Dirac Magnons in a Honeycomb Lattice Quantum $\mathit{XY}$ Magnet ${\mathrm{CoTiO}}_3$

The discovery of massless Dirac electrons in graphene and topological Dirac-Weyl materials has prompted a broad search for bosonic analogues of such Dirac particles. Recent experiments have found evidence for Dirac magnons above an Ising-like ferromagnetic ground state in a two-dimensional (2D) kagome lattice magnet and in the van der Waals layered honeycomb crystal ${\mathrm{CrI}}_3$, and in a 3D Heisenberg magnet ${\mathrm{Cu}}_3{\mathrm{TeO}}_6$. Here, we report our inelastic neutron scattering investigation on a large single crystal of a stacked honeycomb lattice magnet ${\mathrm{CoTiO}}_3$, which is part of a broad family of ilmenite materials. The magnetically ordered ground state of ${\mathrm{CoTiO}}_3$ features ferromagnetic layers of ${\mathrm{Co}}^{2+}$, stacked antiferromagnetically along the $c$ axis. The magnon dispersion relation is described very well with a simple magnetic Hamiltonian with strong easy-plane exchange anisotropy. Importantly, a magnon Dirac cone is found along the edge of the 3D Brillouin zone. Our results establish ${\mathrm{CoTiO}}_3$ as a model pseudospin-$1/2$ material to study interacting Dirac bosons in a 3D quantum $\mathit{XY}$ magnet.

Popular Summary

Many types of particles other than electrons are known to exist in condensed-matter systems. For example, in an ordered magnet, a spin fluctuation can propagate coherently through the lattice just like a particle, known as a “magnon.” We study the energy-momentum (or dispersion) relation for magnons in an ordered antiferromagnet ${\mathrm{CoTiO}}_3$, which has a structure of stacked honeycomb layers analogous to graphene. Remarkably, magnons in ${\mathrm{CoTiO}}_3$ exhibit a very similar dispersion relation to that of electrons in graphene.

Graphene has fascinated condensed-matter physicists for decades because of its intriguing electronic properties. Unlike an ordinary metal, where the energy of an electron increases quadratically with momentum, electrons in graphene have a linear dispersion relation very much like photons, implying that they move through the lattice as if they are massless. Interactions between these unusual electrons in graphene, sometimes called Dirac electrons, are key to explaining its many strange electronic properties.

Unlike electrons, which are fermions respecting the Pauli exclusion principle, magnons are bosons and hence interact very differently with each other. The problem of interacting “Dirac bosons” has been investigated extensively in theory. However, it has not been studied much experimentally because of the lack of suitable model systems.

Our discovery of Dirac magnons in ${\mathrm{CoTiO}}_3$ makes it an ideal platform to study what happens when Dirac bosons interact.

Figure 1

(a) Structure of ${\mathrm{CoTiO}}_3$. A nonprimitive hexagonal unit cell, which is used to describe our data, is indicated by solid black lines. (b) Magnetic structure of the ${\mathrm{Co}}^{2+}$ sublattice. The ${\mathrm{Co}}^{2+}$ magnetic moments (shown by green arrows) order ferromagnetically within each honeycomb plane and antiferromagnetically along the $c$ axis. Red arrows labeled 1, $2^{\prime }$, and $2^{\prime \prime }$ are the magnetic couplings considered in the spin-wave calculation.

Figure 2

(a) Neutron scattering intensity plot on a log scale as a function of energy (in meV) and momentum transfer (in ${\AA }^{-1}$) for an incident neutron energy of $E_i=250\mathrm{meV}$. The data were collected at $T=5\mathrm{K}$. Since there is no directional dependence of the crystal field excitations, the scattering intensity is averaged over all orientations, and only the magnitude of momentum transfer is shown. (b) Momentum integrated intensity for $1.0{\AA }^{-1}<|\mathbf{Q}|<6.0{\AA }^{-1}$ plotted as a function of energy transfer. The inset shows trigonally distorted oxygen octahedra around each ${\mathrm{Co}}^{2+}$ ion. Horizontal and vertical arrows in panels (a) and (b), respectively, mark the positions of observed crystal field excitations. (c) Schematic crystal field levels for ${\mathrm{Co}}^{2+}$ (three $t_{2g}$ holes, with total $L_{\mathrm{eff}}=1$ and $S=3/2$) in the regimes $\lambda \ll {\mathrm{\Delta }}_{\text{trig}}$ (left) and ${\mathrm{\Delta }}_{\text{trig}}\ll \lambda$ (right), where $\lambda$ is the SOC and ${\mathrm{\Delta }}_{\text{trig}}$ is the trigonal distortion. Vertical arrows correspond to the sharp transitions seen in panels (a) and (b). The numbers in the square brackets denote degeneracies of the states. (See Supplemental Material [31] for a further description of the figure.)

Figure 3

(a)–(e) Momentum- and energy-resolved neutron scattering intensity map of magnons in ${\mathrm{CoTiO}}_3$. The data were obtained at $T=5\mathrm{K}$, with an incident neutron energy $E_i=50\mathrm{meV}$ at SEQUOIA at the Spallation Neutron Source (SNS). A pseudocolor intensity scale, where red (blue) denotes large (small) scattering intensity, is used to plot the data. Panels (a)–(d) show magnon excitations along ($H,\mathrm{-H}$) and ($H,H$) within the honeycomb plane at fixed $L=0.5$, while panel (e) shows excitations along $L$ for fixed in-plane momentum (1,1). (f)–(j) Calculated magnon spectra using $J_{\parallel ,1}=-4.41\mathrm{meV}$, $J_{\perp ,1}=0\mathrm{meV}$, and $J_{\parallel ,2^{\prime }=2^{\prime \prime }}=J_{\perp ,2^{\prime }=2^{\prime \prime }}=0.57\mathrm{meV}$ after convolving with the experimental energy resolution of 1 meV using spinW [37]. (k) Schematics of the 3D BZ and (l) projection of 3D reciprocal space onto the 2D honeycomb plane. The $L=0.5$ plane has been shaded in blue in panel (k). Directions of momentum transfers within the $L=0.5$ plane in panels (a) to (d) are denoted by thick blue lines in panel (l). Circles in panel (l) indicate positions of the Dirac point where magnon bands cross, which are denoted by arrows in panels (b)–(d). Circles and arrows of the same line style and color are used for the same Dirac point.

Figure 4

(a)–(c) Magnon spectra close to ${\mathbf{K}}^{\prime }=(-\frac{1}{3},\frac{2}{3})$ along different momentum cuts indicated in panel (d), which lie in the ($H,K$) plane (with $L=0.5$): (I) $(\xi -\frac{1}{3},\frac{2}{3})$, (II) $(\eta -\frac{1}{3},\eta +\frac{2}{3})$, and (III) $(2\epsilon -\frac{1}{3},-\epsilon +\frac{2}{3})$. (d) BZ in the ($H,K$) plane, with the directions of I–III indicated by thick blue bars. The Dirac point ${\mathbf{K}}^{\prime }$ is denoted by a red circle [same as in Fig. 3]. The Dirac cone near ${\mathbf{K}}^{\prime }$ is shown schematically in the inset. Cuts through different directions, which all pass through the Dirac point, should show an X-shaped dispersion (black solid line) where the magnon modes cross only at the Dirac point. (e)–(g) Measured intensity versus energy at fixed momentum along I–III, extracted from the magnon spectra in panels (a)–(c), respectively. The plots at different momenta in the same panel have been offset for clarity. Each cut is fit to two Gaussians with the same widths on top of a fixed constant background. Peak positions obtained from the fit are indicated by blue arrows.

Figure 5

(a,c,e) Constant energy slices of the ${\mathrm{CoTiO}}_3$ magnon spectra in the ($H,K$) plane. The constant energy of each slice is chosen to be at the Dirac point crossing with $\hslash {\omega }^{\star }\sim 8.5\mathrm{meV}$ (c), 1 meV below $\hslash {\omega }^{\star }$ (a), and 1 meV above $\hslash {\omega }^{\star }$ (e). Since magnon spectra in ${\mathrm{CoTiO}}_3$ have almost no $L$ dependence at energies close to $\hslash {\omega }^{\star }$, all spectra have been integrated over all $L$ measured to improve statistics. (b,d,f) Calculated intensity maps using the same parameters as in Fig. 3. The calculated spectra are convolved with an energy resolution of 1 meV. An energy integration range of 0.5 meV has been used for both the experimental data and calculation. White dashed lines in panels (c) and (d) denote the boundaries of 2D BZs. Red triangles in panel (b) are loops in a constant energy slice of the magnon dispersion below the Dirac points. As in Fig. 3, the green circle in panel (d) indicates the position of the Dirac point ${\mathbf{K}}^{\prime \prime }$.

Figure 6

(a) Slice of ${\mathrm{CoTiO}}_3$ magnon spectra in the $(H+\frac{2}{3},-H+\frac{2}{3},L)$ plane at the Dirac energy $\hslash {\omega }^{\star }$. (b) Magnon dispersion along $L$ with fixed $H=K=\frac{2}{3}$. (c) Schematic plot of Dirac nodal lines in the 3D BZ winding near the corners with additional nearest-neighbor interplane antiferromagnetic couplings $J_{\parallel ,3}>0$ and $J_{\perp ,3}<J_{\parallel ,3}$. Arrows depict the vorticity associated with the nontrivial Berry phase winding characterizing the nodal lines (positive is shown in blue; negative is shown in red). See text and Supplemental Material [31] for details.

Figure 7

(a) Calculated magnon density of states (DOS) for open boundary conditions (OBC, black) and periodic boundary conditions (PBC, red). The open boundary terminates on a surface with a zigzag edge. The DOS in both cases exhibits a V-shaped wedge arising from the bulk Dirac nodal lines. For OBC, the DOS is a sum of the surface and bulk magnon contributions. (b) The excess DOS, denoted $\mathrm{\Delta }\mathrm{DOS}$, corresponds to the difference DOS(OBC)-DOS(PBC), and it arises from surface magnons appearing at the Dirac line node energy. The geometry of the zigzag edge of an $\mathit{ABC}$ stacked honeycomb lattice is shown in the inset. (c) Magnon spectra along a cut parallel to the zigzag edge, $k_y$. Edge states can be seen connecting the bulk Dirac nodes at the Dirac energy $E\simeq 8.5\mathrm{meV}$. These states are responsible for the excess $\mathrm{\Delta }\mathrm{DOS}$ in panel (b). These surface magon states are not found for a surface along the armchair edge (see Supplemental Material [31]).