††thanks: Current affiliation: Department of Physics and Astronomy, McMaster
University, Hamilton, Ontario L8S 4M1, Canada

# Field-dependent Magnons in a Honeycomb Antiferromagnet CoTiO3

Bo Yuan  Physics Department, University of Toronto, 60 St. George Street,
Toronto, Ontario M5S 1A7, Canada  Ezekiel Horsley  Physics Department,
University of Toronto, 60 St. George Street, Toronto, Ontario M5S 1A7, Canada
M.B. Stone  Neutron Scattering Division, Oak Ridge National Laboratory, Oak
Ridge, Tennessee 37831, USA  Nicholas P. Butch  NIST Center for Neutron
Research, National Institute of Standards and Technology Gaithersburg,
Maryland, 20899, USA  Guangyong Xu  NIST Center for Neutron Research, National
Institute of Standards and Technology Gaithersburg, Maryland, 20899, USA  Guo-
Jiun Shu  Department of Materials and Mineral Resources Engineering, National
Taipei University of Technology, Taipei 10608, Taiwan  Institute of Mineral
Resources Engineering, National Taipei University of Technology, Taipei 10608,
Taiwan  Taiwan Consortium of Emergent Crystalline Materials, Ministry of
Science and Technology, Taipei 10622, Taiwan  J.P. Clancy  Department of
Physics and Astronomy, McMaster University, Hamilton, Ontario L8S 4M1, Canada
Young-June Kim  Physics Department, University of Toronto, 60 St. George
Street, Toronto, Ontario M5S 1A7, Canada

###### Abstract

We report field-dependent high-resolution inelastic neutron scattering (INS)
measurements on the honeycomb lattice magnet, CoTiO3, to study the evolution
of its magnon excitations across a spin reorientation transition driven by an
in-plane magnetic field. By carrying out elastic neutron scattering in a
magnetic field, we show that the sample transitions from a collinear
antiferromagnetic state with multiple magnetic domains at a low field to a
mono-domain state with a canted magnetic structure at a high field. Concurrent
with this transition, we observed significant changes in both the energy and
the width of the zone center magnon peak. The observed width change is argued
to be consistent with an unusual zero-field state with extended domain walls.
On the other hand, the magnon spectra near the $\mathbf{K}$ point of the
Brillouin zone boundary are found to be largely insensitive to the changes in
the ordered moment directions and the domain configuration. We argue that this
observation is difficult to explain within the framework of the bond-dependent
model proposed in a recent INS study [Elliot et al, Nat. Commun., 12, 3936
(2021)]. Our study therefore calls for alternative explanations for the
observed $\mathbf{K}$-point gap in CoTiO3.

##  I Introduction

Searching for magnetic materials with dominant Kitaev interactions[1], or more
generally bond-dependent interactions, have become an important goal of
today’s research in quantum magnetism due to their ability to support exotic
ground states such as the quantum spin liquid state. After more than a decade
of intensive experimental and theoretical research, two properties, namely,
strong spin-orbit coupling and an edge-sharing bond geometry have been
identified as the key ingredients for hosting such non-trivial exchange
interactions[2]. With these general considerations, a few promising examples
have emerged including $\alpha-$RuCl3, iridates with honeycomb and hyper-
honeycomb structure, and more recently a number of honeycomb cobaltates (Given
the large number of works in this still rapidly developing field, instead of
giving references on specific materials, here we only cite a number of review
articles. See Ref. [3, 4, 5, 6, 7, 8, 9]) A wealth of experimental results
ranging from magnetic order (the so called zigzag order in a honeycomb lattice
has been widely associated with a dominant Kitaev term[10]) to some
unconventional bulk [11, 12, 13]) and thermal transport responses [14, 15, 16,
17]) have been cited as indirect evidence for the existence of such
interactions and possible exotic ground states in these materials. However, a
quantitative understanding of the spin Hamiltonian has not been achieved in
most of them. This is the case even for the most well-known Kitaev candidate
RuCl3 where the magnitude and the form of the dominant bond-dependent
interactions are still under scrutiny (E.g. See Table I of Ref. [18] for a
summary of all bond-dependent interactions proposed for RuCl3)

The main difficulty is a lack of careful measurement of their magnon
excitations (either in the zero-field ordered phase, or a field polarized
phase when the zero-field excitation spectrum is strongly damped such as
$\alpha-$RuCl3[19]) that can be compared with linear spin wave calculations -
the gold standard for parametrization of the spin Hamiltonian in a material.
Compared to $\alpha$-RuCl3 and the honeycomb iridates both with a small moment
size, and in the case of iridates, a large neutron absorption cross-section,
honeycomb cobaltates are clearly better suited for such purposes because of
the strong magnetic scattering by the large Co2+ moments. Inelastic neutron
scattering (INS) measurements carried out on powder samples of honeycomb
cobaltate all show well-defined spin waves with a finite anisotropy gap,
consistent with the existence of some type of exchange anisotropy between the
pseudospins, although the extracted exchange parameters can vary significantly
when trying to fit the powder averaged spectrum[20, 13, 21, 22, 23, 24]. On
the other hand, more detailed INS measurements of magnon excitations using
single crystal samples became available only recently for two of the most
prominent candidates for dominant Kitaev interactions, Na2Co2TeO6[25] and
BaCo2(AsO4)2[26]. Surprisingly, the single crystal magnon spectra for both of
these compounds show large deviations from the predictions of a simple model
with dominant Kitaev interactions, at odds with some early powder INS work[21,
23, 24, 22]. These recent results raise important questions about the
relevance of bond-dependent interactions [27, 28] in real materials, and call
for careful modelling of their single crystal magnon spectra.

To address these questions, we revisit the magnon dispersions in a well-known
honeycomb lattice with edge-sharing bond geometry, CoTiO3. Unlike most Kitaev
candidates with a zigzag order, the ground state of CoTiO3 has a simple A-type
antiferromagnetic order with ferromagnetically ordered honeycomb planes
antiferromagnetically coupled in the out-of-plane direction (See Fig. 1 for
the crystal and magnetic structure of CoTiO3). Simplicity of the magnetic
order in CoTiO3 allowed detailed modelling of the magnon spectra of single
crystal samples [29, 30]. Our first INS measurement[29] found a magnon
spectrum consistent with a dominant XXZ interaction between the pseudospins,
while subsequent high resolution INS[30] and optical measurements[31]
discovered subtle discrepancies that require exchange anisotropies beyond the
XXZ model. First, like the other honeycomb cobaltates, a gap in the Goldstone
mode indicative of an in-plane U(1) symmetry breaking was discovered at the
magnetic zone center, requiring either a quantum order by disorder mechanism
with bond-dependent interactions [30] or a multi-spin ring exchange
interaction [31]. In addition, a gap at the $\mathbf{K}$ point of the
Brillouin zone (also referred to as a Dirac gap) was discovered between the
optical and acoustic magnon branches, which was modelled using a linear spin
wave theory with the bond-dependent interactions [30].

Figure 1:  (a,b) 3-dimensional crystal and magnetic structure of CoTiO3. (c)
ordered moment configuration for three consecutive Co honeycomb layers within
each structural unit cell separated by $\frac{1}{3}c$ at zero field. (d) is
the same as (c) but in the high-field mono-domain state for a field applied
along the crystallographic $\mathbf{b}$ axis or equivalent (-1,2,0) in
reciprocal space. At zero field, the ordered moments on neighbouring honeycomb
layers denoted by $\mathbf{m}_{1}$ and $\mathbf{m}_{2}$, respectively, are
collinear and are antiferromagnetically coupled. Without the knowledge of the
actual ordered moment directions at zero field, the ordered moments are shown
to point along an arbitrary in-plane direction in (b) and (c). At high field,
the two antiferromagnetic sublattices cant away from the collinear
configuration towards the direction of the applied field. The size of the
canting is denoted by $\phi$ in (d). The trigonal unit cell used throughout
the paper has been indicated by the black solid lines. The relevant in-plane
real-space and reciprocal space coordinates are indicated by black solid
arrows in (c) and (d), respectively.

However, there are two difficulties in the modelling of the zero-field spectra
that render the determination of the exchange parameters in CoTiO3 an under-
constrained problem. First, the directions of ordered moments necessary for
any linear spin wave calculations are not known in CoTiO3, which had to be
assumed in Ref [30]. Second, there exist both structural and magnetic domains
at zero field in CoTiO3. The structural domains are related by a two-fold
rotation around (1,1,0), while the configuration of magnetic domains at zero
field is unknown in CoTiO3. Since different domains give rise to distinct
excitation spectra, knowledge of the actual domain configuration is important
for the correct simulation of the zero-field spectra which is an average over
all domains.

In this paper, we address these difficulties by carrying out field-dependent
high resolution neutron scattering measurements with an in-plane magnetic
field. The in-plane field drives a spin reorientation transition from a zero-
field multi-domain state to a high-field mono-domain state with a canted
magnetic structure in CoTiO3, which we characterize using elastic neutron
scattering. The changes in magnon spectra near both the magnetic zone center
(ZC) and the K point of the Brillouin zone boundary were studied by high
resolution INS measurements with a magnetic field. In addition to a change in
the energy of the ZC magnon, we observe a strong suppression of its width
across the spin reorientation transition, which suggests a complex zero-field
state with extended domain walls. On the other hand, we find very similar
magnon spectra near the $\mathbf{K}$ point in both the high-field mono-domain
state and the zero-field multi-domain state, which we argue to be inconsistent
with the behaviour expected for general bond-dependent interactions[30]. We
close by discussing alternative explanations for the observed gap and possible
experimental tests for them.

##  II Experimental Details

The same CoTiO3 crystal grown by the floating zone method from Ref [29, 32]
was used in the measurements reported in this paper. Inelastic neutron
scattering (INS) measurements of the low energy zone center magnons were
carried out at the Disk Chopper Spectrometer (DCS) and the Spin Polarized
Inelastic Neutron Spectrometer (SPINS) at the NIST Center for Neutron Research
(NCNR). In the DCS experiment, the crystal was aligned in the (H,H,L)
scattering plane (Throughout the paper, we use a hexagonal unit cell with
$a=b=5.0662\AA$ and $c=13.918\AA$). Two incident energies,
$\mathrm{E_{i}}=4.04~{}\mathrm{meV}$ and $\mathrm{E_{i}}=5.66~{}\mathrm{meV}$,
were used, which offered an energy resolution of
$\mathrm{\Delta\mathrm{E}}\approx 0.13\mathrm{meV}$ and
$\mathrm{\Delta\mathrm{E}}\approx 0.23\mathrm{meV}$, respectively, at the
elastic line. In the SPINS experiment, the crystal was aligned in the (H,0,L)
plane and a fixed final energy, $\mathrm{E_{f}}=5~{}\mathrm{meV}$, was used.
An energy resolution of $\sim$0.2 meV was achieved with a vertically focussing
pyrolitic graphite (PG) monochromator, a flat PG analyzer, a Be filter and a
collimation setting of guide-open-80’-open. The same 10 T vertical field
magnet was used in the DCS and SPINS experiments to apply a field along the
(1,-1,0) and (-1,2,0) directions, respectively. The high resolution
measurements of the high energy magnon near the $\mathbf{K}$ point were
carried out at the fine-resolution Fermi chopper spectrometer, SEQUOIA, at the
Spallation Neutron Source (SNS) at the Oak Ridge National Laboratory. The
crystal was aligned in the (H,H,L) plane. A T0 chopper and a high-resolution
Fermi chopper, FC2, rotating at frequencies of 30 Hz and 180 Hz, respectively,
were used to select an incident energy of 12 meV with a resolution of
$\sim$0.27 meV at the elastic line. An 8 T vertical field magnet was used to
apply a field along (1,-1,0). Due to the lack of radial collimation at
SEQUOIA, the magnet gave a large background that was removed by taking empty
magnet measurements without the sample. Alignment of the crystal before all
experiments was carried out at the McMaster Alignment Diffractometer (MAD) at
the McMaster Nuclear Reactor at McMaster University. Magnetization
measurements were carried out using a Quantum Design SQUID Magnetic Property
Measurement System. Linear spin wave modelling of the INS data was performed
using the SpinW package[33].

##  III Results

###  III.1 Spin Reorientation

The magnetization as a function of magnetic field is plotted in Fig. 2 (a) for
several field directions. (Note that we use B to denote magnetic field and H
is used to refer to the reciprocal lattice index along a*.) The magnetic
susceptiblity $\frac{dM}{dB}$ is also plotted in the same figure. The spin
reorientation in CoTiO3 induced by an in-plane magnetic field was first
observed through bulk magnetization measurements [34]. As shown in Fig. 2 (a),
the transition is gradual and manifests as a gradual increase in
$\frac{dM}{dB}$ up to $\sim$2 T above which $\frac{dM}{dB}$ approximately
doubles compared to that at zero-field. The observed change is consistent with
a transition from a zero-field multi-domain state favoured by the intrinsic
spin anisotropy to a high field mono-domain state where the ordered moments
are aligned approximately perpendicular to the applied field to maximize the
Zeeman energy gained by canting of the antiferromagnetic sub-lattices [See
Fig. 1 (d)].

Since only the spin component perpendicular to momentum transfer,
$\mathbf{Q}$, is detected in a neutron diffraction experiment, we directly
observed the change in ordered moment direction by tracking the field
dependence of the intensity of a magnetic Bragg peak with a large in-plane
$\mathbf{Q}$ component. In our experiments, a magnetic field was applied along
a direction in the $\mathbf{ab}$ plane with a vertical field magnet by
aligning the crystal in a scattering plane containing (0,0,1). For example, by
aligning the crystal in the (H,0,L)-plane, the field is applied perpendicular
to (1,0,0) along the crystallographic $\mathbf{b}$-axis [This is equivalent to
(-1,2,0) in reciprocal space. In this section, we will use the real-space and
reciprocal space notations interchangeably- See Fig. 1 for their relations.].
Under this field, the ordered moments are mostly along the (1,0,0) direction
in the high-field state [Fig. 1 (d)]. This leads to suppression of the
(1,0,L)-type peaks with a small L, since $\mathbf{Q}$ is almost parallel to
the ordered moments. This is consistent with the measured field dependence of
the (1,0,0.5) magnetic Bragg peak (Fig. 2 (b)) where an intensity drop mirrors
the behaviour of $\frac{dM}{dB}$. On the other hand, the intensity of the
(0,0,1.5) Bragg peak has a very weak field dependence, consistent with the
fact that the spins only rotate in the $\mathbf{ab}$ plane, and are therefore
always perpendicular to $\mathbf{Q}=(0,0,1.5)$. The (0,0,1.5) Bragg peak
intensity decreases quadratically with field, as shown by a fit of its field
dependence to
$I_{(0,0,1.5)}=I_{0}\left[1-(\frac{B}{B_{\mathrm{sat}}})^{2}\right]$ (black
solid line in Fig. 2 (b)), where $B_{\mathrm{sat}}$ stands for the saturation
field. The field dependence could be either due to a reduction in the
magnitude of the staggered magnetization,
$\mathbf{n}~{}=~{}\mathbf{m}_{1}-\mathbf{m}_{2}$, or a sample movement induced
by a large applied field. $\mathbf{m}_{1}$ and $\mathbf{m}_{2}$ are the sub-
lattice magnetizations on two adjacent honeycomb layers as shown in Fig. 1
(d). The sample movement is ruled out by the field dependence of the two
nuclear Bragg peak intensities, (1,0,2) and (0,0,3). Unlike (0,0,1.5), their
intensities increase quadratically with the field. These observations can be
explained by a canting of $\mathbf{m}_{1}$ and $\mathbf{m}_{2}$ away from
their zero-field collinear configuration towards the applied field [see Fig. 1
(d)], leading to a decrease in the size of
$\mathbf{n}=\mathbf{m}_{1}-\mathbf{m}_{2}$ but an increase in the net
magnetization or $\mathbf{m}=\mathbf{m}_{1}+\mathbf{m}_{2}$, the latter of
which contributes to the elastic intensity at nuclear Bragg peak positions.

To estimate the spin canting at high field, we note that
$I_{(0,0,1.5)}(\mathrm{B})~{}\propto~{}m_{0}(\mathrm{B})^{2}~{}\cos^{2}[\phi(\mathrm{B})]$,
where $\phi$ is the canting of sub-lattice magnetizations from $\mathbf{n}$,
and $|\mathbf{m}_{1}|=|\mathbf{m}_{2}|=m_{0}$ is proportional to the size of
the Co2+ ordered moment. The field dependence of $m_{0}(\mathrm{B})$ can be
determined from the high field magnetization measurement above the saturation
field when all moments have been polarized by the field [35]. In Ref. [35],
$m_{0}$ increases linearly with field as
$\frac{m_{0}(\mathrm{B})-m_{0}(0)}{m_{0}(0)}=0.0044\mathrm{B}$ (B is in
Tesla), which can be attributed to a field-induced admixture of the excited
doublet ($J_{z}^{\mathrm{tot}}=\pm\frac{3}{2}$) into the ground doublet
($J_{z}^{\mathrm{tot}}=\pm\frac{1}{2}$). Given
$\frac{I_{(0,0,1.5)}(7.5~{}\mathrm{T})}{I_{(0,0,1.5)}(0~{}T)}=0.87$ from our
data in Fig. 2 (b), we estimate a canting, $\mathrm{\phi}\approx 25.5^{\circ}$
at B=7.5 T, in reasonable agreement $\mathrm{\phi}\approx 27.9^{\circ}$
estimated from the high field data in Ref. [35]. On the other hand,
$B_{\mathrm{sat}}$ is determined to be $\sim$22 T from the quadratic fit to
$I_{(0,0,1.5)}(\mathrm{B})$ (black solid line in Fig. 2 (b)), somewhat bigger
than a $B_{\mathrm{sat}}=16.3~{}\mathrm{T}$ directly measured in Ref. [35]. We
note that a precise determination of $B_{\mathrm{sat}}$ here is probably
difficult given the small variation of $I_{(0,0,1.5)}(\mathrm{B})$ in the
field range covered in our experiment.

Figure 2:  (a) Magnetization, $M$, and magnetic susceptibility,
$\frac{dM}{dB}$, as a function of field strength for different field
directions. The magnetic field was applied in the $\mathbf{ab}$-plane at
different angles with respect to the crystallographic $\mathbf{a}$-axis. For
each measurement, the sample was warmed to room temperature, rotated to the
correct orientation with respect to the field, and cooled to 2 K either in
zero field (ZFC) or in a 5 T field (FC). The M vs B measurement was then
carried out from 0 T to 5 T. In the FC case, the field was first ramped down
to 0 T before the start of the measurement. (b) Field dependence of magnetic
and nuclear Bragg peak intensities for different field directions. Data shown
in solid (open) symbols were obtained at DCS (SPINS) with field applied along
the (1,-1,0) [(-1,2,0)] direction, respectively. For comparison, intensities
of different Bragg peaks were normalized to their zero-field values. The solid
line is a fit of the field dependence of the (0,0,1.5) peak is to a quadratic
function as described in the text.

To understand the effect of varying field directions on the spin reorientation
in CoTiO3, we remounted the crystal in the (H,H,L)-plane, with a field applied
along the (1,-1,0) direction that is 30∘ from the $\mathbf{b}$-axis. The
intensity of the (1,1,1.5) Bragg peak was used to track the moment direction
as a function of the field. As shown in Fig. 2 (b), aside from a different
high field intensity due to a different projection of $\mathbf{Q}$ in the
$\mathbf{ab}$ plane, field dependence at (1,1,1.5) is almost identical to that
at (1,0,0.5). More detailed angle dependence was obtained by carrying out
magnetization measurements for different field directions as shown in Fig. 2
(a), where we observed identical $M(\mathrm{B})$ for all field directions. Our
results here are consistent with a fine angle dependence reported in Ref. [36]
showing a variation of at most 3$\%$ at 1 T. The lack of angle dependence
observed here is quite puzzling: depending on whether the field is along the
easy or hard axis determined by the intrinsic spin anisotropy in CoTiO3, one
should naively expect a difference in the field dependence as has been
observed in other simple collinear magnets[37]. As we will discuss in Section
IV.1, this is naturally explained by the unusual domain structure at zero
field.

###  III.2 Zone Center Magnon

####  III.2.1 Temperature Dependence

As discussed in the last subsection, the in-plane spin reorientation
transition implies the existence of a finite spin anisotropy that prevents the
ordered moments from freely rotating in CoTiO3 at zero field. The same spin
anisotropy opens a gap in the Goldstone mode at the magnetic zone center (ZC).
The ZC gap was not resolved in our first INS measurement[29], but was
determined to be $\sim$ 1 meV by a subsequent high-resolution INS study by
Elliot et al. [30], which was later confirmed by both terahertz and Raman
spectroscopy measurements[31]. Assuming the reported high symmetry crystal
structure to be correct, this gap is forbidden on a mean-field level when only
bilinear spin interactions are considered. Quantum order by disorder[30],
which corrects the mean-field energy by considering zero-point fluctuations of
the high energy magnons, and a sixth-order ring exchange interaction[31] were
proposed as possible mechanisms to account for observed ZC magnon gap.

Figure 3:  (a) Constant $\mathbf{Q}$ scans at the magnetic zone center,
(0,0,1.5), at different temperatures below and above $\mathrm{T_{N}}=38$ K
measured at the SPINS triple axis spectrometer using fixed final energy,
$\mathrm{E_{f}}=$5 meV. Solid lines are fit to a damped harmonic oscillator
model convolved with the SPINS instrumental resolution as described in the
text. The dashed line is the expected line shape at 2 K for a resolution-
limited magnon peak. (b, c) Temperature dependence of (b) the energy,
$\Delta_{0}~{}(\mathrm{T})$, and (c) the width, $\Gamma_{0}~{}(\mathrm{T})$,
of the zone center magnon peak extracted from the fit in (a). Blue lines in
(b) are temperature dependence of different powers of the measured order
parameter, $n(\mathrm{T})$, whose base temperature value has been set to that
of $\Delta_{0}$ for comparison. $n(\mathrm{T})$ shown here was obtained
experimentally by taking the square root of the Bragg peak intensity at
(0,0,1.5) measured at SPINS.

Using cold neutron triple-axis measurements, we confirmed the previously
determined gap size by observing a magnon peak at $\sim$ 1 meV in a constant
$\mathbf{Q}$ scan at the magnetic ZC as shown in Fig. 3 (a). Further
measurements were carried out at higher temperatures to study the evolution of
the ZC magnon gap as magnetic order gets suppressed when T approaches
$\mathrm{T_{N}}$. A clear softening and damping of the ZC magnon peak is
observed as temperature increases, consistent with the typical behaviours
observed in an ordered magnet. We fit the inelastic part of the spectrum
($|\hbar\omega|~{}\geq~{}0.4$ meV) to a damped harmonic oscillator model (DHM)
convolved with the instrumental resolution. The DHM is defined as,

$\displaystyle S(\mathbf{Q},\hbar\omega,T)=$
$\displaystyle\frac{1}{\Omega_{\mathbf{Q}}}\frac{A(\mathrm{T})}{1-e^{-\frac{\hbar\omega}{\mathrm{k_{B}T}}}}\bigg{(}\frac{\frac{1}{2}\Gamma_{0}(\mathrm{T})}{(\hbar\omega-\Omega_{\mathbf{Q}})^{2}+\left[\frac{1}{2}\Gamma_{0}(\mathrm{T})\right]^{2}}$
(1)
$\displaystyle-\frac{\frac{1}{2}\Gamma_{0}(\mathrm{T})}{(\hbar\omega+\Omega_{\mathbf{Q}})^{2}+\left[\frac{1}{2}\Gamma_{0}(\mathrm{T})\right]^{2}}\bigg{)},$

where the phenomenological magnon dispersion near the ZC,
$\Omega_{\mathbf{Q}}$, is taken to be
$\Omega_{\mathbf{Q}}~{}=~{}\sqrt{(v_{\parallel}Q_{\parallel})^{2}+(v_{\perp}Q_{\perp})^{2}+\Delta_{0}(\mathrm{T})^{2}}$.
In Eq. 1, the overall intensity, $A(\mathrm{T})$, the ZC magnon gap,
$\Delta_{0}(\mathrm{T})$, and the width, $\Gamma_{0}(\mathrm{T})$, are treated
as temperature-dependent fitting parameters, whereas the in-plane and out-of-
plane spin wave velocities, $v_{\parallel}$ and $v_{\perp}$, are taken to be
their base temperature values (See Supplemental Materials). The use of
temperature independent spin wave velocities can be justified by a lack of
significant temperature dependence of the magnon spectrum up to 30 K from our
previous low resolution measurement (see Supplemental Materials). Fit to Eq. 1
is only performed for T$\leq$30 K where the peaks in the neutron energy gain
and loss side are clearly separable.

Temperature dependence of $\Delta_{0}(\mathrm{T})$ and
$\Gamma_{0}(\mathrm{T})$ are plotted in Fig. 3 (b) and (c), respectively,
clearly showing a softening and damping of the ZC magnon with increasing
temperature. A quantity of particular interest that can be extracted from Fig.
3 (b) is the scaling of $\Delta_{0}(\mathrm{T})$ with the order parameter,
$n(\mathrm{T})\equiv|\mathbf{n}|$, the latter of which is determined
experimentally by taking the square root of the measured (0,0,1.5) Bragg peak
intensity. Interestingly, by comparing the temperature dependence of different
powers of the measured order parameter, $n(\mathrm{T})^{\alpha}$, with
$\Delta_{0}(\mathrm{T})$ as shown in Fig. 3 (b), we found neither the order
parameter, $n(\mathrm{T})$, nor the order parameter squared (equivalent to the
magnetic Bragg peak intensity), $n(\mathrm{T})^{2}$, describes the temperature
dependence of the ZC gap. Instead, it scales as a relatively high power of
$n(\mathrm{T})$ as
$\Delta_{0}(\mathrm{T})~{}\propto~{}n(\mathrm{T})^{\alpha}$, with
$\alpha=3.6\pm 1.0$. At T = 0, the ZC gap due to quantum order by disorder and
the ring exchange should scale as $n^{\frac{1}{2}}$ [38], and $n^{5}$,
respectively. Assuming the same scaling holds at finite temperature (likely to
be true at least within a random phase approximation), the large $\alpha$
determined here therefore appears to be consistent with a ZC gap opened by the
ring exchange interaction. However, a more careful measurement of
$\Delta_{0}(\mathrm{T})$ is required to confirm this. Aside from more
temperature points for a robust determination of $\alpha$, a more appropriate
functional form than Eq. 1 is required to describe the data at high
temperatures. Strictly speaking, Eq. 1 starts to break down when
$\Gamma_{0}~{}\sim~{}\Delta_{0}$, as evidenced by the poor fit on the neutron
energy gain side at $T~{}\geq~{}20$ K, and a critical scattering form of
$S(\mathbf{Q},\hbar\omega,\mathrm{T})$ [39, 40] is probably needed for a
better fit as $T\rightarrow~{}\mathrm{T_{N}}$.

####  III.2.2 Field Dependence

A remarkable feature from the above analysis is a significant broadening of
the ZC magnon peak already at base temperature, which is evident by comparing
the measured lineshape (black circles in Fig. 3 (a)) and a simulated one with
$\Gamma_{0}=0$ (dashed line in Fig. 3 (a)). To understand the origin of the
observed broadening, we studied the evolution of the ZC magnon across the in-
plane spin reorientation transition driven by an in-plane magnetic field.

Effects of an in-plane magnetic field on the ZC magnon are most clearly seen
in Fig. 4 (a), where we compare the magnon spectra near (0,0,1.5) measured by
cold neutron time-of-flight (TOF) spectroscopy at 0 T and 7.5 T for a field
along (1,-1,0). In addition to an increase in the ZC magnon gap due to the
Zeeman interaction, the width of the ZC magnon shows a clear reduction at high
field compared to zero field. The observed width change is quite striking
especially considering that a larger incident energy is actually used at 7.5 T
than at 0 T which offers poorer energy resolution. To extract the field
dependence of the energy, $\Delta_{0}~{}(\mathrm{B})$, and the width,
$\Gamma_{0}~{}(\mathrm{B})$, we made constant $\mathbf{Q}$ cuts of the TOF
data in Fig. 4 (a-b). The results shown Fig. 4 (c) reveal a clear sharpening
of the magnon mode, which becomes resolution limited at 7.5 T [the horizontal
bar in Fig. 4 (c) indicates the instrumental energy resolution]. As we show in
the Supplemental Materials, the width of the constant $\mathbf{Q}$ cut at 0 T
is independent of the choice of integration range along $(\zeta,\zeta,0)$ and
L, thus ruling out any artificial broadening due to the finite integration
range used in producing these cuts. $\Delta_{0}~{}(\mathrm{B})$ and
$\Gamma_{0}~{}(\mathrm{B})$ determined by fitting the constant $\mathbf{Q}$
cut of the TOF data in Fig. 4 (c) to a Gaussian is shown in Fig. 4 (e) as red
solid and open squares, respectively. Consistent with the expected behaviour
for an ordered antiferromagnet undergoing a spin reorientation transition,
$\Delta_{0}$ stays roughly constant at small fields and shows an almost linear
increase at high fields, signaling a transition from a low-field regime
dominated by the intrinsic spin anisotropy of the system to a high field
regime where the Zeeman coupling to the applied field becomes important. On
the other hand, $\Gamma_{0}(\mathrm{B})$ exhibits B-dependence behavior almost
exactly opposite to that of $\Delta_{0}(\mathrm{B})$ and shows a clear
decrease in the high-field state.

Figure 4:  (a-b) Magnon spectrum near the magnetic zone center (ZC),
(0,0,1.5), (a) in the absence of a magnetic field and (b) with a 7.5 T field
applied along the (1,-1,0) direction. (c) Constant $\mathbf{Q}$ cut of the ZC
magnon spectrum for different field strengths in the (1,-1,0) direction. An
integration range of $\pm 0.1$ rlu and $\pm 0.02$ rlu were used along the L
and the $(\zeta,\zeta,0)$ directions around the magnetic ZC at (0,0,1.5),
corresponding to $\sim~{}\pm 0.05\AA^{-1}$ in both directions. (Effect of
integration ranges on the lineshape of the constant $\mathbf{Q}$ cuts is shown
in the Supplemental Materials.) Solid lines are obtained by fitting the data
to a Gaussian on top of a constant background. Data in (a-c) were obtained at
the DCS TOF spectrometer with the CoTiO3 crystal mounted in the (H,H,L) plane
in a vertical field magnet. Measurements at 0 T,2.5 T and 5 T used an incident
energy, Ei, of 4.04 meV, whereas that at 7.5T used an Ei of 5.66 meV.
Instrumental resolution for these two Ei’s are shown as horizontal bars at the
bottom. (d) Constant $\mathbf{Q}$ scan at (0,0,1.5) with different fields
applied along the (-1,2,0) direction. The data were obtained at the SPINS
triple-axis spectrometer using a fixed, $\mathrm{E_{f}}=$5 meV, by mounting
the sample in the (H,0,L) plane in a vertical field magnet. The solid lines
are fit to the data after convolving with the SPINS resolution function as
described in the text. (e) Field dependence of the energy,
$\Delta_{0}~{}(\mathrm{B})$ (solid symbols), and width,
$\Gamma_{0}~{}(\mathrm{B})$ (open symbols), of the ZC magnon obtained from the
fit to the DCS and the SPINS data in (c) and (d), respectively. The parameters
extracted from the DCS data are shown as red squares, while those extracted
from the SPINS data are shown as blue circles. For the fit of the DCS data in
(c), The peak position of the Gaussian was taken to be $\Delta_{0}$. To
account for the slightly different energy resolution of the 7.5 T data from
the rest, intrinsic width of the magnon peak, $\Gamma_{0}$, was determined by
removing the instrument resolution ($\Gamma_{\text{res}}$) from the full-width
at half maximum extracted from the fit ($\Gamma_{\text{measured}}$) or,
$\Gamma_{0}=\sqrt{\Gamma_{\text{measured}}^{2}-\Gamma_{\text{res}}^{2}}$.

A similar field-dependent study of the ZC magnon for
$\mathbf{B}\parallel(-1,2,0)$ was carried out at the cold neutron triple-axis
spectrometer at SPINS. Constant $\mathbf{Q}$ scans at (0,0,1.5) shown in Fig.
4 (d) for different fields along (-1,2,0) look almost identical to the TOF
data in Fig. 4 (c). On the other hand, the high data quality in Fig. 4 (d) and
a better known instrumental resolution function for SPINS allowed a more
precise determination of $\Gamma_{0}(\mathrm{B})$ and
$\Delta_{0}(\mathrm{B})$. As shown by solid lines in Fig. 4 (d), we fit the
SPINS data by convolving Eq. 1 with the same phenomenological dispersion used
to fit the temperature-dependent data and the instrumental resolution. When
fitting the data at each field, the spin wave velocities are fixed to the
values determined from the TOF data (see Supplemental Materials), while
$\Gamma_{0}(\mathrm{B})$ and $\Delta_{0}(\mathrm{B})$ are allowed to vary. As
shown by the fitting results in Fig. 4 (e), there is a clear quantitative
agreement between the triple-axis and the TOF data, obtained with magnetic
fields along two in-plane directions unrelated by symmetry. Combining the
elastic and inelastic results in Fig. 2 and Fig. 4 respectively shows that
both the static as well as the low energy dynamical properties across the spin
reorientation transition are independent of field direction in CoTiO3.

###  III.3 Dirac Magnon Near the $\mathbf{K}$ Point

####  III.3.1 Zero field

As pointed out by Elliot et al.[30], a key manifestation of the non-trivial
exchange anisotropy in CoTiO3 is the existence of a gap in the magnon
dispersion at the $\mathbf{K}$ point, which would, in a simple XXZ model, host
a Dirac-cone like dispersion with linear crossing. This feature was unresolved
in our previous measurements with a coarse $\sim$1 meV energy resolution [29],
but could be resolved by using a lower incident energy, Ei, of 12 meV offering
a much better energy resolution of 0.27 meV (at the elastic line). High
resolution magnon spectra measured at the SEQUOIA TOF spectrometer along the
$(\zeta,\zeta)$ and $(\eta+\frac{1}{3},-\eta+\frac{1}{3})$ directions are
shown in Fig. 5 (a) and (b), respectively. [Due to the lack of L-dependence of
the magnon at $\hbar\omega\geq 7$ meV (see Supplemental Materials), the data
shown in Fig. 5\- Fig. 8 have been integrated over all L to improve signal to
noise ratio. The $\mathbf{Q}$ transfers shown in these figures are projections
onto the ab plane.] A clear depletion of the magnon intensity, as highlighted
by the red arrows in Fig. 5 (a,b), is observed near the $\mathbf{K}$ point at
$(\frac{1}{3},\frac{1}{3})$ and $(\frac{2}{3},\frac{2}{3})$, consistent with
the data in Ref [30]. This is further illustrated by the constant $\mathbf{Q}$
cut at the $\mathbf{K}$ point in Fig. 5 (c), where the integrated neutron
intensity in a small rectangular box centered at $\mathbf{K}$ was plotted as
function of energy transfer, $\hbar\omega$. A splitting of the magnon peak of
$\sim 1$ meV was observed in the constant $\mathbf{Q}$ cut, confirming the
existence of a gap between the optical and the acoustic magnon branches at the
$\mathbf{K}$ point. To rule out any artifacts caused by using a large in-plane
integration range when producing the constant $\mathbf{Q}$ cut [41, 42, 43,
44], we compared cuts made with different integration range,
$\delta\mathrm{Q}$, in Fig. 5(c). When a large $\delta\mathrm{Q}$ is used, the
apparent width, and more importantly the apparent splitting between the two
magnon peaks increase, leading to an overestimation of the true magnon gap at
the $\mathbf{K}$ point in the study of CrBr3[43, 44] and CrCl3[41]. In CoTiO3,
we found the constant $\mathbf{Q}$ cuts shown in Fig. 5(c) to be almost
unchanged when using a $\delta\mathrm{Q}\lesssim 0.03$, clearly demonstrating
the robustness of the gap at the $\mathbf{K}$ point. The splitting between the
two magnon peaks, $\Delta_{\mathrm{K}}(\delta\mathrm{Q})$, as a function of
$\delta\mathrm{Q}$ obtained by fitting the constant $\mathbf{Q}$ cuts to two
Gaussians is shown in Fig. 5 (d). Clearly,
$\Delta_{\mathrm{K}}(\delta\mathrm{Q})$ plateaus to a non-zero value,
$\Delta_{\mathrm{K}}$, in the limit of $\delta\mathrm{Q}=0$, representing the
true gap size at the $\mathbf{K}$ point. By fitting the data in Fig. 5(d) to
$\Delta_{\mathrm{K}}(\delta\mathrm{Q})=\frac{1}{2}\Delta_{\mathrm{K}}+\sqrt{\frac{\Delta_{\mathrm{K}}^{2}}{4}+(c\delta\mathrm{Q})^{2}}$
where $c$ denotes the magnon velocity near $\mathbf{K}$(see Supplemental
Materials for the derivation of this expression), we determined
$\Delta_{\mathrm{K}}$ to be 0.7(2) meV, in agreement with the value reported
by Elliot et al [30].

Figure 5:  (a-b) Magnon spectrum near the
$\mathbf{K}=(\frac{1}{3},\frac{1}{3})$ point along the (a) $(\zeta,\zeta)$ and
(b) $(\xi,-\xi)$ directions measured at the SEQUOIA TOF spectrometer using an
incident energy, $\mathrm{E_{i}}$, of 12 meV. The data within $\pm$ 0.01 rlu
transverse to the directions shown in (a) and (b) were integrated to produce
these plots. Due to the lack of $L$ dependence in the magnon spectrum near
$\mathbf{K}$, the data were integrated over all $L$ probed in the experiment,
$-4\lesssim L\lesssim 4$ to improve the signal to noise ratio. (c) Constant
$\mathbf{Q}$ cut at the $\mathbf{K}$ point for different integration range,
$\delta\mathrm{Q}$. For each $\delta\mathrm{Q}$ (in rlu), intensity within the
rectangular box defined by
$-\delta\mathrm{Q}+\frac{1}{3}<\zeta<\delta\mathrm{Q}+\frac{1}{3}$ and
$-\delta\mathrm{Q}<\xi<\delta\mathrm{Q}$ were integrated and plotted as a
function energy transfer. $\delta\mathrm{Q}$=0.01 in rlu corresponds to an
integration range of $\pm~{}0.025~{}\AA^{-1}$ and $\pm~{}0.014~{}\AA^{-1}$
along $(\zeta,\zeta)$ and $(\xi,-\xi)$, respectively. Constant $\mathbf{Q}$
cuts using different $\delta\mathrm{Q}$ are vertically offset for clarity.
Solid lines are fit to these cuts using two Gaussians having equal width and
height. (d) Splitting of the two peaks in (c), $\Delta_{\mathrm{K}}$, as a
function integration range, $\delta\mathrm{Q}$. Figure 6:  (a-b) Schematics
comparing the magnon dispersions near the $\mathbf{K}$ point for a (a) gapless
Dirac cone shifted away from $\mathbf{K}$ by $\kappa_{0}$ in the $(\xi,-\xi)$
direction and (b) fully gapped Dirac cone centered at $\mathbf{K}$ (c-g)
Magnon spectra in the vicinity of the $\mathbf{K}$ point in zero magnetic
field. Momentum directions shown here are along
$(\zeta+\xi_{0},\zeta-\xi_{0})$ for fixed $\xi_{0}$’s, corresponding to the
momentum directions in the slices shown schematically in (a-b). An integration
range of $\pm$0.02 rlu in the $(\xi,-\xi)$ direction was used, which was
chosen to improve the signal to noise ratio without introducing distortion to
the spectrum associated with a large integration range [See Fig. 5(c) and Fig.
7]. Red horizontal arrows denote the weak intensity near the main magnon mode
that gives rise to the shoulder in the constant $\mathbf{Q}$ cuts in Fig. 8.
(h-l) are the same as (c-g) but were obtained in the presence of an 8 T field
along the (1,-1,0) direction. Data in (c-l) were obtained at SEQUOIA using an
$\mathrm{E_{i}}=$12 meV.

Understanding the origin of the $\mathbf{K}$ point gap has posed a major
theoretical challenge in the study of CoTiO3. Importantly, the presence of an
inversion center between two adjacent Co2+ layers combined with the simple
collinear antiferromagnetic order preserves the so-called “magnetic inversion”
symmetry of the Bogoliubov Hamiltonian that protects the Dirac nodal line in
this material[30, 45]. This led to the proposal of a shifted Dirac cone
scenario shown schematically in Fig. 6 (a), which could arise in the presence
of bond-dependent interaction in CoTiO3[30]. In Fig. 6 (a), we show a case
where the Dirac cone remains gapless but is slightly shifted away from the
$\mathbf{K}$ point by $\kappa_{0}$ in the (1,-1) direction. Consequently, a
gap given by $2c\kappa_{0}$ appears at the $\mathbf{K}$ point. This is
contrasted with the scenario of a fully gapped Dirac cone centered at
$\mathbf{K}$ shown in Fig. 6 (b). Although the magnon dispersion along
$(\zeta,\zeta)$ exactly passing through $\mathbf{K}$ looks identical in these
two scenarios, they can be distinguished by examining the magnon dispersions
along $(\zeta+\xi_{0},\zeta-\xi_{0})$ with a non-zero $\xi_{0}$. Since the gap
closes at $(\kappa_{0},-\kappa_{0})$ in the first scenario, one should observe
a gapless crossing along the $(\zeta+\xi_{0},\zeta-\xi_{0})$ direction with
$\xi_{0}$ = $\kappa_{0}$. In addition, since the Dirac cone is now displaced
to the left of the $\mathbf{K}$ point, the magnon spectrum along
$(\zeta+\xi_{0},\zeta-\xi_{0})$ should show a large asymmetry for $\pm\xi_{0}$
as shown by the simulated magnon dispersions in Fig. 6 (a).

Although there is a clear distinction between the two scenarios in a mono-
domain state with a well-defined ordered moment direction, attempting to model
the zero-field spectrum using linear spin wave theory is complicated by the
presence of both structural and magnetic domains at 0 T, each with a distinct
excitation spectrum. The presence of two structural domains related by a two-
fold rotation around (1,1,0) has been inferred from the relative intensities
of two inequivalent Bragg peaks related by the two-fold rotation (see
Supplemental Materials). Its effect is to symmetrize the magnon spectrum along
$(\zeta\pm\xi_{0},\zeta\mp\xi_{0})$, independent of the spin model. On the
other hand, the configuration of magnetic domains at 0 T is entirely unknown
in CoTiO3. A simple configuration with discrete magnetic domains related by
the $\mathcal{C}_{3}$ and the time-reversal symmetry was assumed by Elliot et
al [30] when modelling the zero-field spectrum. However, as we discuss in
Section IV.1, a more complicated situation with extended domain walls is
likely present in CoTiO3 at zero field.

Furthermore, the zero-field ordered moment directions, necessary for any spin
wave modelling, could not be experimentally determined in CoTiO3. Elliot et al
[30] addressed this difficulty by assuming an ordered moment direction
determined by a quantum order-by-disorder (ObD) mechanism using the same form
of bond-dependent interaction that also displaces the Dirac cone. However,
this assumption is questioned by a recent optical study[31] which argued that
the ring exchange interaction, instead of quantum ObD, is responsible for the
observed ZC gap

####  III.3.2 8 T Magnetic Field

To address the above difficulties, we carried out high resolution TOF
measurement with an 8 T field applied along (1,-1,0). The crystal was mounted
in the (H,H,L) plane to access the $\mathbf{K}$ point at
$(\frac{1}{3},\frac{1}{3})$. As we showed in Section III.1, an 8 T field along
(1,-1,0) is sufficient to drive a spin reorientation transition in CoTiO3 from
a zero-field multi-domain state to a high field mono-domain state with a known
canted magnetic structure. This resolves the ambiguities in modelling the spin
wave spectra at zero field where both the ordered moment directions and
magnetic domain configuration are unknown. The magnon spectra near the
$\mathbf{K}$ point in the high-field state are shown in Fig. 6 (h-l). By
comparing with the zero-field spectra shown in Fig. 6 (c-g), we arrive at the
important observation that the magnon spectra at 8 T are almost unchanged
compared to those at 0 T. As shown by Elliot et al [30] and our systematic
investigation using linear spin wave theory (See Supplemental Materials), the
shift of the Dirac cone near the $\mathbf{K}$ point with bond-dependent
interactions is strongly dependent on the direction of ordered moments. Given
that there is a clear change of the ordered moment directions across the spin
reorientation transition, it is difficult to explain the observation of almost
identical magnon spectra near the $\mathbf{K}$ point at 0 T and 8 T by a
shifted Dirac cone scenario with bond-dependent interactions.

Figure 7: (a-c) Constant $\mathbf{Q}$ cuts of the zero field magnon spectrum
near $\mathbf{K}=(\frac{1}{3},\frac{1}{3})$ at various points along the
$(\zeta,\zeta)$ direction. The same integration range ($\pm 0.01$ rlu) in the
$(\zeta,\zeta)$ direction was used in all cuts. Different integration ranges
along $(\xi,-\xi)$ used in (a-c) are given in their respective titles. Solid
lines are fit to two Gaussians. Widths of the two peaks are fixed for all
$(\zeta,\zeta)$’s in each panel to a value determined at a $(\zeta,\zeta)$
point just far enough away from the $\mathbf{K}$ point where the two peaks due
to the acoustic and optical magnon branches are clearly separated. (d-f) are
the same as (a-c) but obtained for the magnon spectra with an 8 T field
applied along (1,-1,0). (g,h) Splitting between the acoustic and optical
magnon modes along $(\zeta,\zeta)$, $\Delta_{(\zeta,\zeta)}$, at (g) 0 T and
(h) 8 T. The size of the splitting is determined from the separation between
the two peaks in (a-f) indicated by the downward triangles. The solid lines in
(g) and (f) are fit of the measured splitting along $(\zeta,\zeta)$ to
$\Delta_{(\zeta,\zeta)}=\sqrt{\Delta_{K}^{2}+[2c(\zeta-\zeta_{0})]^{2}}$. In
this expression, $\Delta_{K}$ is the size of the Dirac gap and $c$ is the
magnon velocities near the $\mathbf{K}$ point. Since the Dirac cone is no
longer centered exactly at the $\mathbf{K}$ point at 8 T, we also allowed its
location $\zeta_{0}$ to vary in the fit. $\zeta_{0}$ is $\frac{1}{3}$ and
$\sim 0.37$ at 0T and 8T, respectively.

To look for any small difference in the size of the Dirac gap at 0 T and 8 T,
we made constant $\mathbf{Q}$ cuts of the spectrum in Fig. 6 (e) and (j) at
different points along $(\zeta,\zeta)$. The results are shown in Fig. 7 (a-c)
and Fig. 7 (d-f), respectively. For each field, cuts made with different
ranges of integration in the $(\xi,-\xi)$ direction are presented to check the
robustness of the analyses. Lineshapes at different $(\zeta,\zeta)$’s show
qualitatively similar behaviours in all plots: at $\mathbf{Q}$ sufficiently
far from the $\mathbf{K}$ point, two distinct peaks due to the acoustic and
optical magnon branches can be resolved. The two magnon modes approach each
other and then separate moving past the $\mathbf{K}$ point. To determine the
gap at the $\mathbf{K}$ point, the constant $\mathbf{Q}$ cuts at different
$(\zeta,\zeta)$ points in each panel are fit to a sum of two Gaussians with a
fixed width. The width is determined by fitting to a cut at a $(\zeta,\zeta)$
point just far enough from $\mathbf{K}$ to have two clearly separated peaks.
The splitting between the optical and acoustic branches determined from the
fit is shown in Fig. 7 (g) and (h) for 0 T and 8 T, respectively, as a
function of the $\mathbf{Q}$ transfer. Aside from a small shift in the $\zeta$
value that minimizes the splitting between the two modes, we found no
difference in the size of the Dirac gap at 8 T and 0 T. As shown by data
points with different symbols, this observation is robust for different
integration ranges used to obtain the constant $\mathbf{Q}$ cuts.

Figure 8:  Constant $\mathbf{Q}$ cuts of the measured magnon spectra at 0 T
and 8 T in Fig. 6 at (a) $\mathbf{Q}$=(0.5,0.5), (b) $\mathbf{Q}$=(0.55,0.45)
and (c) $\mathbf{Q}$=(0.6,0.4). The integration ranges used are $\pm 0.02$ rlu
along both the $(\zeta,\zeta)$ and $(\xi,-\xi)$ directions. For (b) and (c),
spectra along $(\zeta\pm\xi_{0},\zeta\mp\xi_{0})$ are first symmetrized to
improve the signal to noise ratio before making the cuts.

Although the magnon spectra near the $\mathbf{K}$ point shows little change
across the spin reorientation transition, subtle differences between the 0 T
and 8 T spectra are observed along the Brillouin zone boundary away from the
$\mathbf{K}$ point. This is clear by comparing constant $\mathbf{Q}$ cuts of
the 0 T and 8 T data at $\mathrm{(0.5+\xi_{0},0.5-\xi_{0})}$ for different
$\xi_{0}$’s. As shown in Fig. 8, the magnon peak at 8 T shows a slight
softening compared to that at 0 T. More interestingly, unlike the constant
$\mathbf{Q}$ cuts at 8 T showing a symmetric lineshape, that at 0 T shows a
shoulder next to the main magnon peak as indicated by the black vertical
arrow. The shoulder in the 0 T originates from a weak mode close to the main
magnon mode as indicated by the red horizontal arrows in Fig. 6 (d)-(f). The
observation that this weak mode is absent in the high-field monodomain state
suggests that it originates from either magnetic domains with different
ordered moment directions or excitations within the extended domain walls (see
Section IV.1).

##  IV Discussions

###  IV.1 Zone Center Magnon

Figure 9: Zero-field magnon spectrum along L near the magnetic zone center,
(0,0,1.5). This is the same data as Fig. 4 (a), but plotted in the reduced
wave-vector along L defined as, $q$=L-1.5. The peak shown in red is the
$q$-scan through the (0,0,1.5) magnetic Bragg peak obtained by integrating the
elastic intensity with $|\hbar\omega|<0.1$ meV.

A puzzling result from Section III.2 is the broadening of the ZC magnon at 0
T. Broadening of a magnon mode is usually attributed to damping caused by
either magnon-magnon interaction or interaction with other excitations such as
phonons. However, this explanation does not apply here as the ZC magnon is
kinematically forbidden from decaying, and there is no known low energy phonon
mode in CoTiO3. Another possible explanation for damping of the ZC magnon is
the finite size effect of magnetic domains at zero field, which could
potentially account for the observed width change of the magnon peak across
the spin reorientation transition. However, this explanation is unlikely due
to the following argument. For a magnetic domain of size $d_{0}$, only magnon
with a wavelength longer than $d_{0}$, or a reduced wave-vector, $q$, smaller
than $\frac{1}{d_{0}}$ will be damped by scattering with the domain boundary,
while those at $q\gtrsim q_{0}\sim\frac{1}{d_{0}}$ are unaffected. An
upperbound for $q_{0}$ is given by the width of the magnetic Bragg peak (red
data points in Fig. 9) to be $\sim$0.05 rlu 111This value is mostly determined
by the instrumental resolution. The actual $q_{0}$ should be much smaller.
High resolution synchrotron measurement have been carried out for the doped
sample, Co0.65Fe0.35TiO3[48], finding $d_{0}\sim 2500~{}\AA$ along both the
in-plane and out-of-plane direction equivalent to a $q_{0}\sim 0.005$ rlu..
However, a close inspection of the ZC magnon spectrum (reproduced in Fig. 9
together with the q-scan through the (0,0,1.5) magnetic Bragg peak) shows that
the magnon is broadened at least up to $q\sim 0.2$, clearly outside the width
of the Bragg peak. This rules out the finite size effect of magnetic domains
as the reason for the observed broadening.

If not damping, the observed broadening can only be reasonably explained by a
distribution of spin anisotropy in the sample, which produces anisotropy gaps
of slightly different sizes and manifests as a large width of the ZC magnon
with finite energy resolution. Given that the broadening is only observed at 0
T and hence obviously related to the multi-domain state (Fig. 4), the most
plausible explanation for such a distribution of spin anisotropy is the
presence of extended magnetic domain walls. Within these domain walls, the
ordered moments are no longer pinned to the easy axes but continuously rotate
between two easy directions to smoothly connect two magnetic domains.
Consequently, spins within the domain walls experience a slightly different
anisotropy field depending on their orientations with respect to the easy
axes, resulting in a slightly different ZC magnon gap. In addition to
explaining the sharpening of the ZC magnon in the high-field monodomain state,
existence of extended magnetic domain walls with a continuous distribution of
ordered moment directions also partially restores the in-plane U(1) symmetry
and explains the lack of angle dependence observed in Fig. 2. Physically,
existence of extended magnetic domain walls is plausible for two reasons.
First, the energy cost for making these domain walls is extremely small as it
arises from a very small in-plane spin anisotropy due to either quantum order
by disorder [30] or ring exchange [31]. Second, in the presence of significant
magnetoelastic coupling in CoTiO3 [36, 35], extended domain walls are
preferred over sharp ones as the latter require a discrete change in lattice
parameter and hence a large elastic energy cost.

Interestingly, a recent Raman scattering measurement observed sharp ZC magnon
peak at zero field[31], which showed no change across the spin reorientation
transition. Given the large difference in the probed volume by neutrons and
photons at optical wavelengths (the penetration depth of the wavelengths used
in Ref. [31] is estimated to be only about 14 nm using the complex refractive
index calculated in Ref. [47]), the magnon peak measured by Raman scattering
is likely associated with only one magnetic domain. The results in Ref. [31]
therefore provide indirect evidence for the proposed explanation involving
domain walls, and strongly rule out any intrinsic damping of magnons at the
zone center. A direct evidence for our explanation could be found through
position sensitive imaging and spectroscopy techniques to correlate the local
spin directions with their low energy excitations.

###  IV.2 Dirac Magnon

Within the bond-dependent model proposed in Ref. [30], the magnon spectra near
the $\mathbf{K}$ point are strongly dependent on the ordered moment
directions. Since the ordered moment directions are clearly altered across the
spin reorientation transition, we argued in Section III.3 that it is hard to
reconcile the observation of very similar magnon spectra at 0 T and 8 T with
large bond-dependent interactions in CoTiO3. As we show in the Supplemental
Materials, although we could not completely rule out the bond-dependent
interactions as the reasons for the observed $\mathbf{K}$-point gap, in order
to explain the observed magnon spectra at both 0 T and 8 T, the bond-dependent
model requires certain ad hoc assumptions that are hard to be justified
physically.

Given the obvious caveats of the bond-dependent model, we now consider an
alternative scenario where a magnon gap at the $\mathbf{K}$ point is created
by fully gapping out the Dirac cone. As shown in Ref. [30, 45], the existence
of Dirac nodal lines is protected by the magnetic inversion symmetry of the
magnon Hamiltonian, which is a combination of spatial inversion across an
inversion center between the two neighbouring Co2+ ions along the
$\mathbf{c}$-axis, followed by time reversal that flips the ordered moment
direction. This symmetry has to be broken to gap out the Dirac cone. This
could happen either by a canting between the two neighbouring honeycomb
planes, or a structural distortion of the crystal that makes the two octahedra
directly on top of each other inequivalent. Since the canting, if existed, is
clearly modified across the spin reorientation transition in CoTiO3,
observation of almost identical spectra at 0 T and 8 T rules out the first
mechanism, and suggests the second mechanism to be the most plausible
explanation for the absence of magnetic inversion symmetry and consequently a
gap at the $\mathbf{K}$ point.

In fact, indirect evidence for a crystal symmetry lower than the reported
$R\bar{3}$ space group is found in a recent angle-dependent magnetic
susceptibility measurement at room temperature, well into the paramagnetic
regime [35]. In their work, the authors found the susceptibility as a function
of in-plane field direction to exhibit two-fold periodicity that is
inconsistent with the $\mathcal{C}_{3}$ symmetry of the reported space group,
which should show an isotropic in-plane susceptibility. Although a structural
distortion has not been directly observed in X-ray or neutron powder
diffraction measurements, it cannot be ruled out considering the low
sensitivity of powder diffraction measurement (especially in the case of X-ray
powder diffraction) to weak symmetry forbidden reflections with small changes
in the structure of the oxygen octahedra. A future single crystal synchrotron
or neutron diffraction experiment is desired to directly observe such a
distortion. If the observed Dirac gap is indeed due a small structural
distortion in CoTiO3, its size should be quite sensitive to a change in the
crystal structure. An inelastic neutron scattering experiment on a crystal
under an in-plane strain, combined with a single crystal diffraction
measurement, can therefore provide direct evidence for the proposed mechanism.

##  V Conclusions

In conclusion, we have carried out field-dependent high-resolution inelastic
neutron scattering on CoTiO3 to study the changes of its zone center and Dirac
magnon across a spin reorientation transition induced by an in-plane magnetic
field. Through elastic neutron scattering measurements, we confirmed that our
sample transitions from a multi-domain state at zero field to a high-field
mono-domain state with a canted magnetic structure. Concurrent with the spin
orientation transition, we found a large change in the energy and the width of
the zone center magnon peak, the latter of which was argued to be consistent
with an unusual zero-field state with extended domain walls. The complex
domain configuration at 0 T uncovered in the present work highlights the
difficulty in the determination of the exchange parameter based solely on the
zero-field data, and the importance of our field-dependent measurements. In
contrast with the behaviours of the zone center magnon, we found the magnon
spectra near the $\mathbf{K}$ point to be almost unaffected by a change in
ordered moment directions and the domain structure. This observation is
difficult to explain within the framework of the bond-dependent model, which
predicts a strong dependence of the magnon spectra on the ordered moment
directions. We argued that a symmetry-lowering structural distortion appears
to be a more likely explanation for the observed gap. Although this structural
distortion is yet to be directly observed in a diffraction experiment, we
believe our work will provide strong constraints for any models proposed for
CoTiO3 in the future.

##  VI Acknowledgements

Work at the University of Toronto was supported by the Natural Science and
Engineering Research Council (NSERC) of Canada through Discovery Grant No.
RGPIN-2019-06449, the Canada Foundation for Innovation, and the Ontario
Research Fund. G.J.S. acknowledges the support provided by MOST-Taiwan under
Project No. 105-2112-M-027-003-MY3. We acknowledge the support of the National
Institute of Standards and Technology, U.S. Department of Commerce, in
providing the neutron research facilities used in this work. This research
used resources at the Spallation Neutron Source, a DOE Office of Science User
Facility operated by the Oak Ridge National Laboratory. Use of the MAD
beamline at the McMaster Nuclear Reactor is supported by McMaster University
and the Canada Foundation for Innovation.

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