Chapter 4 Canted antiferromagnetic magnetic order in honeycomb $\alpha {\text{-Na}}_2{\text{PrO}}_3$revealed by single crystal torque measurements

In the Kitaev model [55] on a honeycomb lattice, spin $1/2$ moments interact with their nearest neighbours by Ising interactions along mutually orthogonal directions for the three bonds meeting at each site in a honeycomb. This model is interesting in that it is exactly solvable, with a quantum spin liquid ground state and exotic excitations, very different from the conventional spin-waves of ordered magnets. Strong spin-orbit coupling is needed in order for a system to host these bond-dependent interactions, this has meant that $4d$ and $5d$ electron systems have been proposed candidates for Kitaev physics [41, 69, 96]. Generalisations beyond this simple model include the Heisenberg-Kitaev (HK) model, which has been used to describe the exotic magnetic phases observed in the layered honeycomb iridates [16, 92]. A further generalisation included the symmetry allowed "symmetric off-diagonal exchange" $\mathrm{\Gamma }$, which was used to try to explain the ground states and excitations in some of these honeycomb materials, and revealed a complex magnetic phase diagram, including a finite region of parameter space for a quantum spin liquid (QSL) phase around the exactly solvable pure-Kitaev limit [81]. Magnetic honeycomb materials have had significant theoretical interest due to their rich phase diagram. Much of this work has been in the context of the honeycomb iridates [51].

Honeycomb $\alpha {\text{-Na}}_2{\text{PrO}}_3$, with networks of edge-sharing PrO₆ octahedra was recently theoretically predicted to host strong antiferromagnetic Kitaev interactions [42], making it a promising candidate for unconventional magnetism. Studies on powder samples of the material confirm antiferromagnetic order at low temperature [21], but reported inelastic neutron scattering data do not find clear evidence of strong Kitaev interactions.

In this section, I report single-crystal X-ray structure studies, a detailed mapping of torque magnetometry with respect to field orientation up to $4\mathrm{T}$ consistent with a non-collinear, canted antiferromagnetic structure, torque magnetometry up to $35\mathrm{T}$ revealing a spin-flop transition, and powder magnetometry up to $65\mathrm{T}$. Finally, I suggest a minimal anisotropic exchange model which stabilises the proposed magnetic structure, is quantitatively consistent with the angular-dependent torque data in three orthogonal planes, and predicts a spin-flop transition for field applied parallel to the ${𝒄}^{\mathbf{\ast }}$ axis, as is seen experimentally.

Samples of $\alpha {\text{-Na}}_2{\text{PrO}}_3$were synthesised by Ryutaro Okuma at the University of Oxford. The polycrystalline samples grown contained many platelet samples with diameters $\sim 80\mathrm{\mu }\mathrm{m}$. Due to the air sensitivity of the samples, they were synthesised, handled, and selected in an inert gas atmosphere, and sealed by flame in an evacuated borosilicate-glass capillary. In order to determine the crystal structure of $\alpha {\text{-Na}}_2{\text{PrO}}_3$, X-ray diffraction was performed at room temperature on single crystal samples of the material with an Mo-source Oxford Diffraction Supernova diffractometer. These measurements were performed with the sample remaining in an evacuated capillary to prevent degradation during data collection. The detailed structure was refined using FULLPROF [83] analysis software, the results of this refinement are summarised in Table 4.1 and Table 4.2, which shows a summary of the structural parameters and atomic fractional coordinates of the material’s ions in the monoclinic setting. Figure 4.2 shows a comparison between observed and calculated peak intensities, the small disagreement of a few points at large ${|F|}^2$ is likely caused by uncorrected absorption from the glass capillary.

Figure 4.1a, b) show a comparison between the experimentally observed single-crystal diffraction pattern and calculated Bragg peak intensities for the $0kl$ reciprocal lattice plane, for a representative, untwinned, plate-like sample of $\approx 80\mathrm{\mu }\mathrm{m}$ diameter. The crystal structure parameters used in the calculation are those in Table 4.1. X-ray diffraction reveals an overall $C2/c$ space group, with two honeycomb networks of edge-sharing PrO₆ octahedra per unit cell separated by layers of ${\text{Na}}^{+}$ions in a hexagonal lattice, in approximate agreement with previously reported structural studies on powders of $\alpha {\text{-Na}}_2{\text{PrO}}_3$[36]. A key difference observed in single crystal data is a significant different in parameters $c$ and $\gamma$, which in powder data are reported as $c=11.749(2)\mathrm{\AA }$, $\gamma =110.09(1)\mathrm{°}$. This difference can be attributed to the fact that powder structure is sensitive primarily to inter-plane distance $c\sin \gamma$, this inter-plane distance agrees very closely with the data observed in single crystals. The structural refinement of this single crystal data showed no evidence of the atomic site-mixing reported in powder samples [36, 21]. The $C$ centring results in a $h+k=2n(n\in \mathbb{Z})$ selection rule, which can be seen in Fig. 4.1a, b) as systematic absences at odd $k$. The broad rings of scattering intensity at small scattering wave-vector $|𝑸|$ is attributed to scattering from the amorphous glass capillaries. The refined crystal structure can be seen in Figure 4.3, where the two inequivalent ${\text{Pr}}^{4+}$sites are coloured different shades of blue. This shows the layer stacking of the two honeycomb layers.

Untwinned single crystals of $\alpha {\text{-Na}}_2{\text{PrO}}_3$could be identified by their sample morphology, shown Fig. 4.1c). Samples grew as natural plate-like elongated hexagons, with three pairs of equal length and parallel sides. In the standard morphology, the ${𝒄}^{\mathbf{\ast }}$ axis is parallel to the plate normal, the $𝒂$ direction within the plate could be identified as parallel to the long edges in the crystal. The largest untwinned single crystals were $\approx 120\mathrm{\mu }\mathrm{m}$ on their long edge. Most samples grown were $\approx 10\mathrm{\mu }\mathrm{m}$ thick.

In addition to untwinned single crystals, batches of $\alpha {\text{-Na}}_2{\text{PrO}}_3$consistently contained twinned samples. Commonly these contained twins rotated $\pm 120\mathrm{°}$ around the ${𝒄}^{\mathbf{\ast }}$ axis. This effect can be seen in the X-ray diffraction data as additional reflections on the $(0kl)$ families of Bragg peaks at $l=n\pm 1/3(n\in \mathbb{Z})$ as observed in Figure 4.1d,e) which shows a comparison between experimental twinned single-crystal diffraction pattern and calculated Bragg peak intensities for a twinned sample with equal weight twins rotated $\pm 120\mathrm{°}$ around the ${𝒄}^{\mathbf{\ast }}$ axis. The crystal structure parameters used in the calculation are those in Table 4.1. These twinned crystals could be identified by their morphology, as they would grow in a much more symmetric hexagonal plate shape than the untwinned sample. A comparison between twinned and untwinned crystal morphologies can be seen in Fig 4.1c,f). All torque measurements reported in this chapter were performed on untwinned single crystals of $\alpha {\text{-Na}}_2{\text{PrO}}_3$.

$\alpha {\text{-Na}}_2{\text{PrO}}_3$is structurally very similar to other layered honeycomb materials such as ${\text{Na}}_2{\text{IrO}}_3$ [18], $\alpha$-${\text{Li}}_2{\text{IrO}}_3$ [29], and $\alpha$-${\text{RuCl}}_3$ [44]. All three materials display rods of diffuse scattering intensity along the layer stacking direction, attributed to layer stacking faults [35]. Surprisingly, the majority of untwinned samples of $\alpha {\text{-Na}}_2{\text{PrO}}_3$showed no diffuse rods of scattering, or other evidence of stacking faults. This is interesting as the honeycomb ${\text{Pr}}^{4+}$layers are separated by a hexagonal layer of ${\text{Na}}^{+}$ions, similar to $\alpha$-${\text{Li}}_2{\text{IrO}}_3$, and so dislocations of the honeycomb layers in the $𝒂𝒃$ plane might be expected as observed in [18, 29, 44]. This is unlike the case of powder samples of $\alpha {\text{-Na}}_2{\text{PrO}}_3$, which do show evidence of stacking faults [21], and site mixing [36]. The apparent much lower rate of stacking faults in untwinned single crystals of $\alpha {\text{-Na}}_2{\text{PrO}}_3$compared to similar materials suggests a very high degree of structural ordering of the layers. This may be related to the different layer stacking in $\alpha {\text{-Na}}_2{\text{PrO}}_3$, with two layers per unit cell interlocked with an offset along $𝒃$, perhaps providing a higher energy of interlayer bonding to ensure stacking faults are less frequent.

An interesting structural motif in $\alpha {\text{-Na}}_2{\text{PrO}}_3$is that the fractional spacing between the ${\text{Pr}}^{4+}$$(1)$ and ${\text{Pr}}^{4+}$$(2)$ sites along the $𝒃$ axis $d_1\approx 0.3327(2)\approx 1/3$. Similarly, the distance between the ${\text{Na}}^{+}$$(2)$ and ${\text{Na}}^{+}$$(3)$ along the $𝒃$ axis $d_2\approx 0.3286(4)\approx 1/3$, a similar pattern exists for the ${\text{O}}^{2-}$sites. The consequence of this is a systematic weakness in the intensity of reflections at $k=6n(n\in \mathbb{Z})$ ($l$ odd) and $k=6n+3(n\in \mathbb{Z})$ ($l$ even). This can be clearly seen in the X-ray diffraction data in Fig. 4.1.

The magnetic ions in $\alpha {\text{-Na}}_2{\text{PrO}}_3$are Kramers $4f^1$ ${\text{Pr}}^{4+}$ions, with a Hund’s rules ground state of $L=3$, $S=1/2$, ${}_{}{}^{2}F$. Spin-orbit coupling is described by the Hamiltonian

$${\widehat{\mathscr{H}}}_{\text{SO}}=\lambda \widehat{𝑳}\cdot \widehat{𝑺}$$

(4.1)

over this 14-fold degenerate manifold of states. Here, $\lambda >0$ because the shell is less than half full. This can then be diagonalised, coupling the $L$ and $S$ states together into $J=|L-S|,|L-S|+1,\mathrm{\dots },L+S$ states. In these ions, this splits the ${}_{}{}^{2}F$ manifold into a ${}_{}{}^{2}F_{5/2}^{}$ sextet with energy $-2\lambda$ and a ${}_{}{}^{2}F_{7/2}^{}$ octet at $\frac{3}{2}\lambda$, which form the free-ion energy levels of ${\text{Pr}}^{4+}$ions.

In a crystal, the effect of the crystal-electric field (CEF) also has to be considered. In $\alpha {\text{-Na}}_2{\text{PrO}}_3$, the ${\text{Pr}}^{4+}$ions are in an octahedral environment of oxygens. If this octahedron is treated as cubic, the octahedral crystal field Hamiltonian ${\widehat{\mathscr{H}}}_{\text{OCF}}$ can be written in terms of Stevens operators $O_k^q$ [1, 86].

$${\widehat{\mathscr{H}}}_{\text{OCF}}=B_4^0(O_4^0+5O_4^4)+B_6^0(O_6^0-21O_6^4)$$

(4.2)

where for ${\text{Pr}}^{4+}$the ratio between the two cubic crystal field parameters $B_6^0\approx -0.004B_4^0$ [43]. The ${}_{}{}^{2}F_{5/2}^{}$ free-ion sextet is split into a ${\mathrm{\Gamma }}_7$ doublet ground state and a ${\mathrm{\Gamma }}_8$ quartet excited state [42], where the multiplets are labelled by the irreducible representation they transform as in the $O_h$ point group. This ground state doublet is a Kramers pair and can be treated as a pseudospin with an effective spin of $1/2$, as illustrated in Fig. 4.4. The transition energy between the ground state and the excited quartet is measured directly at $E\approx 240\mathrm{meV}$ by low $|𝑸|$ inelastic neutron scattering [21]. The ${}_{}{}^{2}F_{7/2}^{}$ manifold will split into three multiplets as ${}_{}{}^{2}F_{7/2}^{}={\mathrm{\Gamma }}_6\oplus {\mathrm{\Gamma }}_7\oplus {\mathrm{\Gamma }}_8$ in order of increasing energy [42, 43] as illustrated in Fig. 4.4. If this cubic octahedral field is treated as a perturbation on the spin-orbit ground state i.e. $\lambda \gg B_4^0$, then the $g$-factor of the ground state doublet is exactly $\frac{10}{7}$ [34]. As the strength of spin-orbit coupling is made weaker, or equivalently the value of $B_4^0$ is made larger, this value of $g$ decreases, as shown in Fig. 4.4b). The value of $B_4^0$ is fixed to $B_4^0\approx 0.42\mathrm{meV}$ to reproduce the $\approx 240\mathrm{meV}$ transition to the ${\mathrm{\Gamma }}_8$ multiplet [21]. With this value fixed, a spin-orbit coupling of $\lambda \approx 160\mathrm{meV}$ reproduces the $g=0.98$ powder average observed in temperature-dependent magnetic susceptibility measurements [21], extracted from a fit to Curie-Weiss behaviour discussed in the following section. This value of $\lambda$ is somewhat larger, but close to, a previously proposed free-ion value of $\lambda \approx 110\mathrm{meV}$ [37].

Additional distortions of this crystal field split the quartets into doublets, but the ground state doublet is protected by Kramers degeneracy so cannot split further. For example, a symmetry allowed trigonal distortion of the form

$${\widehat{\mathscr{H}}}_{\text{trig}}=B_2^0{O}_2^0$$

(4.3)

will split both quartets, as is illustrated in Figure 4.4a). Additionally, this will break the cubic symmetry and introduce single-ion anisotropy in the $g$-factor. The effect of this can be seen in Fig. 4.4c), which shows how a non-zero value of $B_2^0$ will split $g_{\parallel }$, measured along the ${𝒄}^{\mathbf{\ast }}$ trigonal axis, from $g_{⟂}$ measured in the $𝒂𝒃$ honeycomb plane. Note that the powder average is not very sensitive to this value.

Previously published studies on powder samples of $\alpha {\text{-Na}}_2{\text{PrO}}_3$reported structure, calorimetry, magnetometry, neutron diffraction, and inelastic neutron scattering [21, 36].

Reported field-cooled (FC) and zero-field-cooled (ZFC) temperature dependent susceptibility show a sharp anomaly at $T_N\approx 4.6\mathrm{K}$[21], where FC and ZFC data diverge. The strong suppression of ZFC susceptibility is indicative of dominant antiferromagnetic order, however the large enhancement in the field-cooled susceptibility alongside magnetic order suggests that a ferromagnetic component is also present in the magnetic structure. Neutron scattering at low temperatures did not observe any magnetic Bragg peaks beyond the nuclear scattering Bragg peaks. This is suggestive of a $𝒒=0$ magnetic structure.

The published calorimetry data shows a cusp-like anomaly in magnetic entropy at the Néel temperature $T_N$ [21]. This is suggestive of a second-order phase transition, which in turn indicates that any magnetic basis vectors involved in the order are described by the same irreducible representation of the paramagnetic space group [25].

In order to explore the magnetic anisotropy in $\alpha {\text{-Na}}_2{\text{PrO}}_3$, I performed piezocantilever torque magnetometry measurements on $\approx 80\mathrm{\mu }\mathrm{m}$ single crystals. All torque measurements reported in this section were performed untwinned single crystals of $\alpha {\text{-Na}}_2{\text{PrO}}_3$, confirmed by X-ray diffraction measurements.

Figure 4.5 shows an overview of the observed temperature and angular dependent torque measurements in the $𝒂𝒄$ plane between $100\mathrm{K}$ and $2\mathrm{K}$, measured at fixed applied field magnitude of ${\mu }_{0}H=4\mathrm{T}$. At high temperatures, these measurements show an overall sinusoidal shape, with a stable equilibrium near ${\mu }_0𝑯\parallel 𝒂$ and an unstable equilibrium near ${\mu }_0𝑯\parallel {𝒄}^{\mathbf{\ast }}$. This overall shape is consistent with the sinusoidal torque profile expected in a uniaxial paramagnetic sample [107, 113] (See Chapter 2.2):

	$\tau (\theta ,T)$	$=-{\tau }_2(T)\sin 2\theta$		(4.4a)
		$=-{\displaystyle \frac{1}{2}}{B}^2\left({\chi }_{\parallel }(T)-{\chi }_{⟂}(T)\right)\sin 2\theta$		(4.4b)

where ${\chi }_{\parallel }$ is the susceptibility measured parallel to the ${𝒄}^{\mathbf{\ast }}$ out-of-plane direction and ${\chi }_{⟂}$ is the in-plane susceptibility. The angle $\theta$ is measured relative to the crystal ${𝒄}^{\mathbf{\ast }}$ axis. This suggests an order of susceptibilities

(4.5)

in disagreement with the proposed values of the g-factors in [21]. This can be explained as the proposed values come from a fit of crystal-field parameters to temperature-dependant susceptibility of powder samples of $\alpha {\text{-Na}}_2{\text{PrO}}_3$. In a powder, given the number of free parameters this is under-constrained and not very sensitive to the anisotropy present. This torque data presented should be able to provide an additional constraint to the fitted CEF parameters.

These torque data are fitted to Eqn. (4.4a) and the fitted temperature-dependent amplitude ${\tau }_2(T)$ can be seen in Fig. 4.5b). This shows that above $10\mathrm{K}$ the torque decreases monotonically with temperature. There is a sharp anomaly in this torque amplitude at the Néel temperature $T_N\approx 4.6\mathrm{K}$, below which the torque amplitude rapidly increases upon further cooling.

In the low-temperature regime the torque profile is no longer fit well by the sinusoidal profile (4.4a). This can be seen most clearly in Figure 4.6 which shows a comparison between measured angle dependence of torque just above the transition at $5\mathrm{K}$ (black) and well below the transition at $2\mathrm{K}$ (red). Both rotation directions are shown, clockwise ($\theta$ increasing) as a solid line, anticlockwise ($\theta$ decreasing) as a dashed line. Above the transition the torque is sinusoidal, with the unstable equilibrium near the ${𝒄}^{\mathbf{\ast }}$ direction, as discussed above. In this high-temperature case, the clockwise and anti-clockwise torque curves are the approximately the same, except for a small ($\sim 1\mathrm{°}$) shift in $\theta$ caused by backlash in the rotator probe. Below the transition, two main features are observed; There is significant hysteresis between the clockwise and anticlockwise rotations, and a sharp jump in torque is observed as the field crosses the ${𝒄}^{\mathbf{\ast }}$ axis when the projection of the field onto the $𝒂$ axis changes sign. It should be noted that the width of the hysteresis varies significantly between samples and is larger for lower temperatures. The fact that these two effects onset sharply for is suggestive that both are associated with magnetic order.

Figure 4.7 a) shows the field and angular dependence of torque curves in the $𝒂𝒄$ plane at $4.5\mathrm{K}$ between $1\mathrm{T}$ and $4\mathrm{T}$. This temperature was chosen as just below the transition the observed jump is particularly sharp and the hysteresis does not complicate the shape of the data. These data show that there are two contributions to the bandwidth of the torque, the half-height of the torque jump ${\mathrm{\Delta }}_1$ and remaining half-height of the bandwidth ${\mathrm{\Delta }}_2$. I find that ${\mathrm{\Delta }}_1\sim H$ and ${\mathrm{\Delta }}_2\sim H^2$, suggesting that the two contributions have different origins, although both are associated with magnetic order. This scaling can be seen in the field dependence of the peak-to-peak bandwidth, visible in Fig. 4.7b) where $\frac{{\tau }_{p2p}}{H}$ is plotted against ${\mu }_{0}H$. In a paramagnet where (4.4b) is valid, this should scale as $H$, with zero intercept. Instead, a linear fit to the data shows a significant non-zero intercept indicating that up to $4\mathrm{T}$ the torque bandwidth has the approximate form

$${\tau }_{p2p}(H)=A_{1}H+A_2{H}^2$$

(4.6)

To further explore the anisotropy of $\alpha {\text{-Na}}_2{\text{PrO}}_3$, the magnetic torque was measured in the other two orthogonal crystal planes $𝒂𝒃$ and $𝒃{𝒄}^{\mathbf{\ast }}$. The air sensitivity of the sample, and the need for them to be exposed to atmosphere to mount them meant that remounting the same sample in these orientations was not possible. Instead, new samples were mounted. This means that, unlike in the other systems discussed in this thesis, the units of torque between different orientations are not comparable.

Figure 4.8 shows torque magnetometry data measured in the $𝒂𝒃$ plane, on the sample photographed in panel a). In order to mount the sample in this orientation, the sample was mounted to the side of the lever, attached by a small amount of chilled vacuum grease. Precisely aligning the $𝒂$ axis in this case is difficult as the orientation in the rotation plane cannot be seen during mounting. The $\varphi =0$ rotation position is nominally the $𝑯\parallel 𝒃$ orientation, and is tentatively placed within 5 degrees of this.

Figure 4.8b) shows the angular dependence of magnetic torque measured at ${\mu }_{0}H=4\mathrm{T}$ and temperatures below and above $T_N$ at $4.5\mathrm{K}$ and $8\mathrm{K}$ respectively. The high temperature data (black) shows a very small torque signal and no observable hysteresis. Overall at $8\mathrm{K}$, ${\mu }_0𝑯\parallel 𝒃$ is the stable equilibrium and ${\mu }_0𝑯\parallel 𝒂$ is the unstable equilibrium. This is attributed to a small difference in susceptibilities between the directions, caused by g-factor anisotropy , symmetry allowed by the monoclinic structure. Below $T_N$ (red), the torque signal is much bigger, with a clear jump in torque when the projection of the field along the $𝒂$ direction $𝑯\cdot 𝒂$ changes sign, that is, the field crosses the $𝒃$ axis. A small amount of hysteresis can also be seen, such that for anti-clockwise rotations the torque is more positive than for clockwise rotations, as seen above in the $𝒂𝒄$ plane. This indicates that the material feels a torque opposing the rotation in field, and is dissipating energy, attributed to an energy cost of domain reorientation.

Figure 4.8c) shows the field and angular dependence of magnetic torque measured below the transition at $4.5\mathrm{K}$ up to ${\mu }_{0}H=4\mathrm{T}$. The same features are seen at all fields measured, with larger torque signals at higher fields. The peak-to-peak torque bandwidth ${\tau }_{p2p}$ can be seen in Fig. 4.8d), showing that ${\displaystyle \frac{{\tau }_{p2p}}{H}}$ is approximately constant, demonstrating that the size of the jump is almost linear in magnetic field, as was seen in the $𝒂𝒄$ plane. A small negative gradient in the linear fit is attributed to the small difference in $g$-factors between the two orientations, as seen in panel b), causing a difference in differential susceptibility.

The final orientation to consider is the $𝒃{𝒄}^{\mathbf{\ast }}$ plane. The angular and temperature dependence of the magnetic torque in this plane in shown in Figure 4.9 a). This shows an overall torque profile such that there is a stable equilibrium with field parallel to the $𝑯\parallel 𝒃$ axis and an unstable equilibrium with field parallel to the $𝑯\parallel {𝒄}^{\mathbf{\ast }}$ axis. The torque amplitudes were fitted to (4.4a), and the temperature dependence of the amplitude parameter ${\tau }_2(T)$ is shown in Fig. 4.9b). This shows a broad peak centred at approximately $10\mathrm{K}$, and an anomaly at $T_N$, where the clockwise and anti-clockwise curves diverge. There are no jumps seen in the angular dependence of the torque, and a $\sin 2\theta$ dependence is seen at all temperatures.

These results can be naturally explained by a minimal model with predominant antiferromagnetism along the ${𝒄}^{\mathbf{\ast }}$ direction and a small ferromagnetic moment along the $𝒂$ axis. This would have a time-reversed magnetic domain with a ferromagnetic moment along the $-𝒂$ axis. An applied field with a non-zero projection along the $𝒂$ axis can select which magnetic domain is populated, so when this projection changes sign there is a jump in torque.

In the low-field limit, the ferromagnetic contribution to the torque can be calculated by considering the net magnetic moment as fixed in magnitude with value $\pm M$ along $𝒂$ for the two magnetic domains. In this case the energy is

	$E$	$=-𝑴\cdot 𝑩$		(4.7a)
		$=\mp M{B}_x$		(4.7b)
		$=\mp MB\sin \theta$		(4.7c)

where $\theta$ is the angle measured from the ${𝒄}^{\mathbf{\ast }}$ axis, and $\mp$ refer to the two magnetic domains.

The torque is then the derivative with angle

	${\tau }_y$	$=-{\displaystyle \frac{\partial E}{\partial \theta }}$		(4.8a)
		$=\pm MB\cos \theta$		(4.8b)

For $\sin \theta >0$ the lower energy domain has $E=-MB\sin \theta$, for the lower energy domain has $E=+MB\sin \theta$, such that the overall energy can be written

	$E$	$=-MB\sin \theta \mathrm{sign}(\sin \theta )$		(4.9)
	${\tau }_y$	$=MB\cos \theta \mathrm{sign}(\sin \theta )$		(4.10)

with a jump from $-MB$ to $MB$ at $\theta =0$. This can explain the linear $A_1$ term in (4.6), the quadratic $A_2$ term will come from the difference in differential susceptibility between the ${𝒄}^{\mathbf{\ast }}$ and $𝒂$ directions, as in a uniaxial antiferromagnet.

To find a magnetic structure which fits this minimal description, the $𝒒=0$ irreducible representations of the paramagnetic space group were considered. This is obtained from the structural space group augmented time-reversal symmetry $\mathrm{\Theta }$, i.e. $\mathrm{\Theta }\otimes C2/c$. The $𝒒=0$ structures have the same translational symmetry as the structural lattice, as well at the $2$-fold rotation around $𝒃$, the $c$ glide plane, and time-reversal, so there are 8 symmetry operations and 8 irreps. This group has irreps ${\mathrm{\Gamma }}_n^{\pm }$, and $m{\mathrm{\Gamma }}_n^{\pm }$ with characters under time reversal ${\chi }^{({\mathrm{\Gamma }}_n^{\pm })}\left(\mathrm{\Theta }\right)=+1$,${\chi }^{(m{\mathrm{\Gamma }}_n^{\pm })}\left(\mathrm{\Theta }\right)=-1$ respectively, and all other characters the same as the associated ${\mathrm{\Gamma }}_n^{\pm }$ irrep of the $C2/c$ structural space group [24]. Only the $m{\mathrm{\Gamma }}_n^{\pm }$ irreps have magnetic basis vectors, which must change sign under time reversal. These irreps and their magnetic basis vectors are tabulated in Table 4.3. Here the unprimed basis vectors refer to the ${\text{Pr}}^{4+}$$(1)$ sites, and the primed basis vectors refer to the ${\text{Pr}}^{4+}$$(2)$ sites. $F$ refers to ferromagnetic alignment of spins of the same sublattice between layers, while $A$ refers to antiferromagnetism of spins on the same sublattice in adjacent layers.

Previously reported calorimetry data [21] found that entropy was continuous through the transition. This is a signature of a continuous (second-order) phase transition, which in turn suggests that the magnetic order just below $T_N$ should be described by a single irreducible representation of the high-temperature paramagnetic space group. In order to have a ferromagnetic moment along the $𝒂$ axis, an $(F_x,F_x^{\prime })$ contribution is needed, described by the m${\mathrm{\Gamma }}_2^{+}$ irrep. In order to include the dominant antiferromagnetism along the ${𝒄}^{\mathbf{\ast }}$ axis, an additional contribution is needed of the form $(F_z-F_z^{\prime })$. It should be noted that there is no symmetry requirement here that the ferromagnetic moment is exactly parallel to the $𝒂$ axis, the moments can be rotated by any angle in the $𝒂𝒄$ plane and the structure will be described by the $m{\mathrm{\Gamma }}_2^{+}$ irrep. A model of this magnetic structure can be seen in Figure 4.10.

One puzzling feature is that in this structure, with dominant antiferromagnetism along the ${𝒄}^{\mathbf{\ast }}$ axis, one would expect substantial suppression of susceptibility for fields along ${𝒄}^{\mathbf{\ast }}$ axis. This should be observed in torque as an enhancement in torque in the $𝒃{𝒄}^{\mathbf{\ast }}$ plane. That this is not seen may be because the antiferromagnetic axis is rotated away from the ${𝒄}^{\mathbf{\ast }}$ axis. Alternatively this may indicate further complexities of the magnetic structure not captured by this minimal model. Ultimately this can be best investigated by single crystal susceptibility measurements, not currently possible due to the very small available sample size.

An alternative structure to explain dominant antiferromagnetism parallel to the ${𝒄}^{\mathbf{\ast }}$ axis, and ferromagnetism parallel to the $𝒂$ axis is the $(A_z,A_z^{\prime },F_x,F_x^{\prime })$ magnetic structure. This included magnetic basis vectors in different vectors, so from Landau theory [57] would suggest a first order phase transition at $T_N$, which is not observed.

As the two ${\text{Pr}}^{4+}$sites are not symmetry equivalent, there is no symmetry constraint that the moments on the two sites are the same. A ferrimagnetic structure, where these values are different, would naturally have a ferromagnetic and antiferromagnetic component. These components would however be along parallel vectors, for example in the $F_z-\alpha F_z^{\prime }$ structure, with $\alpha \ne 1$. The antiferromagnetic staggered moment $1/2(1+\alpha )\widehat{z}$ and the ferromagnetic component $1/2(1-\alpha )\widehat{z}$ are parallel. This disagrees with the behaviour seen in torque, in which the jumps indicate a net moment along $\pm 𝒂$ but the overall sinusoidal behaviour indicates antiferromagnetism along ${𝒄}^{\mathbf{\ast }}$, and so the model can be discarded.

All calculations reported in this section were performed numerically using the qafm software I developed for this thesis using the principles outlined in Chapter 3.

A minimal model with only anisotropic bilinear exchange on the honeycomb bonds is able to stabilise the $𝒒=0$, $(F_z-F_z^{\prime })+(F_x+F_x^{\prime })$ magnetic structure discussed above.

There are two symmetry inequivalent honeycomb bonds in the $\alpha {\text{-Na}}_2{\text{PrO}}_3$structure, labelled as $𝐙$, and $𝐗$ in Figure 4.11. The bonds are labelled with an arrow to indicate nominal bond orientation. The third honeycomb bond in a single layer of $\alpha {\text{-Na}}_2{\text{PrO}}_3$, labelled $𝐘$, is symmetry equivalent to the $𝐗$ bond by the 2-fold rotation axis around $𝒚$ labelled $2_y$. Here, I am using a Cartesian frame such that $𝒙\parallel 𝒂$, $𝒚\parallel 𝒃$, $𝒛\parallel {𝒄}^{\mathbf{\ast }}$. The $𝐙$ bond is along the $2_y$ rotation axis, and so the exchange matrix is symmetry constrained to the form

(4.11)

where ${\mathrm{\Gamma }}^{\prime }$ parametrises symmetric off-diagonal exchange, and $D^{\prime }$ parametrises a Dzyaloshinskii–Moriya (DM) interaction.

The $𝐗$ bond is not constrained by symmetry at all, and the tensor is completely general

(4.12)

the $𝐘$ bond is symmetry equivalent by the 2-fold axis and has value

(4.13)

in the $𝒒=0$ structure, with two sublattices in each honeycomb layer (four total), the important parameter is the effective interaction between the sublattices, $𝒥=𝐗+𝐘+𝐙$ which evaluates as

	$𝒥$	$=𝐗+𝐘+𝐙$		(4.14)
				(4.15)
				(4.16)

with symmetry allowed off-diagonal terms only in ${𝒥}_{xz}$, including DM and symmetric off-diagonal exchange. It’s worth noting that if there were a 3-fold symmetry through the ${\text{Pr}}^{4+}$site, relating all three bonds, then these off-diagonal terms would be symmetry constrained to be zero. It can therefore be concluded that the asymmetry between these bonds will be important to understanding the microscopics of the physics in any future theoretical studies of this system.

A minimal model including only these nearest-neighbour honeycomb bonds, with $J_x=J_y$, $J_z>J_x>0$, and non-zero $D$, is able to stabilise the magnetic structure discussed in Section 4.6. This has dominant antiferromagnetism along the ${𝒄}^{\mathbf{\ast }}$ axis, caused by the Ising-like antiferromagnetic XXZ exchange terms $J_z>J_x>0$, and a small canting towards the $𝒂$ axis caused by the DM interaction $D$. The effect of the $\mathrm{\Gamma }$ term is to rigidly rotate canted spins in the $𝒂𝒄$ plane, but does not change the angle between them. Because of this, it is set to zero in calculations in this section.

Calculations for this model were performed with the parameters found in Table 4.4. The values of $J_x,J_z$ come from [21], the g-factor values are chosen such that the spin-flop transition discussed later matches the experimentally observed value, under the constraint that the powder-averaged g-factor matches the previously determined $g=0.98$ found in powder susceptibility measurements [21]. These values for $g$ match the order of susceptibilities seen at high temperatures $g_{xy}>g_z$. It is worth noting that the zero-field magnetic structure is insensitive to the values of $g$. The value of $D$ is fit to match the ratio $A_1/A_2$ in (4.6). These values are under-constrained in the absence of absolute units and should be taken as a rough estimate.

Figure 4.12 shows calculated angular dependence of magnetic torque for fields between $1\mathrm{T}$ and $4\mathrm{T}$. Alongside these calculated torque profiles are the experimentally observed torque profiles at the same magnetic fields, measured at $4.5\mathrm{K}$. These show very good qualitative agreement between the calculation and the data, in all orientations measured. All features seen in the data are reproduced, including the field scaling. Fig. 4.12b) shows the calculated torque for the honeycomb $𝒂𝒃$ plane. This shows a ${\tau }_z=MB\cos \theta \mathrm{sign}(\sin \theta )$ angular dependence, caused by the magnetic field selecting the domain with a moment along the $\pm 𝒂$ axis, so the jump occurs when the field projection along $𝒂$, $𝑩\cdot 𝒂$ changes sign. Fig. 4.12c) shows the calculated torque for the $𝒃{𝒄}^{\mathbf{\ast }}$ plane. This has a sinusoidal behaviour with an overall unstable arrangement with $𝑩\parallel {𝒄}^{\mathbf{\ast }}$, caused by the spins aligning along the ${𝒄}^{\mathbf{\ast }}$ axis, as in a uniaxial antiferromagnet. Fig. 4.12a) shows the angular dependence in the $𝒂𝒄$ plane, revealing a combination of both effects. These features are seen in both calculations and data for all three orthogonal rotation planes.

One interesting feature of this model is that a spin-flop transition is expected for field parallel to the ${𝒄}^{\mathbf{\ast }}$ axis, much like in a conventional XXZ model (see Chapter 3). This can be seen in Figure 4.13, which shows field dependence of magnetisation for fields along the $𝒂,𝒃,{𝒄}^{\mathbf{\ast }}$ axes. With field parallel to the $𝑩\parallel {𝒄}^{\mathbf{\ast }}$ axis, for the model parameters in Table 4.4 a spin-flop transition can be seen at $B_{\text{SF}}\approx 23\mathrm{T}$.

Figure 4.14 shows the field phase diagram for this model. The left panel shows the phase diagram for field orientations in the $𝒂,{𝒄}^{\mathbf{\ast }}$ plane. The right panel shows the phase diagram for field orientations in the $𝒃,𝒄$ plane. The colour shows the trace of the differential susceptibility tensor

$${\chi }_{ij}=-\frac{{\partial }^2ℱ}{\partial B_i\partial B_j}$$

(4.17)

with free energy $ℱ$, and magnetic field vector $𝑩$. In Landau theory this is expected to diverge at first-order phase transitions, and have a discontinuity at second-order phase transitions [57]. The extracted phase diagram is shown in panel Fig. 4.14 b), where solid lines indicate sharp phase transitions, and dashed lined indicate smooth crossovers.

For field orientated in the $𝒃,{𝒄}^{\mathbf{\ast }}$ plane, close to the $𝒃$ axis, the only phase transition is a continuous phase transition to the paramagnetic field-polarised phase at $\approx 58\mathrm{T}$, this phase boundary is connected to the spin-flop transition at field close to ${𝒄}^{\mathbf{\ast }}$. The sharp angle this phase boundary makes with the $𝑩\parallel {𝒄}^{\mathbf{\ast }}$ axis means that for field sweeps in the $𝒃,{𝒄}^{\mathbf{\ast }}$ plane, the phase transition field will vary quickly with angle. There is no further transition to polarisation beyond the spin-flop transition field. This is because the spin-flopped phase only has one domain, the presence of DM interactions selects just one. There is therefore no symmetry breaking between the spin-flop and polarised phases. Instead there is a continuous crossover, as the magnetisation of the material approaches its saturation value asymptomatically. This is in contrast to a spin-flop phase without DM, which shows a first-order spin-flop transition at a critical point, with a further continuous transition to polarisation at a higher field.

For field orientated in the $𝒂,{𝒄}^{\mathbf{\ast }}$ plane, no phase transition to a polarised phase is seen in any orientation, except for $𝑩\parallel {𝒄}^{\mathbf{\ast }}$ exactly which shows a spin-flop transition. This is because a finite field component along $𝒂$ selects one of the two time-reversed domains and so breaks the time-reversal symmetry in the material. There is no phase boundary between the canted antiferromagnet phase with some field component along $𝒂$, and the polarised phase. This case is similar to a ferromagnet, where a finite magnetic field breaks the time-reversal symmetry, so no phase transition is seen between the ordered and paramagnetic phase in finite field, just a smooth crossover.

This spin-flop transition also has an effect on the powder-averaged field dependence of magnetisation, shown in Figure 4.15 a) alongside its field gradient in Fig. 4.15 b). The average was estimated by calculating field-dependent magnetisation curves for 256 field orientations evenly distributed over a sphere [33]. This shows an increase in the differential susceptibility, peaking at $24\mathrm{T}$, approximately at the spin-flop transition field. The magnetisation quickly flattens off with field at approximately $60\mathrm{T}$ upon approaching polarisation. This is not a sharp transition as in the presence of anisotropic g-factors the polarisation field is direction dependent. This shows that powder magnetisation in high magnetic fields is a useful tool for exploring this model.

An alternative minimal model which stabilises the same magnetic structure is the $J,\mathrm{\Gamma }>0$ model on Kitaev axes as has been explored in depth for the case of the honeycomb iridates [81], with an additional unbalanced DM interaction along the $𝒃$ axis. At the mean field level, this is indistinguishable from the XXZ case, as can be seen by computing the same effective interaction between sublattices. The orthogonal transfer matrix between Kitaev axes and the $x,y,z$ axes defined above can be written

(4.18)

The $J,K,\mathrm{\Gamma }$ Hamiltonian can then be written in these coordinates

(4.19)

where the sum is over the nearest-neighbour bonds. Each bond has an associated Ising axis $\gamma \in \{x,y,z\}$ and $\alpha \beta (\gamma )$ refer to the orthogonal plane to this axis. Such that the exchange tensor for the Kitaev $z$ axis ${𝒥}^z$ can be written as

(4.20)

The effective interaction between sublattices can then be written in the same coordinates as the XXZ Hamiltonian

	${𝒥}^{\prime }$	$=P^T\left[{𝒥}^x+{𝒥}^y+{𝒥}^z\right]P$		(4.21)
				(4.22)

which with $\mathrm{\Gamma }>0$ is equivalent to an Ising-type XXZ exchange model. In the mean-field limit, these models cannot be distinguished, even in an applied magnetic field, so long as $𝒒=0$ holds, which is true for a large region of the $J,K,\mathrm{\Gamma }$ parameter space [81]. Beyond this, this model is insensitive to the presence of Kitaev interactions $K$. Searching for the effects of these terms will require study beyond the mean-field level, such as comparing spin-wave calculations to observed inelastic neutron scattering data.

Attribution note: Magnetometry measurements reported in this section were performed in collaboration with Matthew Pearce, and Ryutaro Okuma from the University of Oxford, and instrument scientist Yurii Skourski at the HZDR Dresden high-magnetic-field facility.

From the calculations above, it is clear that the situation in $\alpha {\text{-Na}}_2{\text{PrO}}_3$could be clarified with magnetisation measurements in large magnetic fields. The largest available single crystals of $\alpha {\text{-Na}}_2{\text{PrO}}_3$have a longest edge of $\approx 120\mathrm{\mu }\mathrm{m}$, and a thickness of $\approx 10\mathrm{\mu }\mathrm{m}$, too small for single crystal magnetisation. Co-aligning many crystals was considered, but to achieve a suitable filling factor of the extraction coil magnetometer available, several tens of thousands of crystals would need to be simultaneously aligned in a glovebox. Instead, magnetisation was performed on a powder sample of $\alpha {\text{-Na}}_2{\text{PrO}}_3$, grown by Rytaro Okuma at the University of Oxford, which could achieve an ideal filling factor of 1.

Figure 4.16 shows the field dependence of magnetisation and its derivative with field for a powder sample of $\alpha {\text{-Na}}_2{\text{PrO}}_3$with temperatures below $6\mathrm{K}$ and fields up to $60\mathrm{T}$. The differentiation is performed with a second-order Savitzky-Golay [89] filter over a window size of $4\mathrm{T}$. That is, in a moving window over the data of width $4\mathrm{T}$, a quadratic polynomial is fit to the data and the derivative is taken from the slope of the fitted curve at the centre of the window. The data are in arbitrary units. Broadly, this shows at $6\mathrm{K}$ a linear dependence of magnetisation with field up to $\approx 45\mathrm{T}$, where the magnetisation flattens off, perhaps associated with magnetic saturation. Below the transition, at the $1.5\mathrm{K}$ base temperature, an overall upwards curve is observed in the magnetisation. No flattening of the curve is seen at high field, suggesting that the polarisation field is above the $60\mathrm{T}$ maximum available field. No clear anomaly is detected in the differential susceptibility in the region of $25\mathrm{T}$, associated with the spin-flop transition, seen in single crystal data (next section), and predicted by calculations of the minimal model (Fig. 4.15). The absence of a clear anomaly near the transition field value in the powder data may be due to a number of factors, such as a preferred orientation of grains with ${𝒄}^{\mathbf{\ast }}$ perpendicular to the length of the capsule, a lack of transition caused by increased stacking faults and overall lower sample quality in powder samples of $\alpha {\text{-Na}}_2{\text{PrO}}_3$[21, 36], or a more complex Hamiltonian that makes the anomaly less prominent.

Attribution note: Torque magnetometry measurements reported in this section were performed in collaboration with Matthew Pearce and Ryutaro Okuma at the University of Oxford and Alix McCollam, Roos Leenen, and Uli Zeitler at HFML Nijmegen.

In order to search for the predicted spin-flop transition in $\alpha {\text{-Na}}_2{\text{PrO}}_3$, single-crystal piezocantilever torque magnetometry measurements were performed in DC fields up to $35\mathrm{T}$. Figure 4.17 shows field sweeps close to the ${𝒄}^{\mathbf{\ast }}$ axis, in the $𝒂𝒄$ plane for temperatures between $3\mathrm{K}$ and $4.2\mathrm{K}$. Below $10\mathrm{T}$, these curves lie on top of each other, and the torque is approximately linear in field. This linear relationship matches the behaviour seen at low field previously. At higher fields, a sharp phase transition is seen at all measured temperatures. At lower temperature, the feature becomes sharper and increases in magnetic field. The precise location of the transition field ${\mu }_0{H}_{\text{SF}}$ is extracted by taking the derivative of the signal with field $\frac{\partial \tau }{\partial H}$ Fig. 4.17b). The transition is taken as the centre of the peak in these traces, marked with arrow symbols. At the lowest temperature measured, an additional smaller peak in seen at higher field, just above $20\mathrm{T}$, the origin of which is unclear. At low temperatures, a significant hysteresis opens up in the magnetic torque, shown in Fig. 4.17c) with the field dependence of torque for two consecutive field sweeps. Below the transition field, the two upsweeps disagree significantly, but above the transition field the curves lie on on top of each other. Between the up and down sweep directions, hysteresis opens just above the transition field at approximately $20\mathrm{T}$, and closes in zero field. Both down sweeps overlap completely. These features are summarised in a phase diagram in Fig. 4.17d).

A possible interpretation for this complex behaviour is as follows. The first and largest peak in $\frac{\partial \tau }{\partial H}$ is the phase transition field, where most of the sample volume transitions to the spin-flop phase. A portion of the sample is trapped in the metastable low-field AFM phase, and gradually collapses into the magnetic spin-flop ground state. The second peak in $\frac{\partial \tau }{\partial H}$, observed at $3\mathrm{K}$, may be due to the AFM phase becoming unstable, explaining why it is coincident with the hysteresis opening between up and down sweeps. On the down sweep, the sample remains in the metastable spin-flop phase down to lower field, where is gradually collapses into both domains of the stable AFM phase. On the second upsweep at $3.7\mathrm{K}$, both time-reversed domains are populated approximately equally, so the linear contributions to the field dependence of torque cancel out. Above the spin-flop transition there is only one magnetic domain, so there is no hysteresis.

Temperature sweeps in fixed fields were performed to explore the regions of the phase diagram where the phase boundary may be approximately vertical (see Fig. 4.17d)). In order to do temperature sweeps in high magnetic fields, the field-dependence of the sample thermometer has to be corrected. This was done by filling the cryostat sample volume with liquid helium at ambient pressure to fix the temperature at $4.2\mathrm{K}$, the offset from this is measured, and shown in Figure 4.18. This curve is used to correct the measured temperature. While there will be some temperature dependence of this correction factor, the value measured at $4.2\mathrm{K}$ is expected to be valid in the region of the phase transition at $4.6\mathrm{K}$.

Figure 4.19 shows the measured temperature dependence of magnetic torque measured at $5\mathrm{T}$ and $30\mathrm{T}$ below $8\mathrm{K}$. The right panels show the derivative of this with temperature. Panels a, b) show the torque and derive measured in the $𝒂𝒄$ plane, close to ${𝒄}^{\mathbf{\ast }}$, within $1\mathrm{°}$ of the jump in torque observed in this plane. At $5\mathrm{T}$, this b) shows a very sharp peak at $4.56(1)\mathrm{K}$, associated with the onset of magnetic order, this is shown with a yellow point in Fig. 4.17d). At $30\mathrm{T}$, a broad peak is seen in torque centred at approximately $4.5(1)\mathrm{K}$. The peak observed at high field is the opposite sign than the one seen at low field, indicating that the stability of the system for field ${\mu }_0𝑯\parallel {𝒄}^{\mathbf{\ast }}$ has changed. At low field it was observed that the system was unstable for ${\mu }_0𝑯\parallel {𝒄}^{\mathbf{\ast }}$, so it is an energy maxima. At high field, it appears to be a stable equilibrium. The reason for the broad peak at $4.5(1)\mathrm{K}$ may be a smooth crossover from the paramagnetic to the spin-flop phase. Panels c, d) show the equivalent measurements made in the $bc$ plane. This shows a broad peak in differential torque at all fields, however the location of the peak in temperature is strongly field dependent, increasing up to $5.4(1)\mathrm{K}$ at $15\mathrm{T}$. These are plotted as pink points in Fig. 4.17d)

It is worth considering that while the two ${\text{Pr}}^{4+}$sites are symmetry inequivalent, so the spin-flop and paramagnetic phases have the same symmetry. In a more conventional spin-flop phase (example see Chapter 3), there are two spin-flop domains from swapping the spins of the two sites. In this system, there is only one domain because of the inequivalence, the finite value of DM interaction creates an energy gap between these two domains. So as is seen above in calculations, no sharp phase boundary is expected between the spin-flop and paramagnetic phases. The existence of a broad feature in torque may indicating a smooth crossover between these regions of phase space.

This section primarily reports torque magnetometry experiments on high quality single crystals of $\alpha {\text{-Na}}_2{\text{PrO}}_3$up to $35\mathrm{T}$. Based on detailed measurements of torque as a function of field orientation in three orthogonal crystallographic planes, a model magnetic structure was proposed with dominant antiferromagnetism along the ${𝒄}^{\mathbf{\ast }}$ axis and a small ferromagnetic moment along the $\pm 𝒂$ axis. From this, I proposed a minimal anisotropic exchange model compatible with the crystal structure symmetry. Mean-field calculations on this magnetic model with Ising-type XXZ exchange interactions and symmetry allowed Dzyaloshinskii–Moriya interaction on the honeycomb bonds qualitatively reproduce features seen in torque at low temperature up to $4\mathrm{T}$ for all three rotation planes measured, including the field scaling. I show that the magnetic structure can also be explained by a bond anisotropic $J\mathrm{\Gamma }$ model with an unbalanced DMI, and that these two models cannot be distinguished at the mean-field level. The experimentally determined order of susceptibilities at high temperatures suggests anisotropic g-factors , in disagreement with previously proposed values from fits to powder susceptibility data. I propose a stronger spin-orbit coupling term $\lambda$ to reproduce the observed powder average $g$ and trigonal distortion to produce this g-factor anisotropy. High-field toque magnetometry measurements reveal a spin-flop transition for field along the $𝑯\parallel {𝒄}^{\mathbf{\ast }}$ axis at ${\mu }_0{H}_{\text{SF}}\approx 17\mathrm{T}$ at $3\mathrm{K}$. These measurements also reveal significant hysteresis and metastability below this transition. Temperature sweeps in the same orientation reveal a broad peak in torque which may be associated with the expected crossover between paramagnetic and spin-flop regions of the phase diagram. This crossover region has a peculiar field-dependence, with a non-monotonic dependence of the temperature vs field, and maximum temperature of $T\approx 5.4\mathrm{K}$ at $15\mathrm{T}$. This is qualitatively similar to the predicted phase boundary in some highly-frustrated systems [90]. The origin of this non-monotonic behaviour is unclear and deserves future study.

In order to better understand the physics of this material, high-field calorimetry measurements would be well placed to trace out the boundary between the paramagnetic and spin-flop regions. Additionally, the exact shape of a heat capacity anomaly could distinguish between smooth crossover and a sharp phase transition between these regions. The same process applied to the canted AFM, spin-flop phase boundary could clarify its shape and low-temperature maximum field, which would help to further constrain theoretical models of this material. Additional single-crystal torque magnetometry measurements at lower temperatures and higher pulsed magnetic fields could reveal if there is an additional phase transition to a polarised phase at much higher fields. If much larger single crystals can be synthesised, single crystal magnetometry measurements could reveal how the magnetic structure is changing between the antiferromagnet and spin-flop phases.

Previous theoretical work on this material predicted dominant antiferromagnetic Kitaev interactions [42]. Mean-field calculations of magnetic structures show that a large region of the Heisenberg-Kitaev-$\mathrm{\Gamma }$ phase diagram will order as a $𝒒=0$ antiferromagnetic order with spins-along the ${𝒄}^{\mathbf{\ast }}$ axis [81] as is observed experimentally. Inelastic neutron scattering studies on powder samples of $\alpha {\text{-Na}}_2{\text{PrO}}_3$did not find evidence for significant Kitaev interactions, and could fit the data instead with a minimal model with XXZ exchange interactions [21]. I have proposed that an extension of this model to include also $g$-factor anisotropy and DM interactions predicts a canted AFM structure with moments predominantly along ${𝒄}^{\mathbf{\ast }}$ canted towards $\pm 𝒂$. I show that this model can explain well the observed angular dependence of the torque in three orthogonal crystallographic planes up to $4\mathrm{T}$.

This canted antiferromagnetic structure is different from what is observed in other similar materials such as $\alpha$-RuCl₃, $\alpha$-${\text{Li}}_2{\text{IrO}}_3$, or ${\text{Na}}_2{\text{IrO}}_3$and is likely intimately related to the physics of $4f$ ${\text{Pr}}^{4+}$orbitals. In either case, these experimental results should provide further constraints on theoretical models to understand the nature of exchange interactions and cooperative magnetism of $j_{\text{eff}}=1/2$ ${\text{Pr}}^{4+}$$4f^1$ ions and in Kitaev candidate honeycomb materials more broadly.