††thanks: Contributed equally to this work††thanks: Contributed equally to
this work††thanks: Contributed equally to this work

# Revealing Emergent Magnetic Charge in an Antiferromagnet with Diamond
Quantum Magnetometry

Anthony K. C. Tan<EMAIL_ADDRESS>Cavendish Laboratory, University of
Cambridge, J. J. Thomson Avenue, Cambridge, CB3 0HE, UK  Hariom Jani
<EMAIL_ADDRESS>Department of Physics, National University of
Singapore, Singapore, Singapore  Clarendon Laboratory, Department of Physics,
University of Oxford, Oxford, UK  Michael Högen  Cavendish Laboratory,
University of Cambridge, J. J. Thomson Avenue, Cambridge, CB3 0HE, UK  Lucio
Stefan  Center for Hybrid Quantum Networks (Hy-Q), Niels Bohr Institute,
University of Copenhagen, Copenhagen, Denmark  Claudio Castelnovo  Cavendish
Laboratory, University of Cambridge, J. J. Thomson Avenue, Cambridge, CB3 0HE,
UK  Daniel Braund  Cavendish Laboratory, University of Cambridge, J. J.
Thomson Avenue, Cambridge, CB3 0HE, UK  Alexandra Geim  Cavendish Laboratory,
University of Cambridge, J. J. Thomson Avenue, Cambridge, CB3 0HE, UK  Matthew
S. G. Feuer  Cavendish Laboratory, University of Cambridge, J. J. Thomson
Avenue, Cambridge, CB3 0HE, UK  Helena S. Knowles  Cavendish Laboratory,
University of Cambridge, J. J. Thomson Avenue, Cambridge, CB3 0HE, UK  Ariando
Ariando  Department of Physics, National University of Singapore, Singapore,
Singapore  Paolo G. Radaelli<EMAIL_ADDRESS>Clarendon
Laboratory, Department of Physics, University of Oxford, Oxford, UK  Mete
Atatüre<EMAIL_ADDRESS>Cavendish Laboratory, University of Cambridge, J. J.
Thomson Avenue, Cambridge, CB3 0HE, UK

###### Abstract

Whirling topological textures play a key role in exotic phases of magnetic
materials and offer promise for logic and memory applications. In
antiferromagnets, these textures exhibit enhanced stability and faster
dynamics with respect to ferromagnetic counterparts, but they are also
difficult to study due to their vanishing net magnetic moment. One technique
that meets the demand of highly sensitive vectorial magnetic field sensing
with negligible backaction is diamond quantum magnetometry. Here, we show that
the archetypal antiferromagnet, hematite, hosts a rich tapestry of monopolar,
dipolar and quadrupolar emergent magnetic charge distributions. The direct
readout of the previously inaccessible vorticity of an antiferromagnetic spin
texture provides the crucial connection to its magnetic charge through a
duality relation. Our work defines a novel paradigmatic class of magnetic
systems to explore two-dimensional monopolar physics, and highlights the
transformative role that diamond quantum magnetometry could play in exploring
emergent phenomena in quantum materials.

##  I Introduction

Topologically protected states in magnetic materials are promising candidates
for novel spintronics architectures, such as neuromorphic computing [1, 2]. In
particular, topological textures in antiferromagnets (AFMs) could provide
additional advantages over their ferromagnetic counterparts including enhanced
stability, as well as faster and richer dynamics [3, 4, 5, 6, 7, 8, 9, 10, 11,
12, 13, 14, 15, 16]. The vanishing net moment brings forth its own challenges,
particularly making the detection of AFM textures difficult. Synchrotron-based
dichroic X-ray techniques, sensitive to the staggered magnetization, are at
the imaging forefront and have for the first time revealed the existence of
two-dimensional topological AFM spin textures in hematite, $\alpha$-Fe2O3 [17,
18]. This powerful technique, however, comes with the caveat that it is
insensitive to the sign of the staggered magnetization and thus the associated
vorticity, i.e. the actual whirling of the spin textures, is not observed.

When viewed instead through the lens of the canted magnetization, we uncover
weak but finite magnetic fields emanating from the _twists_ of the staggered
moments. The presence of such magnetic fields can equivalently be described by
the magnetic analogue of Gauss’s law [19], therefore pointing to the existence
of a distribution of emergent magnetic charges in a topologically-rich AFM
landscape. The state of the art vectorial weak-field sensing of diamond
quantum magnetometry (DQM), employing a single nitrogen-vacancy (NV) colour
centre as a point field sensor [20, 21, 22, 23, 24], therefore puts it in a
unique position to study the above proposed conceptual idea of emergent
magnetic charges in a completely new class of magnetic materials – canted
AFMs.

In this letter, we demonstrate DQM imaging of topological textures in the
archetypal AFM $\alpha$-Fe2O3 and we show that these AFM textures host a rich
tapestry of magnetic charge distribution. In particular, the duality relation
between staggered vorticity and magnetic charge allows us to associate the AFM
Bloch meron with a spatially extended _emergent magnetic monopole_. Distinct
from emergent magnetic monopoles in other realizations, such as spin ice[25],
we observe here that the positively and negatively charged monopolar textures
are topologically equivalent, while their topological antiparticle has a
magnetic quadrupolar character. In addition to being a promising tool to
investigate topological AFM textures, our results demonstrate the DQM
potential for discovering emergent phenomena in model magnets.

Figure 1:  Signatures of emergent magnetic field in hematite $\alpha$-Fe2O3.
(a) Atomic structure of $\alpha$-Fe2O3 (Fe and O atoms in yellow/green and
grey spheres respectively). (b) Discrete representation of the alternating FM
sublattice magnetisation $\vec{M_{1}}$ (yellow cones) and $\vec{M_{2}}$ (green
cones) with AFM coupling along the $c$-axis shown in (a). (c) Illustration of
the whirling staggered magnetisation, $\vec{l}$ (grey cones), forming an anti-
clockwise a-Bloch meron, and the resultant canted magnetic moment, $\vec{m}$
(red cones). The insert shows the relationship between $\vec{l}$, $\vec{m}$,
$\vec{M_{1}}$, $\vec{M_{2}}$ and the canting angle $\Delta$. (d) A scanning
diamond sensor with a single NV centre maps out the $B$-field generated near
the sample surface. The insert shows the energy diagram of the NV ground state
(GS) $\ket{\pm 1}$ and $\ket{0}$ sublevels. A microwave field drives the GS
spin transition while a 532-nm laser excites the NV to the excited state (ES)
(green arrow). The NV then undergoes a radiative decay to GS (red arrow) or a
non-radiative and spin selective path via the intersystem crossing (ISC) (blue
arrow) enabling ODMR acquisition. (e, f) ODMR along the fast scan direction,
measured on $\alpha$-Fe2O3 thin film at $T=4\leavevmode\nobreak\ \mathrm{K}$
(e) and across $T_{\mathrm{M}}$ at $300\leavevmode\nobreak\ \mathrm{K}$ (f).
The fitted $f_{+}$ ($B_{\mathrm{NV}})$ is plotted as a white line in each
panel. (g, h) $B_{z}$ images retrieved from fitted $B_{\mathrm{NV}}$ maps
reveal distinct field signatures across $T_{\mathrm{M}}$. Dashed lines in (g)
and (h) correspond to fitted $B_{\mathrm{NV}}$ traces in (e) and (f),
respectively. Scale bars are $1\leavevmode\nobreak\ \mu\mathrm{m}$.

##  II Details of $\alpha$-Fe2O3 and DQM

Hematite $\alpha$-Fe2O3 is an AFM oxide insulator, which hosts a wide family
of topological spin textures [17, 26, 27]. Figure 1(a) illustrates the
trigonal corundum atomic structure of $\alpha$-Fe2O3. It comprises a stack of
anti-parallel FM sublattices along the $c$-axis, with magnetisation textures
$\vec{M_{1}}$ (yellow) and $\vec{M_{2}}$ (green), as shown in Fig. 1(b). Spin
re-orientation occurs at the Morin transition temperature, $T_{\mathrm{M}}\sim
200\leavevmode\nobreak\ \mathrm{K}$ [17](see supplementary note S1); below and
above $T_{\mathrm{M}}$, the FM sublattices lie predominantly out-of-plane and
in-plane, respectively. The Néel vector $\vec{l}\leavevmode\nobreak\
=\vec{M_{1}}-\vec{M_{2}}$ characterises the AFM order, while
$\vec{m}\leavevmode\nobreak\ =\vec{M_{1}}+\vec{M_{2}}$ is the net
magnetisation, as shown in Fig. 1(c). Above $T_{\mathrm{M}}$, $\vec{m}$ has a
predominantly in-plane orientation with an average magnitude $m_{\Delta}\sim
2\times 10^{3}\leavevmode\nobreak\ \mathrm{A/m}$. This weak magnetisation is
due to the slight in-plane canting of angle $\Delta\leavevmode\nobreak\ $
$\sim 1.1\leavevmode\nobreak\ \mathrm{mrad}$ (see S1) between $\vec{M_{1}}$
and $\vec{M_{2}}$, as a consequence of the intrinsic bulk
Dzyalonshinskii–Moriya interaction (DMI) vector along the $c$-axis[28](see
S5). Consequently, $\vec{m}$ lies in-plane and satisfies the relation
$\vec{m}\cdot\vec{l}=0$. Since $m_{\Delta}$ is around three orders of
magnitude weaker than $|\vec{l}|$, this canting-induced weak magnetisation has
no discernible effect on the AFM character of $\alpha$-Fe2O3. Finally,
$\vec{m}$ vanishes in situations where $\vec{l}$ turns out-of-plane below
$T_{\mathrm{M}}$ or due to the local formation of AFM spin textures.

We quantify the magnetic-field distribution from these spin textures via a
custom-built cryogenic DQM (see S2). Figure 1(d) illustrates our diamond
cantilever probe hosting a single NV centre, which is scanned at a constant
height above the sample surface. The NV centre is an atomistic single spin
defect with a paramagnetic ground-state manifold and state-selective optical
transitions. This allows the Zeeman splitting between the ground states
$\ket{\pm 1}$ and $\ket{0}$ to be probed with a microwave frequency
$f_{\mathrm{mw}}$ sweep and optical excitation, i.e. via optically detected
magnetic resonance (ODMR). Consequently, in the weak-field approximation[29]
with negligible strain, we infer the local $B$-field projected onto the NV-
axis ($B_{\mathrm{NV}}$) from the energy difference between $\ket{0}$ and
$\ket{+1}$ given by $\Delta
E_{+}=h(|f_{+}-D|-\tilde{\gamma}B_{\mathrm{bias}})=h\tilde{\gamma}B_{\mathrm{NV}}$,
where $h$ is Planck’s constant, $f_{+}$ is the resonant frequency
corresponding to the transition, $D\approx 2.87\leavevmode\nobreak\
\mathrm{GHz}$ and $\tilde{\gamma}=28\leavevmode\nobreak\ \mathrm{MHz/mT}$. A
small bias field $B_{\mathrm{bias}}$ $\approx 0.5\leavevmode\nobreak\
\mathrm{mT}$ is applied along the NV-axis to enable extraction of the field
orientation. Figures 1(e) and (f) illustrate the typical variation in ODMR
frequency across a linescan over the $\alpha$-Fe2O3 surface at
$T=4\leavevmode\nobreak\ \mathrm{K}$ and $300\leavevmode\nobreak\ \mathrm{K}$,
respectively. The colour plot displays the amplitude of the ODMR signal, while
the white curves demarcate $f_{+}$ used to extract $B_{\mathrm{NV}}$. An ODMR
raster scan across the sample surface provides a $B_{\mathrm{NV}}$ image. We
then transform to the laboratory coordinates ($B_{x,y,z}$) via the Fourier
reconstruction technique[30], where $z$ coincides with the $c$-axis of the
crystal. Figures 1(g) and (h) present example images of $B_{z}$ collected at
$4\leavevmode\nobreak\ \mathrm{K}$ and $300\leavevmode\nobreak\ \mathrm{K}$,
respectively. The qualitative differences observed in these images already
reveal distinct magnetic phases for temperatures below and above
$T_{\mathrm{M}}$ [17]. The $B_{z}$ image below $T_{\mathrm{M}}$ is generally
weak, consistent with the absence of net magnetisation. By contrast, the
$B_{z}$ image above $T_{\mathrm{M}}$ shows strong and distinct features,
consistent with the expectations of non-zero net magnetisation in this phase.

## Emergent Properties in $\alpha$-Fe2O3

To gain a physical interpretation of the underlying magnetisation distribution
from the measured $B_{z}$ images, we begin from a thin-film approximation
given by (see S4.1):

$B_{z}=\alpha_{xy}(t,d)*\vec{\nabla}\cdot\vec{m}_{xy}+\alpha_{z}(t,d)*\nabla^{2}m_{z}\leavevmode\nobreak\
,$ (1)

where $*$ indicates convolution, $\alpha_{i}$ ($i=xy,z$) are effective point
spread functions[31], $t$ is the film thickness, $d$ is the height above the
film surface, and $\vec{m}_{xy}$ and $m_{z}$ are the in-plane and out-of-plane
components of $\vec{m}$. $\vec{m}_{xy}$ and $m_{z}$ contribute to $B_{z}$
through the divergence and Laplacian, respectively. The $\alpha_{i}$ account
for the $B$-field decay above the surface, acting as blurring kernels with
their size of order $d$. Hence, the spatial resolution of our DQM imaging
technique is set by the effective NV-sample distance, $d_{\mathrm{NV}}$ (see
S3). Due to the symmetry of the DMI in $\alpha$-Fe2O3(see S5), $m_{z}=0$ and
$\vec{m}_{xy}\neq 0$, rendering the second term in Eq. 1 identically zero.
Therefore, $B_{z}$ images are the _divergence of the canted magnetisation_
$\vec{\nabla}\cdot\vec{m}_{xy}$, convolved with $\alpha_{xy}$. Moreover, for a
$\hat{z}$-oriented DMI, the net magnetisation is also given by
$\vec{m}=\Delta(\hat{z}\times\vec{l})$, where $\Delta$ is the DMI-set canting
angle. This yields an additional expression, namely
$\vec{\nabla}\cdot\vec{m}_{xy}=\Delta[\hat{z}\cdot(\vec{\nabla}\crossproduct\vec{l}\leavevmode\nobreak\
)]$ (see S5). The striking consequence of this expression is that $B_{z}$
images also offer a projected measure of the _staggered vorticity_ , i.e. the
_curl of the Néel vector_ ,
$\vec{\mathcal{V}}=\vec{\nabla}\crossproduct\vec{l}$.

Figure 2:  Classification of topological AFM textures via DQM. (a-c)
Topological AFM textures observed below $T_{\mathrm{M}}$. Distinct $B_{z}$
signature of an ADW simulated (a) and measured (b) above the sample surface.
The reconstructed $\vec{m}_{xy}$ (black arrows) from (b) and its $B_{z}$
distribution are given in (c). (d-o) Topological AFM textures observed above
$T_{\mathrm{M}}$. Simulated (d) and measured (e) $B_{z}$ field signatures of
an anti-clockwise a-Bloch meron and (f) the reconstructed $\vec{m}_{xy}$
(black arrows) of (e) and its associated $B_{z}$. Similarly, the simulated and
measured $B_{z}$ signatures, and the $\vec{m}_{xy}$ reconstruction of a
clockwise a-Bloch meron, an anti-meron and a bimeron is given in panels (g-i),
(j-l), (m-o) respectively. Details of $\vec{m}_{xy}$ reconstruction are given
in Ref. (see S7). Scale bars are $200\leavevmode\nobreak\ \mathrm{nm}$.

## Characteristic Field Signatures and Vorticity Readout

With the gained insight on the relationship between the $B_{z}$ images and
$\vec{m}_{xy}$, we show next the $B_{z}$ images obtained in Fig. 1 are
produced by AFM antiphase domain walls (ADWs), merons, antimerons and bimerons
– consistent with their recent observation in $\alpha$-Fe2O3[17]. Below
$T_{\mathrm{M}}$, we model the $B_{z}$ images with a linear AFM domain wall
ansatz [17], characterised by width $w$ and phase $\xi_{a}$ (see S5.1). The
phase $\xi_{a}$ controls the spatial variation of $\vec{l}$, resulting in an
AFM Néel (a-Néel) or an AFM Bloch (a-Bloch) ADW profile for ($\xi_{a}=0,\pi$)
and ($\xi_{a}=\frac{\pi}{2}$,$\frac{3\pi}{2}$), respectively. For a linear ADW
profile centred at $x=0$ along the $x$-axis,
$\vec{\nabla}\cdot\vec{m}_{xy}=m_{\Delta}\big{(}\frac{\pi}{w}\big{)}\sin(\frac{\pi
x}{w})\sin(\xi_{a})$ for $|x|\leq\frac{w}{2}$, and zero elsewhere [17] (see
S5.1). In other words, we expect ADWs to display a sinusoidal profile in
$\vec{\nabla}\cdot\vec{m}_{xy}$ and hence in $B_{z}$, with zero crossing at
the centre and amplitude and sign modulated by $\sin{(\xi_{a})}$. In
particular, an a-Néel ADW will not yield a $B_{z}$ signal as
$\vec{\nabla}\cdot\vec{m}_{xy}=0$, whereas an a-Bloch counterpart will show
maximal signal. Based on Eq. 1, these characteristics are reflected in the
simulated $B_{z}$ image of an ADW in Fig. 2(a) (see S5.4) assuming a constant
phase $\xi_{a}=\pi$. The measured $B_{z}$ image in Fig. 1(g), as well as a
close-up image in Fig. 2(b), capture the signature zero crossing of an ADW.
DQM also reveals variations of $B_{z}$ along the wall boundary capturing the
spatially varying nature of the phase $\xi_{a}$, as expected in $\alpha$-Fe2O3
[17]. In effect, the full phase dependence of $B_{z}$ allows us to identify
unambiguously the zero-signal sections along the varying $\xi_{a}$ wall
boundary as a-Néel ADWs. Figure 2(c) presents a reconstruction of the
$\vec{m}_{xy}$ distribution of the multi-chiral ADW (illustrated by arrows),
obtained by fitting its $B_{z}$ field to the experimental data in Fig. 2(b)
through systematic regularization strategies (see S7).

In contrast, DQM at $300\leavevmode\nobreak\ \mathrm{K}$ captures relatively
larger spatial features of strong $B_{z}$ signal, as showcased in Fig. 1(h).
This above-$T_{\mathrm{M}}$ regime is where we expect finite net magnetisation
forming a tapestry of localised whirling topological structures, such as
multi-chiral merons, antimerons and bimerons[17], as well as topologically-
trivial in-plane domain walls. Each of these AFM textures produces a
distinctive $B_{z}$ signal, allowing us to develop a systematic procedure to
disambiguate them, see S5. Here we focus on two-dimensional (2D) topological
textures and model isolated merons and antimerons based on a linear ansatz
[17, 32], described by the phase $\xi_{a}$ and the topological winding number
$\mathcal{N}$ (see S5.3). The magnetisation divergence for such an object in
polar coordinates ($r,\leavevmode\nobreak\ \phi$) is
$\vec{\nabla}\cdot\vec{m}_{xy}=m_{\Delta}\sin\big(\phi(1-\mathcal{N})-\xi_{a}\big{missing})f(r)$,
where $f(r)$ is a radial function dependent on the (anti)meron phase (see
S5.3). A meron ($\mathcal{N}=+1$) produces a radially symmetric $B_{z}$
distribution about its core with magnitude and polarity controlled by
$\sin{(\xi_{a})}$. Analogous to ADWs, a-Néel merons ($\xi_{a}=0$, $\pi$) are
divergence-free and exhibit $B_{z}=0$, whereas a-Bloch counterparts
($\xi_{a}=\frac{\pi}{2}$, $\frac{3\pi}{2}$) show maximal $B_{z}$ amplitude. In
contrast, for an antimeron ($\mathcal{N}=-1$), the $B_{z}$ distribution is
two-fold symmetric and $\xi_{a}$ controls a trivial azimuthal offset.
Simulated $B_{z}$ images in Fig. 2(d, g, j) of the a-Bloch meron of both
polarities and the antimeron reinforce these observations (see S5.4). Thus,
DQM unambiguously reveals the topological winding number $\mathcal{N}$ and the
staggered vorticity $\vec{\mathcal{V}}$ for each spin texture. The final
attribute of such a spin texture, the sign of its topological charge, is
however rendered indistinguishable by DQM due to the vanishing canted moment
at its core. Figure 2 (panels e, h, k) presents the measured $B_{z}$ images of
an anti-clockwise a-Bloch meron ($\mathcal{N}=+1,\xi_{a}=\pi/2$), a clockwise
a-Bloch meron ($\mathcal{N}=+1,\xi_{a}=3\pi/2$), and an antimeron
($\mathcal{N}=-1$), respectively, in good agreement with their simulated
counterparts. Further, in Fig. 2 (f, i, l) we once again reconstruct the
corresponding $\vec{m}_{xy}$ distributions and use them to simulate the
measured $B_{z}$ image for each texture discussed above (see S7). Given the
typical density of spin textures evident in Fig. 1(h), the reconstructed
$\vec{m}_{xy}$ approach better captures the finer details of the measured
$B_{z}$ images in the absence of true isolation. We note that the initial
density of (anti)merons can be reduced via meron-antimeron annihilation
mediated by an external in-plane magnetic field [17] (see S8).

Figure 3:  Emergent magnetic charge distributions. (a-d) Three-dimensional
visualisation of $\vec{B}$ in the volume above an assortment of topological
AFM textures. Streamtubes illustrate the magnetic field lines of $\vec{B}$
above (a) an anti-clockwise a-Bloch meron, (b) a clockwise a-Bloch meron, (c)
an ADW and (d) an antimeron. The girth and colour of each streamtube vary with
the magnetic field norm $|B|$ and the z-component of the field $B_{z}$,
respectively. (e-h) Magnetic charge density $\sigma_{m}$ distributions
retrieved from downward continuation of (a-d) reveal a magnetic monopolar,
anti-monopolar, dipolar and quadrupolar charge character associated to an
anti-clockwise a-Bloch meron, a clockwise a-Bloch meron, an ADW and an
antimeron respectively. Scale bars are $200\leavevmode\nobreak\ \mathrm{nm}$.

Finally, a meron and an antimeron in close proximity can also form a stable
bimeron. Figure 2(m) displays the corresponding simulated $B_{z}$ image of an
isolated bimeron, while Fig. 2(n) shows the measured $B_{z}$ image of one such
occurrence. As before, Fig. 2(o) displays the simulated $B_{z}$ image from its
reconstructed $\vec{m}_{xy}$. It is interesting to note that while
(anti)merons are always topologically protected, this is not necessarily true
for meron-antimeron pairs. Labelling a meron-antimeron pair as topologically
protected would require the knowledge of the sign of the topological charge of
its constituents [34, 28].

## Emergent Magnetic Charge

Reconstructing the $\vec{m}_{xy}$ distribution from $B_{z}$ images is useful
in understanding the underlying spin textures, but the fact that DQM actually
provides a direct measure of $\vec{\nabla}\cdot\vec{m}_{xy}$ creates a unique
opportunity to consider a magnetic analog of the electric Gauss’s law. Namely,
the non-zero divergence of the magnetisation manifests the existence of an
areal 2D magnetic charge density via
$\sigma_{m}=-t(\vec{\nabla}\cdot\vec{m}_{xy})$. Here, $t$ is the film
thickness and $\vec{m}_{xy}$ is independent of $t$ – valid in the thin film
limit [17]. Therefore, AFM textures in $\alpha$-Fe2O3 have associated
_emergent magnetic charge_ distributions, which locally act as sources or
sinks of the magnetic field. In fact, we can define a formal _duality
relation_ that connects the 2D magnetic charge density $\sigma_{m}$ to the
staggered vorticity $\vec{\mathcal{V}}$ via

$\sigma_{m}/t=-\vec{\nabla}\cdot\vec{m}_{xy}=\Delta(\hat{z}\cdot\mathcal{\vec{V}}),$
(2)

which scales with $\sin(\xi_{a})$, highlighting the influence of the texture
phase. Crucially, retrieving the full 2D emergent charge density $\sigma_{m}$
only requires the Fourier deconvolution of the measured $B_{z}$ images from
the transfer function $\alpha_{xy}$ (see S4.2). We can also perform a downward
(upward) continuation [35] (see S4.2) of the planar $\vec{B}$ distribution
captured in Fig. 2 (b, e, h, k, n), down to (away from) the sample surface.
This allows a three-dimensional visualisation of $\vec{B}$ ($=\vec{H}$ in
vacuo) in the volume above the emergent magnetic charge distribution.

Figure 3 panels (a) to (d) illustrate the representative field lines of
$\vec{B}$ above a (anti-)clockwise a-Bloch meron, an ADW and an antimeron,
respectively. For the two Bloch merons, $\vec{B}$ is consistent with the
profile expected for spatially extended sources and sinks of magnetic field
emanating from a _monopolar_ distribution, which we retrieved via Eq. 2 in
panels (e) and (f). This implies that a-Bloch merons host a class of emergent
monopoles piggybacking on the whirling AFM textures. Interestingly, such
monopolar magnetic distributions are not observed in ferromagnetic materials,
as the presence of long-range demagnetizing fields favours divergence-free
Bloch textures (e.g. in FM vortices). In our case, we are freed from this
constraint due the presence of a weak demagnetizing contribution relative to
the much stronger AFM exchange in $\alpha$-Fe2O3. In contrast to merons, the
ADW in Fig. 3(c) and the antimeron in Fig. 3(d) are associated with
$\sigma_{m}$ distributions that exhibit _dipolar_ and _quadrupolar_
characters, respectively, as illustrated in panels (g) and (h). Finally, we
emphasise that the observation of emergent monopoles is fully consistent with
the simulations of AFM topological textures in Fig. 2 and does not violate
Maxwell’s equation, $\vec{\nabla}\cdot\vec{B}=0$.

On the basis of the above analysis, it is tempting to attribute to isolated
a-Bloch merons a non-zero net _monopolar charge_ quantified by
$Q_{m}\equiv\int_{S}\sigma_{m}\leavevmode\nobreak\ ds$ within an area $S$. We
pick a circular integration area $S$ of radius $r$ centred on a given spin
texture, as shown in Fig. 4(a and b) as an example in the case of antimerons.
The particular $(1/r)$ dependence of $\vec{\nabla}\cdot\vec{m}_{xy}$ for two-
dimensional magnetic charges hosted by spin textures then yields (see S5.9)

$\displaystyle\scriptsize Q_{m}(r)=\begin{cases}\leavevmode\nobreak\
2\pi\leavevmode\nobreak\ m_{\Delta}\leavevmode\nobreak\
\sin{(\xi_{a})}\leavevmode\nobreak\ \sin{\big{(}\frac{\pi
r}{2R_{M}}\big{)}}\leavevmode\nobreak\ r\leavevmode\nobreak\
t&,\leavevmode\nobreak\ \mathcal{N}=+1,\leavevmode\nobreak\ r\leq R_{M}\\\
\leavevmode\nobreak\ 2\pi\leavevmode\nobreak\ m_{\Delta}\leavevmode\nobreak\
\sin{(\xi_{a})}\leavevmode\nobreak\ r\leavevmode\nobreak\
t&,\leavevmode\nobreak\ \mathcal{N}=+1,\leavevmode\nobreak\ r>R_{M}\\\
\leavevmode\nobreak\ 0\leavevmode\nobreak\ &,\leavevmode\nobreak\
\mathcal{N}=-1\end{cases}\leavevmode\nobreak\ .$ (3)

Figure 4(c) presents the radial dependence of $Q_{m}$ for four measured merons
(light blue dashed curves) and antimerons (light red dashed curves), while the
dark blue (red) dashed curve is the average $Q_{m}$ radial dependence for
merons (antimerons). For an isolated linear meron model, $Q_{m}$ scales
linearly with $r$, and the measured $Q_{m}$ radial dependence is in agreement
with this predicted behaviour. $Q_{m}$ itself is not a topological invariant,
as a smooth transformation of an a-Bloch meron to an a-Néel meron would tune
$Q_{m}$ from non-zero to zero value. For an isolated antimeron, the two-fold
rotational symmetry should ensure $Q_{m}=0$ for all $r$ values (see S5.9) and
at short distances the experimentally obtained $Q_{m}$ value of an antimeron
agrees well with this prediction. Beyond typical measurement-induced
variations, deviation from the strict $Q_{m}=0$ condition arises when an
antimeron is influenced by neighbouring spin textures. This reduces the two-
fold symmetry and reveals finite $Q_{m}$. Figures 4(a) and 4(b) capture this
reduced symmetry and the consequential bias towards negative and positive
$Q_{m}$, respectively. Finally, as $r$ goes to infinity, the integration area
starts would overlap with surrounding spin textures, which leads to a further
deviation from the $Q_{m}=0$ condition for an isolated antimeron model.

Figure 4: Scaling of two-dimensional integrated magnetic charges. (a-b)
Reconstructed magnetic charge distribution, $\sigma_{m}$, of two
experimentally observed antimerons (AM 1 and AM 2) with slightly distorted
quadrupolar characters. AM 1 and AM 2 display a respective bias towards
positive and negative charges, respectively. Dashed circle in (a) and (b)
illustrates the circular integration area $S$ of radius $r$, centred at the
core of the antimeron, to obtain $|Q_{m}|$. Scale bars are
$200\leavevmode\nobreak\ \mathrm{nm}$. (c) Experimentally retrieved magnitude
of the total integrated magnetic charge, $|Q_{m}|$ of multiple merons
($|Q_{m}|_{\mathrm{M}}$, light blue dashed curves) and antimerons
($|Q_{m}|_{\mathrm{AM}}$, light red dashed curves) plotted as a function of
integration radius, $r$. Their average experimental $|Q_{m}|$ profiles,
$|Q_{m}|^{\mathrm{avg}}_{\mathrm{M}}$ and
$|Q_{m}|^{\mathrm{avg}}_{\mathrm{AM}}$, are represented by dark blue and dark
red curves, respectively. Solid black curves plot the theoretically predicted
$Q_{m}$ radial dependence based on Eq. 3 for isolated merons and antimerons.

The assumption that we have a collection of isolated spin textures somewhat
oversimplifies the reality. While the AFM topological textures are
mesoscopically discernible via DQM and X-ray techniques and thus appear
localised, they are in fact the constituents of the complex multi-textural
ensemble that interacts via a 2D magnetic charge canvas. The magnetic charge
per constituent is therefore not just dictated by their nature as merons and
antimerons, but modified through their interaction with other constituents in
the ensemble. For example, an isolated meron and antimeron pair forming an AFM
bimeron, as shown in Fig. 2(n), would have a non-zero $Q_{m}$, whose sign is
determined predominantly by its meron (see S5.9). However, this clearly cannot
be the case for a bimeron embedded in a uniform magnetisation field in the far
field (see S5.10), since the divergence theorem ensures that $Q_{m}=0$. This
goes to show that the interaction among AFM textures produces additional
magnetic charge density away from the cores and highlights the interacting
nature of this multi-textural ensemble.

## Discussion and Outlook

Our ability to identify the duality between topological AFM textures and
magnetic charges is due to the direct readout of staggered vorticity enabled
by DQM, as the NV centre senses both amplitude and orientation of the weak
magnetic field. As such, we go beyond detecting antimerons and merons, to
further distinguish between clockwise and anti-clockwise a-Bloch components,
which otherwise appear indistinguishable in X-ray dichroic images. Our imaging
approach can be easily extended to a wider family of topological textures,
including a-Néel merons and bimerons, skyrmions, and distorted AFM textures
that are otherwise divergence-free (see S5.6-5.8), relevant for topological
AFM circuitry [36, 7, 26].

Although hematite provides especially favourable conditions for DQM imaging
due to DMI-induced spin canting, it is by no means unique among AFMs in
possessing a weak net magnetisation or quasi-isotropic spins in 2D [37, 38,
39, 40]. In fully compensated antiferromagnets without bulk DMI, staggered
spin textures can also generate local net magnetisation, either statically or
dynamically [41, 42, 43]. Moreover, the DQM technique can be useful in
detecting preferential vorticity in ultra-thin films induced by interfacial
interactions –– a key requirement for applications of quantum materials in
topological spintronics [27, 26].

In light of our discovery of the duality between magnetic charges and
topological AFM textures, it is tempting to compare hematite with other
systems that harbour emergent magnetic monopoles, such as the pyrochlore spin
ice [25]. Although intriguing, monopoles in spin ice are intrinsically
distinct, as they have an underlying gauge charge, which is topological and
quantised. On the contrary, the emergent magnetic charges in hametite are two-
dimensional, not quantised and are only topological in the sense that they
dress topological AFM textures underpinning them. In fact, we have
demonstrated that hematite supports a rich tapestry of interacting magnetic
charge distributions which could open up new and complementary ways to detect,
manipulate and functionalise – via their magnetic charge – AFM topological
textures. For example, our capability to classify different AFM spin textures
could be combined with conventional spin manipulation techniques such as
various spin torques, allowing for natural read-out and write-in schemes.
Realizing this in a complex manifold of states endowed with highly non-linear
interactions could be attractive for unconventional computing [1, 2]. Finally,
the intriguing physical insights revealed in $\alpha$-Fe2O3 are a testament to
the relevance and potential of DQM as a versatile table-top platform to
explore emergent phenomena in antiferromagnets and other quantum materials.

##  III Acknowledgements

We like to thank D. M. Kara, A. Soumyanarayanan and J. J. Robinson for their
useful discussions and valuable insights. Work performed at the University of
Cambridge was supported by the Cambridge Nanoscale Sensing and Imaging Suite
(CANSIS) as part of the Henry Royce Institute under Engineering and Physical
Sciences Research Council (EPSRC) grant No. EP/P024947/1. We further
acknowledge funding from EPSRC QUES2T (EP/N015118/1) and the Betty and Gordon
Moore Foundation. Work done at the National University of Singapore was
supported by the Agency for Science Technology & Research (A*STAR) under
Advanced Manufacturing & Engineering Individual Research Grant (A1983c0034).
Work done at the University of Oxford was supported by EPSRC grant
(EP/M020517/1) and the Oxford-ShanghaiTech collaboration project. A.K.C.T.
acknowledges funding from A*STAR, through the National Science Scholarship.
H.J. acknowledges the support of Marie Skłodowska-Curie Postdoctoral
Fellowship under the Horizon Europe Guarantee Funding from UK Research and
Innovation (EP/X024938/1). M.H. acknowledges funding from EPSRC NQIT
(EP/M013243/1). L.S. acknowledges funding from the European Union’s Horizon
2020 research and innovation programme under the Marie Skłodowska-Curie grant
agreement No. 101063285. H.S.K. acknowledges funding from the Royal Society
through a University Research Fellowship.

##  Appendix A Material Growth and Characterization

Epitaxial $\alpha$-Fe2O3 films were pulsed-laser deposited on (0001)-oriented
single crystalline $\alpha$-Al2O3 substrates (CrysTec GmbH), using a KrF
excimer laser at 248 nm. X-ray reflectometry indicated a film thickness
$t=25\leavevmode\nobreak\ \mathrm{nm}$. A detailed fabrication recipe can be
found in our previous studies[17, 33]. No capping overlayer was used in this
study. Bulk magnetometery was performed using a Superconducting Quantum
Interference Device magnetometer (Quantum Design). The temperature dependence
of the canted moment, shown in Fig. S5, reveals a Morin transition in the
range of $200-220\leavevmode\nobreak\ \mathrm{K}$ such that the magneto-
crystalline anisotropy changes from in-plane ($T>T_{M}$) to out-of-plane
($T<T_{M}$).

Figure S5:  Magnetometry characterisation. Temperature dependence
characterisation (M-T) with an in-situ 50 mT in-plane field. Arrows indicate
the direction of the temperature sweep.

##  Appendix B Setup and Imaging Details

The diamond quantum magnetometry (DQM) setup consists of an electrical readout
atomic force microscope in a closed-cycle cryostat (attoDRY1000, Attocube
systems) integrated with a home-built confocal setup. For magnetic field
sensing, CW-ODMR was employed for sensor characterisation, and switched to
pulsed-ODMR for the imaging of $\alpha$-Fe2O3 to improve sensitivity. The
diamond sensor used in our measurements (Qzabre AG) contains Nitrogen-15
($I=1/2$) thereby giving two hyperfine resonances per electron spin resonance
with a 3 MHz splitting that is easily detectable with pulsed-ODMR. The on-axis
field magnitude is determined from the ODMR frequency splitting via the weak
field approximation[29], where the influence of the off-axis field on the
zero-field splitting is neglected. Microwave excitation is delivered via a
copper wire pulled across the sample surface. Imaging is conducted with the
diamond sensor positioned $\sim 50\leavevmode\nobreak\ \mathrm{\mu m}$ away
from the wire antenna. A Tektronix AWG (70002A) was programmed to provide the
pulse sequences necessary for pulsed-ODMR. Typical scans have a pixel size of
$\sim 33.3\leavevmode\nobreak\ \mathrm{nm}$, with an averaging time per pixel
of $50-100\leavevmode\nobreak\ \mathrm{\mu s}$ at both 4 K and 300 K. The
total time for a $5\times 5\leavevmode\nobreak\ \mathrm{\mu m^{2}}$ image is
approximately twelve hours, with the retrace speed being $\leq
0.1\leavevmode\nobreak\ \mathrm{Hz}$ to avoid physical damage to the sample
surface by the diamond pillar.

##  Appendix C Sensor Characterisation

###  C1 NV-axis Orientation

NV-axis orientation with reference to the lab frame is characterized by
applying an external magnetic field $\vec{B}(B_{0},\theta_{B},\phi_{B})$ using
a three-axis Helmholtz coil. We characterize the NV orientation by obtaining
ODMR spectra while holding the $B$-field at a constant magnitude $B_{0}$ and
polar $\theta_{B}$ (azimuthal $\phi_{B}$) angle and sweeping $\phi_{B}$
($\theta_{B}$) (details are found in [44]). The NV-axis orientation
characterisation of the diamond sensor used for imaging in Figs 1-3 in the
main text (fig.S6), shows $\theta_{NV}=120\pm 4^{\circ}$ and a
$\phi_{NV}=96\pm 4^{\circ}$.

Figure S6:  NV orientation characterisation. (a) ODMR with 0.5 mT applied
along the NV axis showing the Zeeman-split resonances with additional
hyperfine splitting due to the host 15N. ODMR spectrum as a function of (b)
$\phi_{B}$ with $\theta_{B}$ fixed at $180^{\circ}$ and (c) $\theta_{B}$ with
$\phi_{B}=96^{\circ}$. The maximum splitting in (b) determines
$\phi_{NV}=96^{\circ}$ which is then used in (c) to identify
$\theta_{NV}=120^{\circ}$.

###  C2 NV-to-Sample Distance

The NV-to-sample distance $d_{NV}$ is determined by measuring the magnetic
field emitted across the edge of an out-of-plane magnetic material with the
sensor of interest. Our calibration sample is a CoFeB magnetic strip (details
in [44]) with a magnetization that remains saturated at remanence. This allows
for accurate $d_{NV}$ value extraction following the proposal in [45]. An
example ODMR linescan across the edge of the magnetic strip, obtained with the
diamond sensor used for imaging in Figs. 1-3 in the main text, is given in
Fig. S7 (a). A distribution of extracted $d_{NV}$ values from multiple ODMR
linescans, such as the one in (a), is given in Figure S7 (b), resulting in an
average $d_{NV}$ value of $70\pm 6\leavevmode\nobreak\ \mathrm{nm}$ which sets
the effective spatial resolution of our imaging. This is sufficient to study
the AFM topological textures and piggybacked charges in the study, whose
length scales are larger.

Figure S7:  NV-to-sample distance characterisation. (a) ODMR spectrum across
the edge of a CoFeB magnetic strip plotted on top of the corresponding
topography. From the fit one can extract the NV-to-sample distance. (b) Height
distribution extracted from 100 line scans across the edge, giving an average
NV-to-sample distance of $70\pm 6\leavevmode\nobreak\ \mathrm{nm}$.

##  Appendix D Magnetic Field Analysis

###  D1 Magnetic Field Components Above Sample Surface

Assuming a density of magnetic dipoles $\vec{m}(x,y)$ of thickness t, the
magnetic field $\vec{B}(x,y)$ generated at a distance d above the sample
surface (source-free region) can be described in Fourier space via the dipolar
tensor [46, 47, 30]:

$\begin{pmatrix}\tilde{B}_{x}(k_{x},k_{y},d)\\\
\tilde{B}_{y}(k_{x},k_{y},d)\\\
\tilde{B}_{z}(k_{x},k_{y},d)\end{pmatrix}=-\frac{\mu_{0}}{2}e^{-kd}(1-e^{-kt})\begin{pmatrix}k_{x}^{2}/k^{2}&k_{x}k_{y}/k^{2}&ik_{x}/k\\\
k_{x}k_{y}/k^{2}&k_{y}^{2}/k^{2}&ik_{y}/k\\\
ik_{x}/k&ik_{y}/k&-1\end{pmatrix}\begin{pmatrix}\tilde{m}_{x}(k_{x},k_{y})\\\
\tilde{m}_{y}(k_{x},k_{y})\\\ \tilde{m}_{z}(k_{x},k_{y})\end{pmatrix}.$ (4)

In the absence of a canted $m_{z}$ component in $\alpha$-Fe2O3, we have:

$\tilde{B}_{z}(k_{x},k_{y},d)=-\frac{\mu_{0}}{2}e^{-kd}(1-e^{-kt})\left(\frac{ik_{x}\tilde{m}_{x}+ik_{y}\tilde{m}_{y}}{k}\right),$
(5)
$B_{z}=\mathscr{F}^{-1}\left[\left(-\frac{\mu_{0}e^{-kd}(1-e^{-kt})}{2k}\right)(ik_{x}\tilde{m}_{x}+ik_{y}\tilde{m}_{y})\right]=\alpha_{xy}(d,t)*\vec{\nabla}\cdot\vec{m}_{xy}.$
(6)

In equation 6 we use the convolution theorem, and define
$\alpha_{xy}(d,t)\equiv\mathscr{F}^{-1}(-\frac{\mu_{0}e^{-kd}(1-e^{-kt})}{2k})$
and $\mathscr{F}$ and $\mathscr{F}^{-1}$ are the Fourier transform and its
inverse. In real space, one can intuitively interpret equation 6 as the
convolution of a point spread function, $\alpha_{xy}$[31], with the divergence
of the in-plane canted magnetization, $\vec{\nabla}\cdot\vec{m}_{xy}$. In
fact, as the distance $d$ above the sample surface increases, spatial features
of $\vec{\nabla}\cdot\vec{m}_{xy}$ get increasingly ‘blurred out’, decreasing
the effective spatial resolution of the NV sensor. A general discussion
involving all components of the magnetization can be found in [31].

###  D2 Downward (Upward) Continuation and Charge Retrieval

Given the constrains set out in section D1 and following equation 4, one can
write the relation [48, 35]:

$\vec{\tilde{B}}(d+\Delta d)=e^{-k\Delta d}\vec{\tilde{B}}(d),$ (7)

where $\vec{\tilde{B}}=\mathscr{F}(\vec{B})$. A positive or negative $\Delta
d$ in equation 7 corresponds to an upward or downward continuation of the
$B$-field distribution captured at $d$. Notice that when $\Delta d$ is
positive, the exponential term $e^{-k\Delta d}$ acts to smooth out high
frequency components and physically translates to losing spatial resolution.
In the negative case, the exponential term tends to amplify high frequency
noise present in the measurements. For this reason, we apply a Hanning window
with a cut-off frequency at $2\pi/d_{NV}$, which is a common method to
suppress the undesired amplification of high-frequency noise [48, 23, 49].

By applying the downward continuation method down to the sample surface, it is
therefore possible to retrieve the $\sigma_{m}=-t\leavevmode\nobreak\
\vec{\nabla}\cdot\vec{m}_{xy}$ given by:

$\sigma_{m}=-t\leavevmode\nobreak\
\mathscr{F}^{-1}\big{(}\tilde{B}_{z}/\tilde{\alpha}_{xy}(t,d=d_{\mathrm{NV}})\big{)},$
(8)

where $\tilde{\alpha}_{xy}=\mathscr{F}(\alpha_{xy})$. This is a deconvolution
process with the function $\alpha_{xy}$.

##  Appendix E Magnetic Models

The presence of $\vec{\mathcal{D}}$ along the $c$-axis in $\alpha$-Fe2O3 (bulk
DMI) produces a small in-plane canting of angle $\Delta$ in the basal
$ab$-plane, generating a net magnetization which can be obtained from the
local Néel vector using
$\vec{m}=\vec{\mathcal{D}}\times\vec{l}=\Delta\hat{\eta}\times\vec{l}$ in the
small angle approximation. In $\alpha$-Fe2O3, the unit vector
$\hat{\eta}=(0,0,+1)$[28] is parallel to the c-axis. Since in our case the
c-axis is along the $z$-axis of the lab frame, the expression is simplified to
$\vec{m}=\Delta\hat{z}\times\vec{l}$. Note that this condition is strictly
enforced in the trigonal symmetry of the paramagnetic phase but is relaxed at
lower temperatures due to magneto-elastic interactions, resulting in a small
out-of-plane moment, which has hence been ignored here [37]. With this, we
will now describe the spatial distributions of the canted moment in various
AFM quasiparticles considering their respective linear ansatz.

Herein, we assume a canted moment $m_{\Delta}\sim 2\times
10^{3}\leavevmode\nobreak\ \mathrm{A/m}$, and a canting angle $\Delta\sim
1.1\leavevmode\nobreak\ \mathrm{mrad}$. This is consistent with values
reported for bulk $\alpha$-Fe2O3 (Table 4.2 of [50]) and comparable to the
magnetic response of our samples, as discussed in our previous studies [17,
33]. In the small angle approximation, the average magnitude of the sublattice
magnetization $M_{s}=m_{\Delta}/(2\Delta)\sim 9.1\times
10^{6}\leavevmode\nobreak\ \mathrm{A/m}$.

###  E1 Magnetic Field and Charge of AFM Antiphase Domain Walls

Assuming a linear ansatz for the antiphase domain wall profile centered at
$x=0$ along the $x$-axis, with phase $\xi_{a}$ and width $w$, we have:

$\displaystyle\vec{l}$ $\displaystyle=2M_{s}\leavevmode\nobreak\
\big{(}\leavevmode\nobreak\ \cos{\theta}\cos{\xi_{a}}\leavevmode\nobreak\
,\leavevmode\nobreak\ \cos{\theta}\sin{\xi_{a}}\leavevmode\nobreak\
,\leavevmode\nobreak\ \sin{\theta}\leavevmode\nobreak\
\big{)}\leavevmode\nobreak\ ,$ (9) $\displaystyle\vec{m}$
$\displaystyle=2\Delta M_{s}\leavevmode\nobreak\ \big{(}\leavevmode\nobreak\
-\cos{\theta}\sin{\xi_{a}}\leavevmode\nobreak\ ,\leavevmode\nobreak\
\cos{\theta}\cos{\xi_{a}}\leavevmode\nobreak\ ,\leavevmode\nobreak\
0\leavevmode\nobreak\ \big{)}\leavevmode\nobreak\ ,$ (10)
$\displaystyle\mathrm{where,\leavevmode\nobreak\ }\leavevmode\nobreak\
\leavevmode\nobreak\ \theta=\begin{cases}-\frac{\pi}{2}&,\leavevmode\nobreak\
x<-\frac{w}{2}\\\ \frac{\pi x}{w}&,\leavevmode\nobreak\ |x|\leq\frac{w}{2}\\\
+\frac{\pi}{2}&,\leavevmode\nobreak\ x>\frac{w}{2}\\\
\end{cases}\leavevmode\nobreak\ .$

Hence,

$\vec{\nabla}\cdot\vec{m}=\begin{cases}2\Delta
M_{s}\big{(}\frac{\pi}{w}\big{)}\sin{\theta}\sin{\xi_{a}}\leavevmode\nobreak\
&,\leavevmode\nobreak\ |x|\leq\frac{w}{2}\\\ 0\leavevmode\nobreak\
&,\leavevmode\nobreak\ |x|>\frac{w}{2}\end{cases}\leavevmode\nobreak\ .$ (11)

###  E2 Magnetic Field and Charge of AFM In-plane Domain Walls

Uniform IP AFM domains do not generate any stray fields since their
magnetisation would be non-divergent (see main text Eq. 1). However, in-plane
domain walls above the Morin transition, which are expected in
$\alpha$-Fe2O3to span $60^{\circ}$ and $120^{\circ}$, can generate stray
fields. Their stray field signature depends on the domain wall width $w$,
winding direction $l_{0}$, internal phase of the domain wall – given by the
angular span $\eta_{0}$, and phase offset $\phi_{0}$:

$\displaystyle\vec{l}$ $\displaystyle=2M_{s}\leavevmode\nobreak\
\big{(}\leavevmode\nobreak\
\cos{\big{(}\Psi+\phi_{0}\big{)}}\leavevmode\nobreak\ ,\leavevmode\nobreak\
\sin{\big{(}\Psi+\phi_{0}}\big{)}\leavevmode\nobreak\ ,\leavevmode\nobreak\
0\leavevmode\nobreak\ \big{)}\leavevmode\nobreak\ ,$ (12)
$\displaystyle\vec{m}$ $\displaystyle=2\Delta M_{s}\leavevmode\nobreak\
\big{(}\leavevmode\nobreak\
-\sin{\big{(}\Psi+\phi_{0}\big{)}}\leavevmode\nobreak\ ,\leavevmode\nobreak\
\cos{\big{(}\Psi+\phi_{0}\big{)}}\leavevmode\nobreak\ ,\leavevmode\nobreak\
0\leavevmode\nobreak\ \big{)}\leavevmode\nobreak\ ,$ (13)
$\displaystyle\mathrm{where,\leavevmode\nobreak\ }\leavevmode\nobreak\
\leavevmode\nobreak\
\Psi=\begin{cases}-\frac{l_{0}\eta_{0}}{2}&,\leavevmode\nobreak\
x<-\frac{w}{2}\\\ \frac{l_{0}\eta_{0}x}{w}&,\leavevmode\nobreak\
|x|\leq\frac{w}{2}\\\ +\frac{l_{0}\eta_{0}}{2}&,\leavevmode\nobreak\
x>\frac{w}{2}\\\ \end{cases}\leavevmode\nobreak\ .$

Hence,

$\vec{\nabla}\cdot\vec{m}=\begin{cases}-2\Delta
M_{s}\big{(}\frac{l_{0}\eta_{0}}{w}\big{)}\cos{\Psi+\phi_{0}}\leavevmode\nobreak\
&,\leavevmode\nobreak\ |x|\leq\frac{w}{2}\\\ 0\leavevmode\nobreak\
&,\leavevmode\nobreak\ |x|>\frac{w}{2}\end{cases}\leavevmode\nobreak\ .$ (14)

IP domain walls generate different stray fields depending on their internal
phase as illustrated in Fig. S9.

###  E3 Magnetic Field and Charge of AFM Merons and Antimerons

Assuming a linear ansatz for an (anti)meron centered at the origin, with phase
$\xi_{a}$, topological winding number $\mathcal{N}$, and radial size $R_{M}$,
we have:

$\displaystyle\vec{l}$ $\displaystyle=2M_{s}\leavevmode\nobreak\
\big{(}\leavevmode\nobreak\
\sin{\theta}\cos{(\mathcal{N}\phi+\xi_{a})}\leavevmode\nobreak\
,\leavevmode\nobreak\
\sin{\theta}\sin{(\mathcal{N}\phi+\xi_{a})}\leavevmode\nobreak\
,\leavevmode\nobreak\ \cos{\theta}\leavevmode\nobreak\
\big{)}\leavevmode\nobreak\ ,$ (15) $\displaystyle\vec{m}$
$\displaystyle=2\Delta M_{s}\leavevmode\nobreak\ \big{(}\leavevmode\nobreak\
-\sin{\theta}\sin{(\mathcal{N}\phi+\xi_{a})}\leavevmode\nobreak\
,\leavevmode\nobreak\
\sin{\theta}\cos{(\mathcal{N}\phi+\xi_{a})}\leavevmode\nobreak\
,\leavevmode\nobreak\ 0\leavevmode\nobreak\ \big{)}\leavevmode\nobreak\ ,$
(16) $\displaystyle\mathrm{where,\leavevmode\nobreak\ }\leavevmode\nobreak\
\leavevmode\nobreak\ \theta=\begin{cases}\frac{\pi
r}{2R_{M}}&,\leavevmode\nobreak\ r\leq R_{M}\\\
\frac{\pi}{2}&,\leavevmode\nobreak\ r>R_{M}\\\ \end{cases}\leavevmode\nobreak\
.$

Hence,

$\displaystyle\vec{\nabla}\cdot\vec{m}=\begin{cases}2\Delta
M_{s}\sin{((1-\mathcal{N})\phi-\xi_{a})}\Big{(}\frac{\mathcal{N}}{r}\sin{\theta}+\frac{\pi}{2R_{M}}\cos{\theta}\Big{)}&,r\leq
R_{M}\\\ 2\Delta
M_{s}\sin{((1-\mathcal{N})\phi-\xi_{a})}\Big{(}\frac{\mathcal{N}}{r}\Big{)}&,r>R_{M}\end{cases}\leavevmode\nobreak\
.$ (17)

The last term is referred to as a radial function $f(r)$ in the main text.

###  E4 Magnetic Field Simulations of Various AFM Entities

We have simulated the magnetic fields generated $d=70\leavevmode\nobreak\
\mathrm{nm}$ above $\vec{m}$ grids containing antiphase domain walls in Fig.
S8, $60^{\circ}$ and $120^{\circ}$ in-plane domain walls in Fig. S9, and
merons and antimerons in Fig. S10.

Figure S8:  Magnetic field simulation of one-dimensional topological textures.
Simulated $B_{z}$ exhibited by ADWs upon varying $\xi_{a}$. Black arrows
illustrate $\vec{m}_{xy}$. Figure S9:  Magnetic field simulation of trivial
in-plane domain walls. Simulated $B_{z}$ exhibited by 60∘ and 120∘ in-plane
domain walls with various phase offset. Black arrows illustrate
$\vec{m}_{xy}$. Figure S10:  Magnetic field simulation of two-dimensional
topological textures. Simulated $B_{z}$ exhibited by merons ($\mathcal{N}=+1$)
and antimerons ($\mathcal{N}=-1$) upon varying $\xi_{a}$. Black arrows
illustrate $\vec{m}_{xy}$.

###  E5 Distinguishing Topological and non-topological AFM textures

Above the Morin transition, topological textures are always intertwined with
IP domain walls – in fact, each topological texture is a ‘pinch-point’ of IP
domain walls [17, 18]. It is therefore crucial to be able distinguish them
apart. Whilst IP domain walls can generate stray fields (fig. S9), their
strength, distribution, and evolution are markedly different from counterparts
generated by merons and antimerons (Fig. S10). In the following we outline the
salient differences and how one can distinguish them:

  * •

Locally, topological merons and antimerons are two-dimensional entities which
produce confined stray fields, characteristic of monopolar and quadrupolar
distributions. Alternatively, IP domain walls are deconfined quasi-one-
dimensional trivial texturesThis is illustrated in Fig. S11 using the linear
meron ansatz.

Figure S11: Linear charge profile merons and IP domain walls. The linear
charge profile $\lambda_{M}$ is the integrated charge along the y-axis given
as a function of the x-axis of the meron in Fig. S10 and the IP domain walls
(IPDW) in Fig. S9. The $\lambda_{M}$ of the meron and IPDWs reveal their
respective confined and deconfined nature.

  * •

The maximum stray field magnitude generated by merons and domain walls is (i)
inversely proportional to their characteristic length scale (e.g., meron core
radius; IP domain wall width, $W$) and (ii) directly proportional to the total
angle they span in spin space ( $\pi$ for meron; $\eta_{0}$ for IP domain wall
angle) and is given by:

$\displaystyle
B^{\mathrm{max}}_{z,\mathrm{meron}}\propto|\vec{\nabla}\cdot\vec{m}_{xy}|^{\mathrm{max}}_{\mathrm{meron}}=2\Delta
M_{s}\left(\frac{\pi}{R_{M}}\right)$ (18) $\displaystyle
B^{\mathrm{max}}_{z,\mathrm{IPwall}}\propto|\vec{\nabla}\cdot\vec{m}_{xy}|^{\mathrm{max}}_{\mathrm{IPwall}}=2\Delta
M_{s}\left(\frac{\eta_{0}}{W}\right)$ (19)

Since merons trap out-of-plane cores, the anisotropy energy penalty causes
these cores to be small[17]. On the other hand, IP domain walls are much wider
as they do not have out-of-plane regions and are only limited by the very weak
basal anisotropy in $\alpha$-Fe2O3[51]. This causes $R_{M}<W$, at temperatures
well above the Morin transition, which is the case at 300 K. Moreover, since
IP domain walls are expected to have $\eta_{0}\leq\pi$. Consequently, the
magnitude of the stray field generated by a-Bloch merons at their core is
significantly more intense, by a factor of $\sim 2$ compared to trivial
textures. This allows unambiguous identification of topological meron cores.

  * •

Although both topological and trivial textures are present above the Morin
transition, application of in-plane magnetic fields results in the large-scale
annihilation of merons and antimerons only. This triggers the selective
elimination of strong and confined monopolar stray fields, leaving behind
trivial textures, as illustrated in SI section H and our previous work[17,
18]. This experimentally evinces the different evolution of topological and
non-topological entities.

  * •

As for antimerons, the quadrupolar stray field distribution with a zero-field
cross pattern (Fig. S10) while distinct from IP domain walls (Fig. S9), can be
challenging to disambiguate in isolation. For this reason, we restrict
antimeron identification and fitting to regions in the immediate proximity of
strong monopolar field distributions generated by merons. This is justified,
given that merons and antimerons occur with equal probability after executing
a Kibble-Zurek transition[17, 18] and that, based on our X-ray work, meron-
antimeron pairs are extremely common.

###  E6 Invisibility of Divergence-free Textures

For a divergence-free texture (i.e.
$ik_{x}\tilde{m_{x}}+ik_{y}\tilde{m_{y}}=0$), the z-component of the field in
Eq. 5 vanishes, rendering it invisible to magnetic field detection. This is
further illustrated in Figs. S12 and S13, where ADWs and merons turn invisible
when $\xi_{a}=0,\pi$. In reality, textures are never perfectly isolated, and
the distortion produced by the proximity of other textures will make even
a-Néel meron visible through the resulting magnetic field. This is illustrated
in figure S14, which display the magnetic field of an a-Néel meron undergoing
progressive geometrical distortion, and will be further discussed in Fig. S19
in the context of a-Néel bimerons.

Figure S12:  Field magnitude of ADWs with different phase $\xi_{a}$. Field
magnitude $|\vec{B}|$ obtained from the magnetic field simulations of ADWs
with different $\xi_{a}$. Black arrows illustrate $\vec{m}_{xy}$. Figure S13:
Magnetic field magnitude of merons with different phase $\xi_{a}$. Field
magnitude $|\vec{B}|$ obtained from the magnetic field simulations of merons
with different $\xi_{a}$. Black arrows illustrate $\vec{m}_{xy}$. Figure S14:
Distortion of a divergence-free texture. Field magnitude $|\vec{B}|$ exhibited
by an a-Néel meron with increasing distortion to the surrounding
$\vec{m}_{xy}$distribution from (a) to (d). Black arrows illustrate
$\vec{m}_{xy}$.

###  E7 Magnetic Field and Charge of AFM Bimerons

An AFM bimeron in a uniform $\vec{m}_{xy}$ background can be described
analytically as a skyrmion model rotated by $90^{\circ}$ along an in-plane
axis. Assuming a phase $\xi_{a}$ and radial size $R_{B}$, we have:

$\vec{l}=2\leavevmode\nobreak\ M_{s}[\leavevmode\nobreak\
\cos{\theta},\leavevmode\nobreak\
\sin{\theta}\sin(\phi+\xi_{a}),\leavevmode\nobreak\
-\sin{\theta}\cos(\phi+\xi_{a})]\leavevmode\nobreak\ ,\\\ $ (20)
$\vec{m}=2\Delta M_{s}\leavevmode\nobreak\ [-\leavevmode\nobreak\
\sin{\theta}\sin(\phi+\xi_{a}),\leavevmode\nobreak\
\cos{\theta},\leavevmode\nobreak\ 0\leavevmode\nobreak\ ]\leavevmode\nobreak\
,\\\ $ (21) $\displaystyle\mathrm{where,\leavevmode\nobreak\
}\theta=\begin{cases}\frac{\pi r}{R_{B}}&,\leavevmode\nobreak\ r\leq R_{B}\\\
\pi&,\leavevmode\nobreak\ r>R_{B}\\\ \end{cases}\leavevmode\nobreak\ .$

Hence,

$\displaystyle\vec{\nabla}\cdot\vec{m}_{xy}=\begin{cases}2\Delta
M_{s}\big{[}-\frac{\pi}{R_{B}}\big{(}\cos{\theta}\cos{\phi}\sin{(\phi+\xi_{a})+\sin{\theta}\sin{\phi}}\big{)}&\nobreak\leavevmode\textstyle\\\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
+\frac{1}{r}\sin{\theta}\sin{\phi}\cos{(\phi+\xi_{a})}\big{]}&,\leavevmode\nobreak\
r\leq R_{B}\\\ 0&,\leavevmode\nobreak\ r>R_{B}\\\
\end{cases}\leavevmode\nobreak\ .$ (22)

Equation 22 allows us to model the $\sigma_{m}$ distribution of bimerons, see
Fig. S15. Here, we observe that individual monopolar and quadrupolar
$\sigma_{m}$ distributions contributed by the meron and antimeron components,
albeit significantly distorted in comparison to their isolated counterparts,
can be discerned in the $\sigma_{m}$ distribution of an isolated AFM bimeron.
This is slightly more evident in the case of a-Bloch bimerons
($\xi_{a}=\pi/2$, $3\pi/2$). For a-Néel bimerons the $\sigma_{m}$ distribution
only reflects a strongly distorted quadrupolar character as the meron
component is relatively divergence-free.

More importantly, we observe a non-zero $\sigma_{m}$ distribution,
irrespective of the $\xi_{a}$ value, implying that bimerons of all characters
generate distinct and detectable $B_{z}$ signatures, see Fig. S16. This
enables DQM to play a versatile role in the future discovery and study of
isolated AFM bimerons in engineered AFM systems with a defined in-plane
anisotropy and inhomogeneous DMI. Interestingly, while the integrated charge
$Q_{m}$ over a sufficiently large area centred at the core of the bimeron
vanishes for all $\xi_{a}$, the asymmetric $\sigma_{m}$ distribution suggests
that it may be possible to spatially manipulate AFM bimerons via external
magnetic fields.

Figure S15:  Magnetic charge density $\sigma_{m}$ of an AFM bimeron with
varying phase $\xi_{a}$. Black arrows illustrate $\vec{m}_{xy}$. Figure S16:
Magnetic field distribution of an AFM bimeron with varying phase $\xi_{a}$.
Black arrows illustrate $\vec{m}_{xy}$.

###  E8 Magnetic Field and Charge of AFM Skyrmions

We can adapt the treatment from the previous section to construct the magnetic
distribution in AFM skyrmions. Assuming a phase $\xi_{a}$ and radial size
$R_{S}$, we have:

$\vec{l}=2\leavevmode\nobreak\ M_{s}[\leavevmode\nobreak\
\sin{\theta}\cos(\phi+\xi_{a}),\leavevmode\nobreak\
\sin{\theta}\sin(\phi+\xi_{a}),\leavevmode\nobreak\
\cos{\theta}]\leavevmode\nobreak\ ,\\\ $ (23) $\vec{m}=2\Delta
M_{s}[\leavevmode\nobreak\
-\sin{\theta}\sin(\phi+\xi_{a}),\leavevmode\nobreak\
\sin{\theta}\cos(\phi+\xi_{a}),\leavevmode\nobreak\ 0]\leavevmode\nobreak\
,\\\ $ (24) $\displaystyle\mathrm{where,\leavevmode\nobreak\
}\theta=\begin{cases}\frac{\pi r}{R_{S}}&,\leavevmode\nobreak\ r\leq R_{S}\\\
\pi&,\leavevmode\nobreak\ r>R_{S}\\\ \end{cases}\leavevmode\nobreak\ .$

Hence,

$\vec{\nabla}\cdot\vec{m}_{xy}=\begin{cases}2\Delta
M_{s}\sin{\xi_{a}}\big{(}\frac{-\sin{\theta}}{r}-\frac{\pi\cos{\theta}}{R_{S}}\big{)}&,\leavevmode\nobreak\
r\leq R_{S}\\\ 0&,\leavevmode\nobreak\ r>R_{S}\\\
\end{cases}\leavevmode\nobreak\ .$ (25)

AFM Néel skyrmions ($\xi_{a}=0,\pi$) have vanishing $\sigma_{m}$, as they host
divergence-free $\vec{m}_{xy}$. Other skyrmions with $\xi_{a}\neq 0,\pi$ have
a non-zero $\sigma_{m}$ distribution, see Fig. S17. As expected, $B_{z}$
mirrors the $\sigma_{m}$ trend, Fig. S18. While AFM Néel skyrmions are
invisible, DQM can still play a crucial role in imaging AFM skyrmions with
mixed or a-Bloch characters, with the added advantage of reading out the
projected vorticity via the strength of $B_{z}$. In actuality, Néel skyrmions
undergo deformation/distortion [52], which could result in detectable magnetic
field signatures, akin to what is expected for distorted a-Néel merons, as
illustrated in Fig. S14.

Figure S17:  Magnetic charge density $\sigma_{m}$ distribution of an AFM
skyrmion with varying phase $\xi_{a}$. Black arrows illustrate $\vec{m}_{xy}$.
Figure S18:  Magnetic field distribution of an AFM skyrmion with varying phase
$\xi_{a}$. Black arrows illustrate $\vec{m}_{xy}$.

###  E9 Integrated Magnetic Charge of AFM Merons and Antimerons

The integrated charge $Q_{m}$ of an isolated (anti)meron with phase $\xi_{a}$
and winding number $\mathcal{N}$ as a function of radial distance $r$ from the
core can be analytically written as:

$\displaystyle Q_{m}(r)=\begin{cases}\leavevmode\nobreak\ 4\pi
M_{s}\leavevmode\nobreak\ \Delta\leavevmode\nobreak\
\sin{(\xi_{a})}\leavevmode\nobreak\ \sin{\big{(}\frac{\pi
r}{2R_{M}}\big{)}}\leavevmode\nobreak\ r\leavevmode\nobreak\
t&,\leavevmode\nobreak\ r\leq R_{M},\leavevmode\nobreak\ \mathcal{N}=+1\\\
\leavevmode\nobreak\ 4\pi M_{s}\leavevmode\nobreak\ \Delta\leavevmode\nobreak\
\sin{(\xi_{a})}\leavevmode\nobreak\ r\leavevmode\nobreak\
t&,\leavevmode\nobreak\ r>R_{M},\leavevmode\nobreak\ \mathcal{N}=+1\\\
\leavevmode\nobreak\ 0\leavevmode\nobreak\
&,\mathcal{N}=-1\end{cases}\leavevmode\nobreak\ .$ (26)

For isolated symmetric textures, $Q_{m}$ is either zero for antimerons
($\mathcal{N}=-1$) and a-Néel merons ($\mathcal{N}=+1$, $\xi_{a}=0,\pi$), or
accumulates radially ($\mathcal{N}=+1,\xi_{a}\neq 0,\pi$), as illustrated in
Fig. S19(d).

Moreover, the integrated charge $Q_{m}$ of an isolated bimeron with phase
$\xi_{a}$ can be written as:

$\displaystyle Q_{m}(r)=\begin{cases}\leavevmode\nobreak\ 2\pi
M_{s}\leavevmode\nobreak\ \Delta\leavevmode\nobreak\
\sin{(\xi_{a})}\leavevmode\nobreak\ \sin{\big{(}\frac{\pi
r}{R_{B}}\big{)}}\leavevmode\nobreak\ r\leavevmode\nobreak\
t&,\leavevmode\nobreak\ r\leq R_{B}\\\ \leavevmode\nobreak\
0&,\leavevmode\nobreak\ r>R_{B}\end{cases}\leavevmode\nobreak\ .$ (27)

While isolated bimerons with $\xi_{a}\neq 0,\pi$ accumulate some charge at
short distances near their cores, see Fig. S19(d), the total integrated
charge, outside $R_{B}$, for all isolated bimerons embedded in a uniformly
aligned magnetic background is identically is zero.

Figure S19:  Charge accumulation of isolated textures. Numerical simulation of
charge density $\sigma_{m}$ distribution of (a) an a-Bloch meron, (b) an
antimeron and (c) a bimeron. Arrows represent $\vec{m}_{xy}$. (d) The total
charge $|Q_{m}|$ accumulated in a circular area $S$ of increasing radius $r$,
for each numerical simulation in (a-c), is given by the solid dots. $Q_{m}$
from analytical equations 26 and 27 are represented by solid curves. The
analytical $Q_{m}$ for a a-Bloch meron and an antimeron ansatz agree well with
the numerically obtained $Q_{m}$.

###  E10 Charge Variability of AFM Bimerons

The total charge $Q_{m}$ of a bimeron in a fixed area $S$ can be varied with
different $\vec{m}_{xy}$ boundary conditions. Figs. S20(a-e) illustrate
bimerons with various boundary conditions, alongside their $\vec{m}_{xy}$ and
$\sigma_{m}$ distributions. Figure S19(f) shows the variability of $|Q_{m}|$
corresponding to panels (a-e), where each $|Q_{m}|$ is obtained from the
integration of their respective $\sigma_{m}$ distribution over the whole
region shown. Notably, in panel (e), the bimeron is embedded in a uniform
magnetization with a boundary condition that satisfies:

$Q_{m}=\int_{S}\sigma_{m}\leavevmode\nobreak\
ds\propto\oint_{C}\vec{m}_{xy}\cdot\hat{e}\leavevmode\nobreak\ dc=0,$ (28)

where $C$ is the boundary of the area of integration $S$ and $\hat{e}$ is the
unit normal to $C$.

Figure S20:  Charge variability of a bimeron. (a-e) Numerical simulations of
charge density $\sigma_{m}$ distribution of bimerons with various
$\vec{m}_{xy}$ boundary conditions. Arrows represent $\vec{m}_{xy}$. (f) Total
charge $|Q_{m}|$ accumulated in a fixed area $S$ of various bimeron simulation
with different $\vec{m}_{xy}$ boundary conditions in (a-e). For convenience,
$S$ is defined as the square area covering the entirety of each panel, and is
centered at the midpoint between the core of the meron and the antimeron.

##  Appendix F Meron Core Size Analysis

The average meron radial profile shown in Fig. S21 is obtained from the merons
found in main text Fig. 1(h), by averaging their azimuthally–averaged radial
$B_{z}$ profiles. In general, there is good qualitative agreement with the
simulated $B_{z}$ profiles obtained from the structures in section E. Any
quantitative deviations are expected to result from the limitations of the
linear meron ansatz and of a comparison between experimental data obtained
from a composite textural fabric and an isolated meron model. We estimate
$R_{M}\sim 150\pm 40$ nm based on the maximum value of $B_{z}$, which
corresponds reasonably well with our previous work [17].

Figure S21:  Average meron core size. The average experimental $B_{z}$ profile
of six merons as a function of radial distance $r$ from the center of the
meron core (solid black dots, shaded area is the standard deviation). The
experimental profile is compared with the simulated $B_{z}$ profiles of merons
with various core sizes $R_{M}$ (coloured lines).

##  Appendix G Retrieving $\vec{m}_{xy}$

The inverse magnetostatic problem of retrieving $\vec{m}$ from $\vec{B}$ is in
general ill posed. However, in our case, we can take advantage of existing
constrains to approximate a solution. We first simplify the problem, as
discussed in section G1, by utilizing the fact that the magnetic field
components are linearly dependent, and that the out-of-plane magnetization
component vanishes in $\alpha$-Fe2O3. Next in section G2, we adopt appropriate
regularisation strategies to obtain $\vec{m}_{xy}$ from $B_{z}$. Finally, some
examples are discussed in sections G3 and G4.

###  G1 Simplification of Retrieval Problem

Retrieving $\vec{m}$ from $\vec{B}$ at distance $d$ above the sample surface
(i.e., inversion of Eq. 4 ) is an ill-posed problem and is therefore not
uniquely solvable. The problem can be simplified to one field component, since
the other two are linearly dependent on the first one. We chose to work with
$B_{z}$ since it intuitively reveals a low-pass filtered
$\vec{\nabla}\cdot\vec{m}_{xy}$ (eqn. 6). The inverse of the map $\bm{f}$ such
that $B_{z}=\bm{f}\left(\vec{m}\right)$, is not still not unique, even within
the restricted class of problems represented by the dipolar tensor (eqn. 4).
However, a choice of appropriate constraints reduces the degrees of freedom,
thereby allowing us to estimate $\vec{m}$ from a DQM scan. In the present
case, we set the out-of-plane magnetization component $m_{z}=0$. The inversion
problem is thus reduced from three to two dimensions, but the reduced
dimensionality is still not sufficient to estimate the in-plane magnetization
vector. This is clear when considering equation 5, where we show that any
divergence-free $\vec{m}_{xy}$ , including trivial ordering, produces the same
field distribution ($|B|=0$, section E6).

###  G2 Regularization

We adopt several regularisation strategies developed by the Lorentz
transmission electron microscopy community to approximate a solution. Closely
following the strategy in [53], we replace the ill-posed inversion problem
with the following minimization problem:

$\min\left(\|\mathbf{F}\mathbf{m}-\mathbf{B}_{z}\|^{2}+\|\mathbf{\Gamma}\mathbf{m}\|^{2}\right)\leavevmode\nobreak\
,$ (29)

where, $\mathbf{F}$, $\mathbf{B}_{z}$ and $\mathbf{m}$ are the matrix
representation of the map $\bm{f}$, the observed $B_{z}$ and the
reconstruction of $\vec{m}$, respectively. The spatial discretisation,
implicit to the minimisation equation and also to a DQM scan, reduces the
problem to finite dimensions and limits the maximum resolvable signal
frequency to the inverse of the spatial resolution (i.e.
$<\frac{1}{33}\leavevmode\nobreak\ \mathrm{GHz}$ in our images). The first
term in equation 29 is the squared error needed for the base minimisation. The
second term, known as the Tikhonov regularisation, is used to impose a
smoothness constraint to the solution $\mathbf{m}$. This is achieved via
$\mathbf{\Gamma}\mathbf{m}$ where $\mathbf{\Gamma}=\lambda\nabla$ is the
matrix representation of the gradient $\nabla$ weighted by the constant
$\lambda$. We constrain the problem further by fixing the magnitude of the
canted magnetization $|\vec{m}_{xy}|$. Given that $\vec{m}_{xy}$ can be
written as $|\vec{m}_{xy}|(\cos{\phi_{m}},\sin{\phi_{m}})$, one only needs to
reconstruct the angle $\phi_{m}$. Note that the reduction of the problem to
one dimension alone does not guarantee a unique solution as we have a non-
trivial null space. Finally, to quicken the minimisation and to avoid local
minima, we initialize the problem with a $\mathbf{m}$ guess based on a priori
information of our system [17]. $\phi_{m}$ is initialized based on the
expected topological texture associated with the characteristic $B_{z}$
distribution shown analytically and via simulations in Fig. 2 in the main
text, whereas $|\vec{m}_{xy}|$ is fixed based on the linear meron ansatz
(section E) with an average core size of 150 nm (section F), and the linear
ADW ansatz with a similar domain wall width (section 10). In this respect, we
only attempt to reconstruct $\vec{m}_{xy}$ for AFM textures and their
immediate neighbourhood with characteristic $B_{z}$ signatures. Additionally,
we minimize the problem with $B_{z}$ at $z=d_{NV}$ rather than $z=0$ to avoid
using low pass filters for the propagation, and with Neumann boundary
condition to reduce edge effects.

###  G3 Fitting of Antiphase Domain Walls

We assume $|\vec{m}_{xy}|=0$ outside the ADW. Inside the ADW, $|\vec{m}_{xy}|$
varies according to the linear ADW model (section 10). It is easy to separate
these areas since out-of-plane AFM domains have negligible divergence while
the ADWs have a distinct $B_{z}$ signature. We show an example of the
reconstruction process from the $B_{z}$ image of a simulated multi-chiral ADW
$\vec{m}_{xy}$ distribution in Fig. S22(a). The $|\vec{m}_{xy}|$ mask is
obtained by estimating the binary mask from the simulated $B_{z}$, and then
modulating it as per the linear ansatz, see Fig. S22(b). Here we initialize
the problem with a trivial $\phi_{m}(x,y)=0$ and the corresponding
$\vec{m}_{xy}$ guess is given in Fig. S22(c). The minimized $\vec{m}_{xy}$
solution given in Fig. S22(d) faithfully reproduces the simulated
$\vec{m}_{xy}$ distribution, with errors localized at the edges as indicated
by the fit residues in Figs. S22(e) and (f). Next, we apply the same protocol
to the experimentally obtained $B_{z}$, with its respective $w\sim 2R_{m}$ and
minimized $\vec{m}_{xy}$ solution given in Figs. S23.

Figure S22:  Simulated ADW fit. (a) Simulated $\vec{m}_{xy}$ of an AFM ADW and
the resultant $B_{z}$ at $d_{NV}=70$ nm. (b-c) (b) Fixed $|\vec{m}_{xy}|$ mask
and (c) guess $\vec{m}_{xy}$ distribution used to initialize the minimisation
problem. (d-f) (d) Minimized $\vec{m}_{xy}$ solution and respective (e)
$B_{z}$ and (f) $\vec{m}_{xy}$ residues. Black arrows illustrate
$\vec{m}_{xy}$. Figure S23:  Experimental ADW fit. (a) Measured $B_{z}$ of an
AFM ADW at $d_{NV}=70$ nm. (b-c) (b) Fixed $|\vec{m}_{xy}|$ mask and (c) guess
$\vec{m}_{xy}$ distribution used to initialize the minimisation problem. (d-f)
(d) Minimized $\vec{m}_{xy}$ solution and respective (e) $B_{z}$ residue.
Black arrows illustrate $\vec{m}_{xy}$.

###  G4 Fitting of (Anti)Merons and Bimerons

Similar to the procedure in section G3, $|\vec{m}_{xy}|=1$ is assumed
everywhere except in the (anti)meron core, which is modelled with a linear
ansatz with $R_{M}=150$ nm (section F). We show the $\vec{m}_{xy}$
reconstruction of a simulated a-Bloch AFM meron and an AFM antimeron in Figs.
S24 and S25, respectively. In both cases, we start off with $\xi_{a}$ that
deviates from the simulated texture, and the corresponding $\vec{m}_{xy}$
guess is given in panel (c). The resultant minimized solution given in panel
(d) closely reproduces the simulated $\vec{m}_{xy}$ distribution in panel (a).
The respective residues are given in panels (e-f).

Figure S24:  Simulated a-Bloch meron fit. Simulated $\vec{m}_{xy}$ of an AFM
a-Bloch meron and the resultant $B_{z}$ at $d_{NV}=70$ nm. (b-c) (b) Fixed
$|\vec{m}_{xy}|$ mask and (c) guess $\vec{m}_{xy}$ distribution used to
initialize the minimisation problem. (d-f) (d) Minimized $\vec{m}_{xy}$
solution and respective (e) $B_{z}$ and (f) $\vec{m}_{xy}$ residues. Black
arrows illustrate $\vec{m}_{xy}$. Figure S25:  Simulated antimeron fit.
Simulated $\vec{m}_{xy}$ of an AFM antimeron and the resultant $B_{z}$ at
$d_{NV}=70$ nm. (b-c) (b) Fixed $|\vec{m}_{xy}|$ mask and (c) guess
$\vec{m}_{xy}$ distribution used to initialize the minimisation problem. (d-f)
(d) Minimized $\vec{m}_{xy}$ solution and respective (e) $B_{z}$ and (f)
$\vec{m}_{xy}$ residues. Black arrows illustrate $\vec{m}_{xy}$.

Finally, we apply the same reconstruction protocol to experimental images,
from which we obtain the $\vec{m}_{xy}$ fits of an anti-clockwise a-Bloch
meron (Fig. S26), an antimeron (Figure S27) and a composite structure
consisting of one meron and two antimerons (Fig. S28).

Figure S26:  Experimental a-Bloch meron fit. Measured $B_{z}$ of an AFM right
a-Bloch meron at $d_{NV}=70$ nm. (b-c) (b) Fixed $|\vec{m}_{xy}|$ mask and (c)
guess $\vec{m}_{xy}$ distribution used to initialize the minimisation problem.
(d-e) (d) Minimized $\vec{m}_{xy}$ solution and respective (e) $B_{z}$
residue. Black arrows illustrate $\vec{m}_{xy}$. Figure S27:  Experimental
antimeron fit. Measured $B_{z}$ of an AFM antimeron at $d_{NV}=70$ nm. (b-c)
(b) Fixed $|\vec{m}_{xy}|$ mask and (c) guess $\vec{m}_{xy}$ distribution used
to initialize the minimisation problem. (d-e) (d) Minimized $\vec{m}_{xy}$
solution and respective (e) $B_{z}$ residue. Black arrows illustrate
$\vec{m}_{xy}$. Figure S28:  Experimental fit for composite texture. Measured
$B_{z}$ of a composite texture consisting of two antimerons and an a-Bloch
meron at $d_{NV}=70$ nm. (b-c) (b) Fixed $|\vec{m}_{xy}|$ mask and (c) guess
$\vec{m}_{xy}$ distribution used to initialize the minimisation problem. (d-e)
(d) Minimized $\vec{m}_{xy}$ solution and respective (e) $B_{z}$ residue.
Black arrows illustrate $\vec{m}_{xy}$.

###  G5 Field imaging and Magnetisation Reconstruction of $\alpha$-Fe2O3

Unique magnetisation reconstruction is a general problem in the stray-field
imaging. Nonetheless, the rigorous identification procedure (see E5), together
with the implementation of systematic regularization protocols, has helped
constrain the magnetic solution space and provided meaningful canted moment
reconstructions. While these reconstructions may not always be unique, they
are in practice entirely consistent with AFM textures reported via X-ray
imaging[17].

Although X-ray linearly dichroic imaging is not affected by the same
uniqueness problem, it suffers from an intrinsic $180^{\circ}$ ambiguity which
would make reconstructions of the canted moment more ambiguous than in the
present case. Additionally, the canted magnetisation in $\alpha$-Fe2O3 is too
weak to produce circular dichroic X-ray contrast[17]. Therefore, field imaging
via DQM provides an invaluable view of the rich magnetic charge phenomenology
in $\alpha$-Fe2O3.

##  Appendix H Reversal Field Imaging

Figure S29:  Reversal field imaging at remanence. Remanent state of
$\alpha$-Fe2O3 after applying an in-situ in-plane magnetic field with
magnitude varying from (a) 150mT, (b) 200mT and (c) 250mT. Figure S30:
Annihilation of topological spin entities. Remanent state of $\alpha$-Fe2O3
after applying an ex-situ in-plane magnetic field of $\sim
400\leavevmode\nobreak\ \mathrm{mT}$.

Upon application of strong in-plane magnetic fields, it is possible to drive
AFM domain realignment which results in meron-antimeron annihilation [17]. To
confirm this, two permanent magnet poles were aligned parallel to the sample
surface resulting in a homogeneous field across the sample, variable via
micrometer stages. Since imaging under high off-NV-axis field is not possible,
the external field was applied for a few minutes followed by NV imaging in
zero field, see Fig. S29. Although the scans are approximately in the same
area, open-loop scanning requires post processing, wherein the images have
been registered using distinct field features in the scans. Several merons and
antimerons are initially present. As the magnitude of the external field
increases, annihilation processes occur, accompanied by an alteration of the
surrounding trivial AFM textures. Due to the limited field magnitude that can
be applied in-situ, large-scale annihilation of textures was driven by an ex-
situ 400 mT field [17], see Fig. S30. Due to the probabilistic nature of the
generation of topological textures via the Kibble-Zurek mechanism, the
occurrence of merons with different projected vorticities (red vs blue cores)
can vary across the sample [18]. It is noted that while Fig. S30 shows
predominately anti-clockwise a-Bloch merons (red cores), additional large area
scans (see S9) confirm approximately similar distributions of merons with
different projected vorticities (red vs blue cores) are obtained due to the
absence of chiral interactions in $\alpha$-Fe2O3[17].

##  Appendix I Additional Data

Figure S31:  Large Area RT Images Taken After Thermal Cycling. Field imaging
over a large area showing presence of both right and left a-Bloch merons
indicated by blue and red arrows respectively.

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