Chapter 5 Magnetic order and high-field magnetic phase transitions in hyperhoneycomb $\beta {\text{-Na}}_2{\text{PrO}}_3$

A generalisation of the honeycomb lattice, discussed in Chapter 4, into three dimensions gives the hyperhoneycomb and harmonic honeycomb family of lattices [68], some of which have been realised as polymorphs of ${\text{Li}}_2{\text{IrO}}_3$. These 3D structures show the same three-fold coordination around each site as the planar honeycomb case, and if they host the same Kitaev exchange interactions on these bonds they can be exactly solved through the same methods [65, 53]. Much like in the honeycomb case discussed previously, more complex models with additional Heisenberg $J$ [60, 53] and symmetric-off-diagonal exchange $\mathrm{\Gamma }$ [59] with only nearest-neighbour interactions were also considered to try to explore the possible physics of hyperhoneycomb and harmonic honeycomb iridates. These revealed a rich phase diagram of exotic magnetic phases depending on the precise balance between the three terms in the model.

A promising candidate material for unconventional magnetism is the hyperhoneycomb $\beta {\text{-Na}}_2{\text{PrO}}_3$, which was first synthesised in 1988 [105]. Its promise as a three-fold coordinated $4f$ magnet was first identified in literature only recently [42, 43], where large, antiferromagnetic Kitaev interactions were theoretically predicted.

In this section I report AC-calorimetry, torque magnetometry, and X-ray structural studies of hyperhoneycomb $\beta {\text{-Na}}_2{\text{PrO}}_3$, revealing non-collinear magnetic ordering at $T_N=5.20(2)\mathrm{K}$ to a phase stabilised by bond-anisotropic exchange of the symmetric off-diagonal type. I also report torque magnetometry up to $67\mathrm{T}$, revealing a metamagnetic phase transition at $26\mathrm{T}$ for fields applied close to the crystal $𝒂$ axis.

Samples of $\beta {\text{-Na}}_2{\text{PrO}}_3$were synthesised by Ryutaro Okuma at the University of Oxford. Like its polymorph $\alpha {\text{-Na}}_2{\text{PrO}}_3$, $\beta {\text{-Na}}_2{\text{PrO}}_3$is very air sensitive so samples 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 $\beta {\text{-Na}}_2{\text{PrO}}_3$, X-ray diffraction was performed at room temperature on $\sim 100\mathrm{\mu }\mathrm{m}$ single-crystals with an Oxford Diffraction Supernova diffractometer. These measurements were performed with the sample remaining in the evacuated capillary. The detailed structure was refined using FULLPROF [83] analysis software. The results of this refinement are summarised in Table 5.1 and Table 5.2, which show structural parameters and atomic fractional coordinates of $\beta {\text{-Na}}_2{\text{PrO}}_3$’s ions in the orthorhombic setting.

Figure 5.1 shows a comparison between the experimentally observed diffraction pattern and calculated Bragg peak intensities for the reciprocal lattice planes $(h1l)$, $(hk0)$. The intensities are calculated from the refined atom positions in Table 5.1. The $Fddd$ space group imposes a number selection rules for scattering. The $F$ centring has the scattering rule $h,k,l$ all odd or all even. The systematic absences due to this selection rule can be seen in the data in the $(h1l)$ plane as no peaks are seen at $h$ even or $l$ even. In the $(hk0)$ plane, there is an additional selection rule $h+k=4n(n\in \mathbb{Z})$ which is seen in the data. This arises from the diamond glide planes with mirror perpendicular to the $𝒄$ axis, and translation by $\frac{1}{4}\left(𝒂+𝒃\right)$. Finally, for the family of peaks with $h,k,l$ all odd, the peaks at $l=6n+3$ are systematically weak. This is caused by the fact that the ${\text{Pr}}^{4+}$and ${\text{Na}}^{+}$ sites exist in pairs, approximately translated by $\frac{𝒄}{6}$ which causes destructive interference on $l=6n+3$ peaks, clearly seen in Fig. 5.1. Figure 5.2 shows a comparison between observed and calculated peak intensities, the small disagreement at large ${|F|}^2$ is likely caused by uncorrected absorption from the glass capillary the sample was held in to prevent air exposure.

This analysis revealed that $\beta {\text{-Na}}_2{\text{PrO}}_3$has a complex three-dimensional crystal structure, isostructural to hyperhoneycomb $\beta$-Li₂IrO₃ [98], with an $Fddd$ space group. This hyperhoneycomb structure contains edge-sharing PrO₆ octahedra containing the ${\text{Pr}}^{4+}$ion at the centre, arranged to appear locally like a honeycomb structure, but with the bond directions alternating along the $𝒄$-axis. The structure, visualised in Figure 5.3, contains zigzag chains of ${\text{Pr}}^{4+}$ions along the $𝒂+𝒃$ and $𝒂-𝒃$ directions, connected by bonds parallel to the $𝒄$ axis. These chains can be seen in Figure 5.3c). When projected into the $𝒂𝒃$ plane, the chains form a tiling pattern of diamond shapes, with the long axis parallel to $𝒃$ and the short axis parallel to $𝒂$. This shape is reflected in the crystal morphology, visible in Figure 5.3d), as the crystals grow with natural edges parallel to these chains into diamond shapes with the long axis parallel to $𝒃$ and the short axis parallel to $𝒂$. This $Fddd$ space group is in disagreement with the previously reported lower symmetry $C2/c$ structure [105]. This is due to a mischaracterisation of the material as lower symmetry than the space group observed here.

The primitive unit cell of $\beta {\text{-Na}}_2{\text{PrO}}_3$is triclinic, and contains 4 ${\text{Pr}}^{4+}$ions, 8 ${\text{Na}}^{+}$ ions, and 12 ${\text{O}}^{2-}$ ions. The primitive cell lattice vectors $𝑨,𝑩,𝑪$ can be written in terms of the centred cell lattice vectors $𝒂,𝒃,𝒄$ as

(5.1)

The magnetic ions in $\beta {\text{-Na}}_2{\text{PrO}}_3$are Kramers $4f^1$ ${\text{Pr}}^{4+}$ions in an octahedral environment of oxygens, just as in $\alpha {\text{-Na}}_2{\text{PrO}}_3$, so up to different values for the strength of the CEF parameters the single-ion physics will be the same as that discussed in Chapter 4. This results in the same energy level spitting diagram shown in Figure 5.4, and a $j_{\text{eff}}=1/2$ ground state.

In order to determine if the material orders magnetically at low temperatures, specific heat capacity was measured using the AC method described in Section 2.3. Measurements were first performed on a $0.12(2)\mathrm{mg}$ collection of single crystals in zero-field, revealing a clear peak in heat capacity at $5.2\mathrm{K}$, shown in Fig 5.5b). Measurements were then performed on a single crystal of $\beta {\text{-Na}}_2{\text{PrO}}_3$to determine how the transition responded to applied magnetic field, shown Fig. 5.5a). The single-crystal sample was too small to accurately measure the mass on a weighing balance, but by comparing the size of the heat capacity anomaly to that of the multi-crystal sample, the mass could be estimated to $m\approx 0.07\mathrm{mg}$. The sample was mounted with $𝑯\parallel 𝒄$, for $0\mathrm{T}\le {\mu }_{0}H\le 16\mathrm{T}$.

Figure 5.5 shows the evolution of heat capacity with temperature between $2\mathrm{K}$ and $6\mathrm{K}$. The trace in red shows the evolution of heat capacity $C_p$ with temperature at zero field, revealing a finite peak, with a sharp anomaly at the ordering temperature $T_N=5.20(2)\mathrm{K}$. This is a signature of a second-order phase transition [25], 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.

The evolution of specific heat capacity with temperature at the maximum experimentally available field of ${\mu }_{0}H=16\mathrm{T}$ can be seen in the blue curve of Fig. 5.5a). This shows a sharp peak with an anomaly at $5.01(4)\mathrm{K}$, suppressed from the zero-field value. The fact that the transition is sharp in a large magnetic field indicates that there is no spontaneous ferromagnetic component to the zero-field magnetic structure, if there were the finite field would break the symmetry and the transition would be a smooth crossover between low and high temperature regimes. The field suppression of the critical ordering temperature is suggestive of antiferromagnetic ordering and is intuitively understood as the fact that the field induces a uniform magnetisation, so less of the moment orders spontaneously in antiferromagnetic structure. Therefore the mean-field energy from ordering is reduced and the critical temperature $T_N$ is reduced.

In order to determine the temperature dependence of magnetic entropy, the heat capacity of the collection of single crystals was fit to a phenomenological form [106]

$$C_{\text{mag}}(T)=A{T}^n\exp \left(-\frac{\mathrm{\Delta }}{k_{B}T}\right)+C_{\text{bg}}$$

(5.2)

with $n=-1$ and the background term $C_{\text{bg}}$ was subtracted off. The fitted magnetic gap $\mathrm{\Delta }\sim 1\mathrm{meV}$ is very sensitive to the value of $n$ used and the temperature fitting window. This is likely an artefact of the small temperature window available. The magnetic entropy was then computed as

$$S_{\text{mag}}(T)={\int }_0^T\frac{C_{\text{mag}}(T^{\prime })}{T^{\prime }}𝑑T^{\prime }$$

(5.3)

where the fitted $C_{\text{mag}}(T^{\prime })$ is extrapolated for temperatures below the measurement base temperature, and the measured data itself is used above the base temperature. This computed $S_{\text{mag}}(T)$ is shown in Fig. 5.5c). The high-temperature magnetic entropy produced is $S\approx 0.5k_B\ln 2$, less than the expected $k_B\ln 2$ value for a $j_{\text{eff}}=1/2$ paramagnet. This value is also less than the $S\approx 0.76k_B\ln 2$ value found in $\alpha {\text{-Na}}_2{\text{PrO}}_3$[21]. That this may be evidence of strong fluctuations in the ordered state [98] is an interesting prospect, however given the unavailability of an accurate lattice background to subtract, significant uncertainties in the sample mass and low-temperature extrapolation, and the challenge of determining absolute values of heat capacity in the AC calorimetry technique, the uncertainty on the absolute value of this is large, and it should be taken as a rough estimate.

In order to explore the magnetic anisotropy in $\beta {\text{-Na}}_2{\text{PrO}}_3$, I performed piezocantilever torque magnetometry measurements on $\sim 200\mathrm{\mu }\mathrm{m}$ single crystals. Torque measurements reported in this section were performed on the crystal pictured in Fig. 5.3d). Figure 5.6 shows measured magnetic torque for rotations in the $𝒂𝒄$ and $𝒃𝒄$ planes for temperatures between $2\mathrm{K}$ and $100\mathrm{K}$. Figure 5.6a) shows a waterfall plot of torque against rotation angle in the $𝒂𝒄$ plane, where an angle of $0\mathrm{°}$ implies field parallel to the crystallographic $𝒄$ axis. At all temperatures measured, the torque is fit well by a $\tau =-{\tau }_2(T)\sin 2\theta$ form, with ${\tau }_2>0$. When the field is approximately parallel to the $𝒄$ axis ${\mu }_0𝑯\parallel 𝒄$, the torque crosses zero with a negative gradient of torque with angle is observed, indicating a stable equilibrium in both the paramagnetic and ordered phases. Conversely, for ${\mu }_0𝑯\parallel 𝒂$ torque crosses zero with a positive gradient, suggesting that this is an unstable equilibrium. This implies that the susceptibility along the $𝒄$ direction is larger than the susceptibility along the $𝒂$ direction at all temperatures. Figure 5.6b) shows torque against rotation angle in the $𝒃𝒄$ plane, where an angle of $0\mathrm{°}$ implies field parallel to the crystallographic $𝒄$ axis. Here the situation is very similar, at all temperatures the torque is fit well by a $\sin 2\theta$ form and the sign of the signal indicates ${\mu }_0𝑯\parallel 𝒄$ is a stable equilibrium, while ${\mu }_0𝑯\parallel 𝒃$ is an unstable equilibrium.

At each temperature, torque versus angle curves were fitted to the sinusoidal form

$$\tau (\theta ,T)=-{\tau }_2(T)\sin 2\theta$$

(5.4)

where ${\tau }_2(T)>0$ indicates a stable equilibrium when ${\mu }_0𝑯\parallel 𝒄$.

Figure 5.6c) shows how these coefficients ${\tau }_2$ vary with temperature for both orientations. The trace in blue shows the values extracted for the $𝒂𝒄$ plane, while the trace in red shows values for the $𝒃𝒄$ plane. These torque measurements in both planes were performed on the same crystal and the same lever so the units are comparable. In both planes the evolution of this amplitude ${\tau }_2$ with temperature are qualitatively similar. At high temperatures, above $\sim 20\mathrm{K}$ the torque decreases monotonically, as is expected for a Curie-Weiss paramagnet. Here the magnitude of the torque signal is about twice as large in the $𝒂𝒄$ plane as in the $𝒃𝒄$ plane, with the sign of the signal indicating that ${\chi }_c$ is the largest susceptibility. This suggests that the difference in susceptibilities between the $𝒄$ axis and $𝒂$ axis is about twice as large as the difference between the $𝒄$ axis and $𝒃$ axis. From this, the order of susceptibilities can be inferred as

$${\chi }_c>{\chi }_b>{\chi }_a$$

(5.5)

In both orientations there is a broad peak in torque centred at approximately $10\mathrm{K}$. Low dimensional systems of weakly coupled chains or planes have been observed to display a broad peak in susceptibility associated with short range magnetic order at a temperature $\tilde{T}$ dependent on the dominant interactions, and an anomaly at a lower temperature with the onset of long-range order $T_N\ll \tilde{T}$ [22, 70, 31]. Whether this could indicate short-range order along the zigzag chains in $\beta {\text{-Na}}_2{\text{PrO}}_3$is an interesting prospect which should be investigated further.

At the Néel temperature, determined from heat capacity as described in Section 5.4, $T\simeq 5.2\mathrm{K}$, there is a pronounced anomaly in torque. In both orientations, the fitted torque amplitude grows quickly as temperature drops deeper into the ordered phase. This is consistent with a suppression of susceptibility in the $𝒂$ and $𝒃$ directions. The size of the torque anomaly in the $𝒂𝒄$ plane (4.8 a.u.), is much larger than the size of the anomaly in the $𝒃𝒄$ plane (1.6 a.u.), suggesting that the susceptibility is suppressed much more strongly in the $𝒂$ direction than the $𝒃$ direction. This is consistent with an antiferromagnetic structure with spins predominantly along the $𝒂$ axis, but with a finite component also along the $𝒃$ axis.

Figure 5.6d) shows how the value of $\frac{{\tau }_2}{{({\mu }_{0}H)}^2}$ varies with temperature for applied fields between $2\mathrm{T}$ and $16\mathrm{T}$. In the paramagnetic regime, magnetic torque is expected to scale with the field squared $\tau \sim H^2$ [107, 113] (See Chapter 2.2), and indeed the curves $\frac{\tau }{H^2}$ lie on top of each other. This can be seen for all fields measured with temperature $T>10\mathrm{K}$. For temperatures below the transition this relationship holds for small fields up to $6\mathrm{T}$. At larger applied fields, the value of $\frac{{\tau }_2}{{({\mu }_{0}H)}^2}$ drops, indicating that the torque is growing slower than $H^2$. This is further evidence of magnetic order, as the paragenetic torque-field relationship no longer holds.

The most straightforward interpretation of this torque data is a collinear antiferromagnetic order with moments in the $𝒂𝒃$ plane, in-between the $𝒂$ and $𝒃$ axes. To test this torque was measured in the $𝒂𝒃$ plane of the same crystal, so the sample had to be mounted with the lever against the side of the diamond shape visible in the crystal morphology. The absence of a flat face for mounting makes the alignment of this challenging. For this mounting, the lever was aligned parallel to the $𝒂+𝒃$ direction, along the direction of a zigzag chain. The $mmm$ point group of the high-temperature paramagnetic phase means that the torque is required by symmetry to be zero for ${\mu }_0𝑯\parallel 𝒂$ or ${\mu }_0𝑯\parallel 𝒃$, as such this can be used to correct any misalignment caused by the mounting difficulties. The $\varphi =0$ position is taken as the angle with $\tau =0$ at high temperature closest to the estimated $𝑯\parallel 𝒃$ position from crystal geometry. In the measurement reported, the corrected alignment was $8\mathrm{°}$ off the estimated alignment.

Figure 5.7a) shows torque versus angle in the $𝒂𝒃$ plane for temperatures between $2\mathrm{K}$ and $100\mathrm{K}$. The angle in plane $\varphi$ is measured from the nominal $𝒃$ axis. Much like in the other orientations, these show an $\tau =-{\tau }_2\sin 2\varphi$ shape. In the $𝒂𝒃$ plane, the stable equilibrium is near ${\mu }_0𝑯\parallel 𝒃$ and the unstable equilibrium is near ${\mu }_0𝑯\parallel 𝒂$, consistent with the order of susceptibilities shown in Eqn. (5.5).

Fig. 5.7b) shows the evolution of the fitted torque amplitudes with temperature, showing the same overall shape as seen in both other orientations.

If the torque can be explained simply by anisotropic susceptibilities with principal axes aligned with the crystallographic $𝒂$, $𝒃$, $𝒄$ axes then for small magnetic fields

	${\tau }_{2,ab}$	$=-{\displaystyle \frac{1}{2}}{B}^2\left({\chi }_b-{\chi }_a\right)$		(5.6)
		$={\tau }_{2,ac}-{\tau }_{2,bc}$		(5.7)

Overlaid on the data points in red is the curve of ${\tau }_{ac}(T)-{\tau }_{bc}(T)$, which agrees qualitatively and quantitatively very well with the measured data points of ${\tau }_{ab}(T)$. It is worth noting that these data are not scaled to match, but the data was measured on the same sample and lever for all three datasets. The subtracted data is over-plotted with no scaling parameters. This quantitative agreement provides further confirmation of the internal consistency of the dataset. The torque data provides additional confirmation for the non-collinear picture as follows; torque in the $𝒂𝒄$ and $𝒃𝒄$ planes is consistent with suppression of susceptibility along both $𝒂$ and $𝒃$ directions. If the structure was collinear with moments in the $𝒂𝒃$ plane, away from crystallographic axes, one would expect an unstable equilibrium when the field is along the moment axis and susceptibility is close to zero, and a stable equilibrium when the field is normal to this axis. The principal axes of the susceptibility tensor would not be aligned with the crystallographic axes and (5.7) would not hold. However the data in Fig. 5.7 does not show this. As such, the collinear model can be discarded, and the suppression of susceptibilities along both the $𝒂$ and $𝒃$ directions must be because of a more complex non-collinear magnetic order with the dominant (or full) moment components in the $𝒂𝒃$ plane.

Together with heat capacity, the torque data reported above provides clear evidence supporting a non-collinear magnetic structure with an ordering temperature of $5.2\mathrm{K}$, motivating further research into the magnetic structure.

The magnetic ${\text{Pr}}^{4+}$sites are at the $16g$ site in the orthorhombic cell, there are four equivalent ${\text{Pr}}^{4+}$sites in the primitive setting. These are shown in Figure 5.8a), with the four sublattices numbered using the convention developed for $\beta$-Li₂IrO₃ [10]. The magnetic basis vectors can be written in terms of the relative orientation of the moments on the ${\text{Pr}}^{4+}$sites as

(5.8)

This is summarised in Table 5.3, where +, - symbols indicate a $\pm 1$ entry in the matrix above. These magnetic basis vectors are illustrated in Figure 5.8.

For neutron diffraction, the relevant structure factor for a magnetic reflection at Bragg peak position $𝑸$ is

$$𝓕(𝑸)=f_F\sum _{n=1}^4{𝝁}_n{e}^{i𝑸\cdot {𝒓}_n}$$

(5.9)

where the sum runs over the sites in the primitive unit cell, and ${𝒓}_n$ is the position of the $n$^th  site [10]. The prefactor $f_F=1+e^{i\pi (h+k)}+e^{i\pi (k+k)}+e^{i\pi (k+h)}$ is due to the $F$-centring of the orthorhombic setting and takes the value

(5.10)

Each magnetic basis vector has a fixed phase relationship between each site, constraining the structure factors $𝓕$, and imposing a symmetry constrained extinction ($𝓕=\mathrm{𝟎}$) on some Bragg peak positions.

Writing the Bragg peak position as

$$𝑸=h{𝒂}^{\ast }+k{𝒃}^{\ast }+l{𝒄}^{\ast }$$

(5.11)

and the position of the ${\text{Pr}}^{4+}$ions as

	${𝒑}_1$	$=({\displaystyle \frac{1}{8}},{\displaystyle \frac{1}{8}},-{\displaystyle \frac{3}{8}}+\delta )$		(5.12a)
	${𝒑}_2$	$=({\displaystyle \frac{1}{8}},{\displaystyle \frac{1}{8}},-{\displaystyle \frac{3}{8}}-\delta )$		(5.12b)
	${𝒑}_3$	$=(-{\displaystyle \frac{1}{8}},-{\displaystyle \frac{1}{8}},{\displaystyle \frac{3}{8}}-\delta )$		(5.12c)
	${𝒑}_4$	$=(-{\displaystyle \frac{1}{8}},-{\displaystyle \frac{1}{8}},{\displaystyle \frac{3}{8}}+\delta )$		(5.12d)

with $\delta =0.0838(1)\approx \frac{1}{12}$, equivalent to the structure of $\beta$-${\text{Li}}_2{\text{IrO}}_3$[98], the explicit structure factors evaluate as

	${𝓕}_F(𝑸)=$			(5.13a)
	${𝓕}_A(𝑸)=$			(5.13b)
	${𝓕}_C(𝑸)=$			(5.13c)
	${𝓕}_G(𝑸)=$			(5.13d)

where $\xi$ is defined to be

$$\xi :=\frac{\pi }{4}\left(h+k+l\right)$$

(5.14)

These structure factors create some additional selection rules. Where $h$,$k$,$l$ are all even, if $h+k+l=4n$, where $n$ is an integer $\xi =n\pi$. So ${𝓕}_C=0$, and ${𝓕}_G=0$. In the other case if $h+k+l=4n+2$ then $\xi =\left(n+\frac{1}{2}\right)\pi$. So ${𝓕}_F=0$, and ${𝓕}_A=0$. Additionally, in $\beta {\text{-Na}}_2{\text{PrO}}_3$$\delta \approx \frac{1}{12}$ (see Table 5.1), so at $l=6p+3$, ${𝓕}_F=0$, ${𝓕}_C=0$ and at $l=6p$, ${𝓕}_A=0$, ${𝓕}_G=0$. These selection rules are summarised in Table 5.4.

Attribution note: $\beta {\text{-Na}}_2{\text{PrO}}_3$powder sample synthesis, powder neutron diffraction measurements, and magnetic structure fits using FULLPROF [83] were performed by Rytaro Okuma at the University of Oxford and reported in [78]. These data are reproduced in this section for completeness.

Powder neutron diffraction data measured below $T_N$ was reported in [78], showing clear magnetic scattering below $T_N$. The magnetic peaks are shown by taking the difference between base temperature ($1.3\mathrm{K}$) and well above the transition ($10\mathrm{K}$). The use of a centred cell for indexing the peaks can make it less clear what the magnetic unit cell is, given a set of reflections. This is because a larger magnetic cell than the structural primitive cell could still have the translational symmetry of the centred cell, so magnetic reflections would still have integer Miller indices. Table 5.5 shows a number of expected peaks for $𝒒=0$ magnetic reflections in $\beta {\text{-Na}}_2{\text{PrO}}_3$, alongside the associated lattice plane spacing.

All of the magnetic reflections detected can be indexed by Miller indices of the primitive structural cell, indicating a $𝒒=0$ magnetic propagation vector.

The presence or absence of magnetic reflections at different Miller indices can be compared to the selection rules to determine which magnetic basis vectors are present. Of particular interest are the peaks $(020)$, $(006)$, and $(200)$. From the selection rules in Table 5.4 it can be seen that these reflections are in the $h$, $k$, $l$ all even, $h+k+l=4n+2$, $l=6p$ family of peaks, which only have intensity from the $C$ basis vector. The presence of finite intensity on the $(006)$ and $(200)$ peaks indicates that a $C$ basis vector must be present, the zero intensity at $(020)$ means that this must be a $C_y$ component, as this will not contribute to $𝑸\parallel 𝒚$ peaks due to the neutron polarisation factor. There is also finite intensity at the $(004)$ peak which can have contributions from $A$ and $F$ basis vectors. $F$ can be ruled out as this is a ferromagnetic contribution, and non-zero ${\mu }_{tot}$ would be detected as a ferromagnetic anomaly in susceptibility, and have a clear signature in torque as is seen in $\alpha {\text{-Na}}_2{\text{PrO}}_3$and discussed in Chapter 4. This indicates that there is an $A_x$ or $A_y$ contribution, which must come from $A_x$, as the combination $A_y,C_y$ cannot maintain the fixed lengths of the spin vectors at each site. Together, these observed peaks with these selection rules imply a magnetic structure made up of $A_x$ and $C_y$ magnetic basis vectors.

Table 5.6 shows the irreducible representations (irreps) for $𝒒=0$ magnetic structures. The relevant group is $Fddd\otimes \mathrm{\Theta }$, where $\mathrm{\Theta }$ is the time-reversal operator. 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 $Fddd$ structural space group [24]. All magnetic basis vectors exist in the $m{\mathrm{\Gamma }}_n^{\pm }$ irreps, as they all change sign under time-reversal.

From this table, it can be seen that the $A_x$ and $C_y$ magnetic basis vectors are members of the same $m{\mathrm{\Gamma }}_4^{-}$ irrep. The natural interpretation is that these are the only magnetic basis vectors present in the magnetic structure. This is consistent with the behaviour seen in heat-capacity at the transition, discussed in Section 5.4, as second order phase transition at $T_N$ suggests that the magnetic ordering will be described by a single irrep.

The time-reversal symmetry operation of the paramagnetic phase maps the magnetic structure $\alpha A_x+\beta C_y$ to $-\alpha A_x-\beta C_y$, so they must have the same energy. These are expected as two magnetic domains. Because $A_x$ and $C_y$ are both basis vectors for the same $m{\mathrm{\Gamma }}_4^{+}$ irrep, there are no symmetry operations which change the relative phase between the $A_x$ and $C_y$ components. This phase can be determined from the intensities of reflections where both $A_x$ and $C_y$ states contribute. These peaks are the $h$, $k$, $l$ all odd, $l\ne 6p+3$ family, as per Table 5.4

The intensity of magnetic scattering of a Bragg peak at $(hkl)$ can be determined from the structure factor as

	$I(𝑸)=$	${\left|{𝓕}_{⟂}(𝑸)\right|}^2$		(5.15a)
	$=$	${𝓕}_{⟂}(𝑸)\cdot {𝓕}_{⟂}^{\ast }(𝑸)$		(5.15b)

where ${𝓕}_{⟂}(𝑸)$ is the component of the structure factor normal to the scattering direction $𝑸$. If the vector moments are taken out of the structure factors

$$𝓕(𝑸)={𝝁}_A{ℱ}_A(𝑸)+{𝝁}_C{ℱ}_C(𝑸)$$

(5.16)

then the perpendicular component can be extracted as

$${𝓕}_{⟂}(𝑸)={𝝁}_A^{⟂}{ℱ}_A(𝑸)+{𝝁}_C^{⟂}{ℱ}_C(𝑸)$$

(5.17)

where

$${𝝁}_n^{⟂}={𝝁}_n-\frac{{𝝁}_n\cdot 𝑸}{𝑸\cdot 𝑸}𝑸$$

(5.18)

The intensity at $𝑸$ can then be written

(5.19)

where $\mathrm{\Re }$ indicates the real part of the expression. The last term is a ‘cross’ term which is sensitive to the difference between $(A_x,C_y)$ and $(A_x,-C_y)$.

A fit of the powder neutron diffraction data to each of these structures showed that a $(A_x,-C_y)$ structure has good quantitative agreement with the intensities of the observed peaks. The resulting magnetic structure is non-collinear, with spins in each zigzag chain close to the direction of the chain and a large angle between spins in adjacent chains. This structure can be seen in Figure 5.9, where the black sites indicate the primitive/magnetic unit cell.

The magnetic phase diagrams of hyperhoneycomb lattices has been the explored in detail theoretically. The case with nearest-neighbour Kitaev interactions has been solved exactly, with a quantum spin liquid ground state [65], which is expected to have a finite transition temperature [71]. Extensions to this such as the Heisenberg-Kitaev model have also been studied [61]. Another proposed minimal model for hyperhoneycomb is the $J$-$K$-$\mathrm{\Gamma }$ model [59]. This includes only nearest-neighbour interactions and has the form

(5.20)

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. In the idealised $Fddd$ crystal structure with all PrO₆ octahedra cubic, the Pr-O-Pr-O superexchange planes have normal axes, called Kitaev axes, which each are a local Ising axis for the associated Pr-Pr bond [41, 59]. These Kitaev axes are mutually orthogonal and can be written in terms of the orthorhombic crystal axes

(5.21)

The Hamiltonian can also be written in tensor notation

(5.22)

such that the Hamiltonian for an individual bond ${\mathscr{H}}_{i,j}^{\gamma }={𝑺}_i^T{𝒥}^{\gamma }{𝑺}_i$. For the $z$-bond, the Hamiltonian ${\mathscr{H}}_{i,j}^z=J{𝑺}_i\cdot {𝑺}_j+K{S}_i^z{S}_j^z+\mathrm{\Gamma }(S_i^x{S}_j^y+S_i^y{S}_j^x)$ exchange tensor can be written in Kitaev axes as

(5.23)

and in orthorhombic axes this takes the diagonal form

(5.24)

In previous theoretical work [59], the $x$,$y$,$z$ bonds were treated as being related by pseudo-symmetry under a 3-fold rotation around the normal to the local honeycomb plane, which contains all three bonds coming from each site. This approximation is reasonable for the idealised $Fddd$ crystal structure with cubic PrO₆ octahedra. However, $\beta {\text{-Na}}_2{\text{PrO}}_3$shows some local distortions which may invalidate this simplification. The important symmetries constraining the nearest-neighbour bonds are a 2-fold axis through the $z$ bonds, which relate the $x$ and $y$ bonds together, and an inversion centre at the $x$ and $y$ bond midpoints. The $J,K,\mathrm{\Gamma }$ parameters on $z$ and $x$ bonds are not required to be the same by symmetry, so the values for an $x$ bond on the $𝒂+𝒃$ chain are written as $J^{\prime },K^{\prime },{\mathrm{\Gamma }}^{\prime }$ respectively as

(5.25)

and the $y$ bond on the same zigzag chain is related by a two-fold rotational symmetry around the $(1/8,1/8,z)$ axis

(5.26)

The previous work on this model found that with $J=J^{\prime },K=K^{\prime },\mathrm{\Gamma }={\mathrm{\Gamma }}^{\prime }$ it could stabilise many different classical magnetic structures including a number of $𝒒=0$ non-collinear magnetic structures. This section seeks to find a minimal model for the observed magnetic structure, investigating various parts of the $J^{\prime }=J,K^{\prime }=K,{\mathrm{\Gamma }}^{\prime }=\pm \mathrm{\Gamma }$ family of Hamiltonians. In all the models discussed in this section, an isotropic g-factor $g=1$ is used consistent with the value determined from powder susceptibility measurements are high temperatures [78], equal to the powder average value determined for the honeycomb polymorph $\alpha {\text{-Na}}_2{\text{PrO}}_3$[21]. Changes in the $g$-factor will affect the quantitative magnetic fields where features occur but not the qualitative features themselves or the overall phase diagram.

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.

The presence of magnetic field induced phase transitions along certain crystallographic directions, and their evolution as the field is rotated in different planes, allows these models to be tested by using high-field magnetometry measurements. The presence of magnetic phase transitions will cause anomalies in magnetic torque and magnetisation, which could allow different magnetic phase diagrams and therefore different models, to be distinguished. In order to achieve this, torque magnetometry was performed at high magnetic fields at HZDR Dresden, allowing access to fields larger than $65\mathrm{T}$. These were performed using a $150\mathrm{\mu }\mathrm{m}$ piezoresistive cantilever on a rotator probe. Additional measurements in DC fields up to $35\mathrm{T}$ will be presented in the subsequent Section 5.9.

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

As the zero-field magnetic structure is predominantly antiferromagnetism along the $𝒂$ axis, a metamagnetic phase transition is likely when the magnetic field is applied along the $𝑯\parallel 𝒂$ axis, regardless of the details of the model which stabilised the structure. Because of this, initial measurements were performed for fields close to the crystal $𝒂$ axis.

An $\approx 150\mathrm{\mu }\mathrm{m}$ long sample was prepared such that the experimental rotation stage would rotate the sample in the $𝒂𝒄$ plane. Here, $\theta$ is defined as the angle between the field direction and the $𝒂$ axis, in the $𝒂𝒄$ plane. Initially, the sample was rotated such that the field would be applied close to the $𝒂$ axis, and was then rotated until it made a $\theta \simeq -15\mathrm{°}$ angle with the axis. In between each field sweep, the sample was rotated by $4.5\mathrm{°}$ in the $𝒂𝒄$ plane, such that $\theta$ increased. Because of the $𝒂𝒄$ rotation plane, the measured torque is the projection of the full vector torque $𝝉=𝑴\times 𝑩$ onto the crystal $𝒃$ axis. A photograph of the mounted sample can be seen in Figure 5.16, note that the photograph shows the sample in the $𝒃𝒄$ plane, as this image shows the sample morphology most clearly.

Figure 5.17 shows the evolution of magnetic torque with applied magnetic field up to 67$\mathrm{T}$ for fields close to the crystallographic $𝒂$ axis measured at $1.5\mathrm{K}$. Fig. 5.17a) shows these torque data for . All of these curves show a peak feature at ${\mu }_0{H}_{\text{SF}}=26.2(1)\mathrm{T}$. This peak is sharper for fields closer to the $𝒂$ axis, and broadens at the angle made with this axis increases. This behaviour is consistent with a spin-flop transition. The angle $\theta$ is defined by taking the sweep with the sharpest peak as $\theta =0\mathrm{°}$, the angle of other sweeps is determined by measuring the angle offset from this point. This is challenging as the angle appears to change when the probe is thermally cycled, as such angles reported here are estimates. Fig. 5.17b) shows the up- and down-sweeps for $\theta =9\mathrm{°}$ in yellow and black respectively. In addition to the peak feature attributed to a spin-flop transition, an additional cusp feature, marked with a downward arrow, can be seen in the up-sweep at ${\mu }_0{H}_{\text{P}}=58.0(1)\mathrm{T}$. In the down-sweep, this feature cannot be seen, instead at this point significant hysteresis opens and closes again at zero-field. I attribute the anomaly at $H_{\text{P}}$ to a polarisation transition.

Figure 5.18 shows magnetic torque up-sweeps for temperatures between $0.67\mathrm{K}$ and $4.2\mathrm{K}$ with applied field close to the $𝒂$ axis. Fig. 5.18a) shows these curves for $\theta \simeq 0\mathrm{°}$. These show a sharp peak in torque at the spin-flop field for the measured temperatures below $4\mathrm{K}$, which appears to be approximately temperature independent. Fig. 5.18b) shows these curves for $\theta \simeq 13\mathrm{°}$. These show a much broader peak in torque at the spin-flop field for the measured temperatures below $4\mathrm{K}$, which appears to be approximately temperature independent. Additionally, these show a cusp feature at higher field. At applied fields above this feature, all temperature curves lie on top of each other. This cusp feature is strongly temperature dependent, consistent with a polarisation transition. From the these torque measurements, a phase diagram is produced, shown in Fig. 5.18c).

Figure 5.19 shows magnetic torque up-sweeps at $1.5\mathrm{K}$ for sweeps in the $𝒂𝒄$ plane, rotating from near the $𝒂$ axis towards the $𝒄$ axis for . Fig. 5.19a) shows the raw measured torque, while Fig. 5.19b) shows the calculated derivative of this signal. For small angles, with the field close to the $𝒂$ axis, the peak in torque associated with the spin-flop field is prominent. This can be seen as a sharp feature in the derivative at the same field. As $\theta$ increases, this spin-flop peak becomes broader and smaller. In the largest angle shown of $27\mathrm{°}$, the spin-flop feature can not be seen in the raw data, but is still visible as a flattening of the derivative at ${\mu }_{0}H\simeq 26\mathrm{T}$. This shows that the field where the feature occurs is independent of rotation angle. The cusp features associated with polarisation can also be seen in these data. These can also be seen as a step-change in the derivative, and are shown with arrows above the data. As angle increases, these appear to decrease in field. It is worth noting though that the high sensitivity of this transition to temperature makes it also sensitive to precise cryogenic conditions in the cryostat during the pulse. There is therefore some uncertainty in the temperature during the pulse, and so also uncertainty in this field. The measured polarisation field drops from $58\mathrm{T}$ at $9\mathrm{°}$ to $55\mathrm{T}$ at $23\mathrm{°}$. At larger angles, the size of this feature drops in magnitude, making it unclear in both the raw torque and the derivative at the largest angle shown of $27\mathrm{°}$.

To determine the magnetic phase diagram close to the $𝒃$ axis the sample was remounted such that the rotator rotated in the $𝒃𝒄$ plane. The mounted sample can be seen in Figure 5.16, with the sample mounted on a flat growth plane normal to the $𝒄$ axis. The sample was then rotated until the field was close to this $𝒃$ axis. These measurements were performed in a ${}_{}{}^4\text{He}$ flow cryostat, with a base temperature of $\simeq$$1.3\mathrm{K}$.

Figure 5.20 shows magnetic torque vs field for field applied close to the crystallographic $𝒃$ axis for temperatures between $1.33\mathrm{K}$ and $4.2\mathrm{K}$. In all temperatures below the transition, these show a cusp-like feature, with increases in field as the temperature decreases. Fig. 5.20b) shows both up and down sweep for the base temperature measurement at $1.33\mathrm{K}$. This shows that a hysteresis opens at the cusp feature, much like the cusp feature observed close to the $𝒂$ axis. This feature can be seen clearly as a sharp peak in the differential of torque with field, shown in Fig. 5.20c), this sharp peak allows the field where the feature occurs to be extracted. The strong temperature dependence of this feature, combined with the fact that this is the only feature seen up to $67\mathrm{T}$, leads me to attribute this to the transition to the paramagnetic phase. The temperature evolution of the extracted polarisation field allows a phase diagram to be constructed for fields close to the crystal $𝒃$ axis Fig. 5.20d). The dotted line on the phase diagram is a fit to the functional form

$${\left(\frac{H_{\text{P}}(T)}{H_{\text{P}}(0\mathrm{K})}\right)}^{\alpha }+{\left(\frac{T}{T_c}\right)}^{\alpha }=1$$

(5.29)

and is intended as a guide to the eye. From this fit, the zero-temperature polarisation field is placed in the region of $53\mathrm{T}$, and $\alpha \simeq 2$.

The evolution of torque-field dependence as the field is rotated towards the $𝒄$ axis can be seen in Fig. 5.21. This shows that as the field is rotated towards the $𝒄$ axis, the observed features in torque broaden and become harder to distinguish from the background. Especially prominent in the $\theta =23\mathrm{°}$ sweep, an additional feature can be seen above $60\mathrm{T}$, appearing as a flattening of the torque at high field. This can be seen at steeper angles, although it has a broader onset. The physical origin of this feature is unclear, if it is present close to the $𝒃$ axis, it must be at above the $67\mathrm{T}$ maximum field. If this is the case, it would suggest a more complex phase diagram than the one proposed. Despite the uncertainty in the origin and precise location of the feature, the associated anomaly in the derivative of torque can be seen in all the angles shown, including with fields close to the $𝒄$ axis.

The last high-symmetry plane explored with torque is the $𝒂𝒃$ plane. Sample morphology means that precise alignment on a cantilever in this plane is challenging. To overcome this an $\approx 170\mathrm{\mu }\mathrm{m}$ long single crystal sample of $\beta {\text{-Na}}_2{\text{PrO}}_3$was mounted on a metal plate with natural growth plane normal to $𝒄$ flush with the plate. This was then placed in a ThermoFisher Dual Beam Focused Ion Beam Scanning Electron Microscope (FIB-SEM). The sample was then cut in the $𝒂𝒄$ plane using the ion beam (FIB) providing a flat surface with the $𝒄$ axis orthogonal to the surface-normal. Figure 5.22 shows an image of the sample, taken with SEM after this cut, showing clear sample morphology covered by a layer of non-crystalline material. Note especially that the sample is mounted on a flat growth face, measured with X-ray diffraction to be normal to $𝒄$, providing confidence that this alignment is precise. This sample was mounted for torque with the cantilever flush to the cut face, allowing the sample to be rotated in the $𝒂𝒃$ plane.

Figure 5.22b,c) show the torque-field dependence and the corresponding derivative for fields close to the $𝒂$ axis. These show a very sharp transition at $26.6(2)\mathrm{T}$, matching closely with the spin-flop field measured previously. The higher field behaviour is less clear, but the rapid change in torque, and corresponding peak in $\frac{\partial \tau }{\partial H}$ at approximately ${\mu }_{0}H\simeq 62\mathrm{T}$ may be associated with the polarisation transition observed in the $𝒂𝒄$ plane torque measurements. The absence of a sharp cusp at polarisation, seen in other orientations, is likely caused by the different projections of the torque vector. In calculations for some models a cusp feature is only expected to be seen in particular projections of the torque, this can be seen in Fig. 5.12.

Figure 5.23 shows the torque vs field and the corresponding derivative for fields applied in the $𝒂𝒃$ plane. The sweep at each angle shows rich behaviour, including low-field peak-trough features in torque even close to the $𝒃$ axis. Arrow markers above the curves show inflection points in the data. Unlike in previous cases in this section, the derivative is not particularly illuminating, as the behaviour is too complex to select individual features of note. For the smallest angle, there is a sharp peak-trough feature associated with the spin-flop, and a drop in torque which may be associated with transition to polarisation. For angles close to the $𝒃$ axis, marked $60\mathrm{°}$ and $80\mathrm{°}$, there is a peak-trough-peak-trough feature. Close to the $𝒂$ axis, the spin-flop feature is selected as a sharp peak in the derivative of torque, if this is applied to the broad peak seen close to the $𝒃$ axis, it suggests a broad low field feature, marked with an arrow, in the region of $30\mathrm{T}$. This feature is marked in Fig. 5.20 b) with a yellow cross symbol. The fact that no feature can be seen in the $𝒃𝒄$ plane measurements is likely caused by one of two scenarios. The anomaly could be only present in the ${\tau }_z$ projection of torque, not the ${\tau }_x$ projection measured in the $𝒃𝒄$ plane. The second possibility is that the feature is related to a projection of field into the $𝒂$ axis, which is zero in the $𝒃𝒄$ plane. Ultimately these scenarios cannot be practically distinguished with torque magnetometry. The high-field inflection point noted with an arrow is in the region of $60\mathrm{T}$. The prominence of both features decreases as the field approaches the $𝒃$ axis. At the intermediate angle of $40\mathrm{°}$ there is a large and broad peak-trough feature at around $40\mathrm{T}$.

A limitation of pulsed magnetic fields is that long magnet cool-down times limit the density of data that can be taken. The data also has a tendency to be noisier than that available in DC fields. In order to better understand the magnetic phase diagram in the $𝒂𝒃$ plane, magnetic torque was measured up to $35\mathrm{T}$ in DC fields. These experiments were performed using capacitive cantilever torque magnetometry such that a jig could be used to hold the sample in the $𝒂𝒃$ plane.

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.

Figure 5.24 shows magnetic torque in the $𝒂𝒃$ plane up to $35\mathrm{T}$. The top panels show raw cantilever capacitances, proportional to torque, the bottom panels show the derivative of this with field. Each measurement was taken at $4.0\mathrm{K}$, the highest temperature which kept all observed transitions below the $35\mathrm{T}$ maximum field available. Every torque vs field trace shown in this plane shows a cusp-like transition. Near the $𝒂$ axis, the transition occurs at $21.5(1)\mathrm{T}$ and is sharp, consistent with the behaviour seen in pulsed fields at these temperatures. In this section, angle $\theta$ is the angle made in the $𝒂𝒃$ plane between the field direction and the crystal $𝒂$ axis. As the field rotates towards the $𝒃$ axis, for , the transition remains sharp and the transition field varies strongly with angle, up to $34.9(2)\mathrm{T}$ at $\theta =45\mathrm{°}$. Towards the $𝒃$ axis, the anomaly associated with the transition is much less pronounced, although it can still be clearly seen in both raw and derivative plots. Close to the $𝒃$ axis, the transition field is approximately $30.0(2)\mathrm{T}$, in agreement with the value determined in pulsed fields at the same temperature of $4\mathrm{K}$. This can be seen in the pink circles overlaid on Fig 5.20d). This increases monotonically with angle up to $\theta =45\mathrm{°}$. In addition to the transition, fields near the $𝒃$ axis show a broad peak-trough feature that was also observed in pulsed fields.

Figure 5.25 shows the derivative of torque with field for all the angles measured. Overlaid is a guide to the eye, indicating where magnetic phase transitions occur. This shows a suppression of the transition field when the magnetic field is applied close to the $𝒂$ axis. It is important to note that at this temperature, $4.0\mathrm{K}$, only one transition is seen when the field is applied along the $𝒂$ axis. One possible interpretation of this suppression is that at low temperatures, there is a phase boundary connecting the spin-flop along $𝒂$ and the polarisation transition in the $𝒂𝒃$ plane. This would be expected if the spin-flop phase magnetic state was in a different irrep to the zero-field $A_x,-C_y$ ground state. These are tabulated in Table 5.7, when the magnetic field is in the $𝒂𝒃$ plane, the ground state is in the ${\mathrm{\Gamma }}_1^{-}$ irrep, a possible spin-flop structure in a different irrep would be $A_z,C_z$ in the ${\mathrm{\Gamma }}_2^{-}$ irrep.

Figure 5.26 shows evolution of magnetic torque near the $𝒂$ axis with field for temperatures between $1.4\mathrm{K}$ and $4.0\mathrm{K}$. These show a sharp peak in the derivative at the spin-flop transition. At base temperature, the spin-flop field is ${\mu }_0{H}_{\text{SF}}=25.5(2)\mathrm{T}$, this is slightly lower than the value determined in pulsed-fields, but overall in good agreement. This can be seen in the phase diagram including both sets of data, in Figure 5.26c).

Attribution note: 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.

In order to understand the high-field magnetic phase transitions observed in $\beta {\text{-Na}}_2{\text{PrO}}_3$, magnetisation measurements were attempted using coil extraction magnetometry up to $67\mathrm{T}$. A collection of 7 single-crystals were co-aligned such that field would be applied along the $𝒂$, shown in Figure 5.27. The total mass of this collection is estimated at $m\sim 5\mathrm{mg}$. In order to allow time for co-alignment without them degrading in air, the samples were coated in paraffin in a glovebox and sealed in vacuum. In the experiment however, no magnetic signal was detected above the experimental noise level. While it is possible that more samples could be co-aligned to fit into the sample space, it is clear that given the small moment a substantially larger amount of material, at least an order of magnitude, would be needed to measure a significant signal. As such, it is unclear if co-aligning crystals is a realistically viable method of achieving a suitable filling factor of the measurement coil with the sample sizes available. If larger samples are able to be grown, or the largest available samples today can be produced more consistently, then this could be reattempted.

This chapter reported on the hyperhoneycomb magnet $\beta {\text{-Na}}_2{\text{PrO}}_3$. I have shown detailed single-crystal structural refinement, and attempted to characterise the magnetic properties of the material for the first time. The material was shown to order at $5.20\mathrm{K}$ with AC calorimetry measurements. Single crystal torque magnetometry in low fields suggested non-collinear antiferromagnetic order, with suppression of susceptibilities in the $𝒂$ and $𝒃$ directions. This was confirmed by neutron diffraction measurements, performed by others, which showed an $(A_x,-C_y)$ magnetic ground state. A minimal $J,\mathrm{\Gamma },{\mathrm{\Gamma }}^{\prime }$ model was developed, and the behaviour of this model in magnetic fields was explored using mean-field calculations, showing that a spin-flop transition was expected for fields along the $𝒂$ axis. A symmetry analysis considering what magnetic states could be reached with and without crossing a phase boundary for fields applied in the high symmetry planes showed that this predicted phase boundary was symmetry-protected in the $𝒂𝒄$ plane, giving a clear signature to look for.

The magnetic torque data presented in this chapter shows rich behaviour in all orientations measured. Close to the $𝒂$ axis, a sharp spin-flop phase transition is observed in both pulsed and DC fields. As the field is rotated toward the $𝒄$ axis it broadens rapidly. This is unlike the behaviour expected from the minimal $J,\mathrm{\Gamma },{\mathrm{\Gamma }}^{\prime }$ model, in which a symmetry-protected first-order transition between $(A_x,-C_y)$ and $(A_y,-C_x)$ states is expected. One possible explanation is that the phase boundary between these states is close to the $𝒂$ axis, such that field sweeps only cross it for very small angles against the axis. Substantial broadening of the transition is seen at an angle of , so some effort would be needed to understand why this would be the case in this scenario. An alternative explanation is that the observed anomaly is a transition to a different spin-flop phase in the same irrep, such as $A_z$ or predominant $C_y$ order. In the $𝒂𝒃$ plane, these phases are in different irreps. In the $A_z$ case, this would imply a symmetry protected transition between the zero-field $(A_x,-C_y)$ and spin-flop $A_z$ states.