we will show the general feature of $F^{(s)}_{\bm{q}}$ in Sec. III and the specific examples in Sec. IV, the quantity $F^{(s)}_{\bm{q}}$ gives a microscopic condition of nonreciprocal magnons irrespective of ferromagnets and antiferromagnets.

In this section, we discuss a general behavior of $F^{(s)}_{\bm{q}}$ independent of the lattice structures and the exchange interactions. We show the microscopic processes contributing to nonreciprocal magnons in the multisublattice systems with $n=1$-4: one-sublattice case in Sec. III.1, two-sublattice case in Sec. III.2, three-sublattice case in Sec. III.3, and four-sublattice case in Sec. III.4. It is noted that the present scheme can be also applied to the systems with the sublattice $n>4$ in a straightforward way.

In this section, we apply the expression in Eq. (20) to noncentrosymmetric ferromagnets and antiferromagnets to host nonreciprocal magnons. As the ferromagnets, we consider the ferromagnetic ordering in the breathing kagome lattice structure in Sec. IV.1. As the antiferromagnets, we consider three types of antiferromagnetic orderings: the staggered collinear antiferromagnetic state in the honeycomb lattice structure in Sec. IV.2, the up-up-down ferrimagnetic state in the breathing kagome lattice structure in Sec. IV.3, and the noncollinear 120^∘ antiferromagnetic state in the breathing kagome lattice structure in Sec. IV.4. In each section, we first show the Bogoliubov Hamiltonian and then we discuss magnon spectra and essential model parameters.