Magnonic Weyl states in Cu2OSeO3

TL;DR

This paper explores the topological magnon excitations in Cu2OSeO3, revealing Weyl magnon nodes, their splitting due to Dzyaloshinsky-Moriya interactions, and potential measurable thermal Hall effects.

Contribution

It provides a systematic analysis of magnon Weyl states in Cu2OSeO3, including theoretical predictions and experimental verification of topological properties.

Findings

Weyl magnon nodes with topological charges ±2 identified at high-symmetry points

Dzyaloshinsky-Moriya interactions cause splitting of Weyl points

Thermal Hall conductivity linked to Weyl points and tunable by crystal symmetry

Abstract

The multiferroic ferrimagnet Cu$_2$OSeO$_3$ with a chiral crystal structure attracted a lot of recent attention due to the emergence of magnetic skyrmion order in this material. Here, the topological properties of its magnon excitations are systematically investigated by linear spin-wave theory and inelastic neutron scattering. When considering Heisenberg exchange interactions only, two degenerate Weyl magnon nodes with topological charges $\pm$2 are observed at high-symmetry points. Each Weyl point splits into two as the symmetry of the system is further reduced by including into consideration the nearest-neighbor Dzyaloshinsky-Moriya interaction, crucial for obtaining an accurate fit to the experimental spin-wave spectrum. The predicted topological properties are verified by surface state and Chern number analysis. Additionally, we predict that a measurable thermal Hall conductivity can be associated with the emergence of the Weyl points, the position of which can be tuned by changing the crystal symmetry of the material.

Tables

Table 1. Table 1.: Values of the parameters entering the Heisenberg Hamiltonian (1). Five exchange interactions are listed together with the corresponding interatomic distances which are very similar to previous works. The nearest-neighbor DMI vector was chosen so as to reproduce the experimental spin-wave dispersion.

Parameters	Distance (Å)	$J$ (meV) Portnichenko et al. (2016)	$𝐃$ (meV)
$J_{\text{w}}^{\text{FM}}$	3.039	–4.2	(–0.458, 2.011, 0.565)
$J_{\text{s}}^{\text{AFM}}$	3.057	12.3	0
$J_{\text{s}}^{\text{FM}}$	3.22	–14.5	0
$J_{\text{w}}^{\text{AFM}}$	3.30	2.33	0
$J_{\text{O..O}}^{\text{AFM}}$	6.35	3.88	0

Table 2. Table S2.: The bond vector and the DMI vector for the nearest neighbor interaction.

Atom $i$ position	Atom $j$ position	${𝐑}_{ij}$	${𝐃}_{ij}$
$r_5$, $(a,b,c)$	$r_{12}$, $(0.5-b,1-c,0.5+a)$	$(0.5-b-a,1-c-b,0.5+a-c)$	$(D_x,D_y,D_z)$
$r_5$, $(a,b,c)$	$r_{16}$, $(-0.5+c,0.5-a,1-b)$	$(-0.5+c-a,0.5-a-b,1-b-c)$	$(-D_z,D_x,D_y)$
$r_6$, $(b,c,a)$	$r_{13}$, $(1-c,0.5+a,0.5-b)$	$(1-c-b,0.5+a-c,0.5-b-a)$	$(D_y,D_z,D_x)$
$r_6$, $(b,c,a)$	$r_{14}$, $(0.5-a,1-b,-0.5+c)$	$(0.5-a-b,1-b-c,-0.5+c-a)$	$(D_x,D_y,-D_z)$
$r_7$, $(c,a,b)$	$r_{11}$, $(0.5+a,0.5-b,1-c)$	$(0.5+a-c,0.5-b-a,1-c-b)$	$(D_z,D_x,D_y)$
$r_7$, $(c,a,b)$	$r_{15}$, $(1-b,-0.5+c,0.5-a)$	$(1-b-c,-0.5+c-a,0.5-a-b)$	$(D_y,D_z,D_x)$
$r_8$, $(1-a,0.5+b,1.5-c)$	$r_9$, $(0.5+b,1.5-c,1-a)$	$(-0.5+b+a,1-c-b,-0.5-a+c)$	$(-D_x,D_y,-D_z)$
$r_8$, $(1-a,0.5+b,1.5-c)$	$r_{10}$, $(1.5-c,1-a,0.5+b)$	$(0.5-c+a,0.5-a-b,-1+b+c)$	$(D_z,D_x,-D_y)$
$r_9$, $(0.5+b,1.5-c,1-a)$	$r_{10}$, $(1.5-c,1-a,0.5+b)$	$(1-c-b,-0.5-a+c,-0.5+b+a)$	$(D_y,-D_z,-D_x)$
$r_{11}$, $(0.5+a,0.5-b,1-c)$	$r_{15}$, $(1-b,-0.5+c,0.5-a)$	$(0.5-b-a,-1+c+b,-0.5-a+c)$	$(D_x,-D_y,-D_z)$
$r_{12}$, $(0.5-b,1-c,0.5+a)$	$r_{16}$, $(-0.5+c,0.5-a,1-b)$	$(-1+c+b,-0.5-a+c,0.5-b-a)$	$(-D_y,-D_z,D_x)$
$r_{13}$, $(1-c,0.5+a,0.5-b)$	$r_{14}$, $(0.5-a,1-b,-0.5+c)$	$(-0.5-a+c,0.5-b-a,-1+c+b)$	$(-D_z,D_x,-D_y)$

Equations

$$H=\sum _{⟨ij⟩}{\mathbf{S}}_i^{†}{\hat{J}}_{ij}{\mathbf{S}}_j^{}\text{vspace}-1ex,$$

$$\hat{J}_{ij}={\left(\begin{array}[]{ccc}J_{ij}^{x}&D_{ij}^{z}&-D_{ij}^{y}\\ -D_{ij}^{z}&J_{ij}^{y}&D_{ij}^{x}\\ D_{ij}^{y}&-D_{ij}^{x}&J_{ij}^{z}\end{array}\right)}$$

$$\alpha_{l}(\mathbf{k})=\left(\begin{array}[]{cc}a_{l}(\mathbf{k})\\ a_{l}^{\dagger}(\mathbf{-k})\end{array}\right)=\frac{1}{\sqrt{N}}\sum_{m}e^{-\mathrm{i}\mathbf{k}\mathbf{R_{m}}}\left(\begin{array}[]{cc}a_{lm}\\ a_{lm}^{\dagger}\end{array}\right),$$

$$\mathrm{i}\frac{\mathrm{d}{\alpha }_l}{\mathrm{d}t}=[{\alpha }_l,H_2]=\sum _m{D}_{lm}{\alpha }_m,$$

$${\mathrm{\Omega }}_n(\mathbf{k})=-\sum _{m\ne n}\frac{\mathrm{Im}\left[⟨V_{n\mathbf{k}}^L⟩{\partial }_{\mathbf{k}}D(\mathbf{k})V_{m\mathbf{k}}^R\times ⟨V_{m\mathbf{k}}^L⟩{\partial }_{\mathbf{k}}D(\mathbf{k})V_{n\mathbf{k}}^R\right]}{({ϵ}_{n\mathbf{k}}-{ϵ}_{m\mathbf{k}}{)}^2},$$

$$Q_i=\frac{1}{2\pi }{\int }_{S_i}\mathrm{\Omega }(\mathbf{k})\cdot \mathbf{n}d{S}_i,$$

$$C(P)=\frac{1}{2\pi }{\int }_P\mathrm{\Omega }(\mathbf{k})\cdot \mathbf{n}dP,$$

$${\kappa }^{ij}(ϵ)=-\frac{k_{\text{B}}^{2}T}{(2\pi {)}^3\hslash }\sum _n{\int }_{\text{BZ}}\delta ({ϵ}_{n\mathbf{k}}-ϵ)C_2(f_n^{\text{B}}){\Omega }_n^{ij}(\mathbf{k})d\mathbf{k},$$

$$C_2(x)=(1+x){\left(\ln \frac{1+x}{x}\right)}^2-{\ln }^{2}x-2{\mathrm{Li}}_2(-x),$$

$$LW(k,j)=\sum _i{V}_L^i(k,j)V_R^i(k,j)(R_z^i-0.5),$$

Full text

Weyl point, spin waves, B20 cluster, Heisenberg model, neutron scattering

I Introduction

Topological insulators and Weyl semimetals attracted tremendous attention as the most prominent realizations of topologically nontrivial electronic matter Bernevig (2013); F. Ortmann, S. Roche, S. O. Valenzuela (2015) (eds.); Bansil et al. (2016); Armitage et al. (2018); J. Zang, V. Cros, and A. Hoffmann (2018) (eds.). In recent years, topologically protected band touching points, known as Weyl nodes, were observed in electronic Xu et al. (2015); Lv et al. (2015); Lu et al. (2015), photonic Lu et al. (2014), phononic Li et al. (2018), and magnetic excitation spectra Chisnell et al. (2015); Yao et al. (2018); Bao et al. (2018). In relationship to magnetically ordered materials, new concepts of topological magnon insulators Zhang et al. (2013); Owerre (2016); Nakata et al. (2017); Li and Kovalev (2018), topological spinon semimetals Schaffer et al. (2015), Dirac and Weyl magnon states Li et al. (2016); Mook et al. (2016); Li et al. (2017); Jian and Nie (2018); Owerre (2018a) were introduced, offering promising new applications in the emerging field of spintronics Šmejkal et al. (2017, 2018); Rückriegel et al. (2018); Wang et al. (2018).

Experimentally, topologically nontrivial magnon states were recently identified in a two-dimensional spin-$\frac{1}{2}$ kagome-lattice ferromagnet Chisnell et al. (2015) and in the three-dimensional (3D) antiferromagnet Cu3TeO6 by two independent groups Yao et al. (2018); Bao et al. (2018) using inelastic neutron scattering (INS). On the theory side, it has been realized that chiral magnets offer a generic route to the realization of topological magnon states, representing a magnon analog of topological insulators. As a result of antisymmetric exchange, known as Dzyaloshinsky-Moriya interaction (DMI) Dzyaloshinsky (1958); Moriya (1960), the bulk magnon spectrum of a chiral magnet can acquire a topological energy gap that supports a topologically protected gapless Dirac cone in the surface magnon spectrum Li and Kovalev (2018). A similar mechanism based on DMI was also proposed for the formation of magnonic Weyl crossing points in the spin-wave spectrum of the noncoplanar antiferromagnetic (AFM) state on a breathing-pyrochlore lattice Li et al. (2016); Mook et al. (2016); Jian and Nie (2018).

The cubic copper(II)-oxoselenite Cu2OSeO3 is a multiferroic ferrimagnet with a chiral crystal structure that came under the focus of recent attention owing to the emergence of skyrmion order in this material Seki et al. (2012a, b); Langner et al. (2014, 2017); Müller et al. (2017). Its crystal structure is cubic (space group $P2_{1}3$) with the lattice constant $a=8.925$ Å Effenberger and Pertlik (1986). The magnetic sublattice of Cu2+ ions can be approximated as a distorted breathing-pyrochlore lattice, consisting of slightly deformed tetrahedral Cu4 clusters in a face-centered cubic (fcc) arrangement Portnichenko et al. (2016). Magnetic interactions within the tetrahedron lead to a ferrimagnetic ground state, in which one of the Cu2+ spins is antiparallel to the other three, resulting in the total spin $S=1$ of the cluster Yang et al. (2012); Romhányi et al. (2014); Janson et al. (2014). Weaker interactions between the clusters lead to a long-range spin-spiral order that sets in below $T_{\text{C}}\approx 57$ K. Existing magnetic models Yang et al. (2012); Romhányi et al. (2014); Janson et al. (2014); Chizhikov and Dmitrienko (2015) consider up to 5 Heisenberg exchange interactions and up to 5 DMI vectors. These models were used to describe the INS spectrum of spin-wave excitations in a broad energy range and in the whole reciprocal space Portnichenko et al. (2016), as well as electron spin resonance (ESR) that probes spin-wave excitations at the zone center Ozerov et al. (2014). However, the DMI was initially neglected in these studies.

This simplified description, that involves only Heisenberg interactions, provides a qualitatively good fit to the experimental spin-wave dispersion over the entire Brillouin zone Portnichenko et al. (2016) with the exception of the zone corner ($\mathbf{R}$ point), where the magnon bands remain degenerate for any values of the exchange parameters. Tucker et al. Tucker et al. (2016) recently showed that this degeneracy is removed by DMI, leading to a clearly resolved spin gap of $\sim$1.6 meV in the magnon spectrum, which they observed by neutron spectroscopy. These observations are a strong indication for the existence of topological magnon states in Cu2OSeO3, which motivated our present study.

In the following, we present spin-dynamical calculations of the magnon spectrum in the presence of DMI that was adjusted to provide the best fit to the experimental spin-wave dispersion in the vicinity of the $\mathbf{R}$ point. Using linear spin-wave theory (LSWT) in combination with high-resolution neutron spectroscopy, we show that, in the absence of DMI terms, two pairs of degenerate Weyl nodes with the topological charge $+2$ and $-2$ are located at the zone center ($\mathbf{\Gamma}$ point) and at the zone boundary ($\mathbf{R}$ point). Consideration of the nearest-neighbor DMI is sufficient to lift the degeneracy of these Weyl nodes, so that they are shifted away from the high-symmetry points into a position that sensitively depends on the direction and magnitude of the DMI vector. A direct observation of the resulting Weyl points would offer a possibility to accurately extract the DMI from INS measurements. We verify the predicted topological properties by the Chern number analysis and give quantitative predictions for the location of magnonic Weyl points in the spin-wave spectrum. We also analyze topologically protected magnon surface states and estimate the magnetic contribution to the thermal Hall conductivity that may serve as robust hallmarks of the emergent topological states in Cu2OSeO3, awaiting a direct experimental verification.

II Results

II.1 Magnetic model, experimental result and magnon spectrum