Chapter 1 Introduction to Quantum Magnetism

In an insulator, the electrons on each ion are localised in orbitals around the ion. These electrons can be treated as moving in a spherically symmetric potential from the nucleus, and interacting with every other electron in the ion. This system has an electronic ground state configuration consisting of degenerate energy levels $E_{nl}$ with quantum numbers $n,l$ such that and degeneracy $2(2l+1)$. These make up shells, which fill from lowest energy to form an overall energy configuration.

For each electron in the ion, spin and orbital angular momentum operators can be defined $\widehat{𝒔}$ and $\widehat{𝒍}$ respectively. These contribute to a magnetic moment, the operator of which can be defined

$${\widehat{𝝁}}_e={\mu }_B\left(g_l\widehat{𝒍}+g_s\widehat{𝒔}\right)$$

(1.1)

where $g_l=1$, $g_s=2+𝒪(\alpha )$ including additional contributions from the fine-structure constant $\alpha \approx 1/137$.

For the ion, total spin and angular momentum operators can be defined

	$\widehat{𝑺}$	$={\displaystyle \sum _i}{\widehat{𝒔}}_i$		(1.2)
	$\widehat{𝑳}$	$={\displaystyle \sum _i}{\widehat{𝒍}}_i$		(1.3)

and so can a total magnetic moment operator

$$\widehat{𝝁}={\mu }_B\left(g_l\widehat{𝑳}+g_s\widehat{𝑺}\right)$$

(1.4)

For every full shell in the electronic configuration, the angular momenta and spins sum to zero, so they do not contribute to the total spin or orbital angular momentum for the ion. Instead, the only electrons which contribute are those in partially filled electron shells. Because of this, reported electronic configurations normally specify just the outermost, partially filled (valence) shell of electrons. For example, the ${\text{Co}}^{2+}$ion has a $3d^7$ valence shell, specifying that there are 7 electrons in the $n=3,l=2$ outermost shell. As free ions are spherically symmetric, the energy cannot change under a rotation. The degenerate ground state manifold is then best described by quantum numbers $L,S$ such that $\widehat{𝑳}\cdot \widehat{𝑳}|{\varphi }_{L,S}⟩=L(L+1)|{\varphi }_{L,S}⟩$ and similarly with $S$. These manifolds of states with a given $L$, $S$ have degeneracies $(2L+1)(2S+1)$.

If the electrons in the outermost valence shell are treated as non-interacting, all the combinations of $L$ and $S$ will be degenerate in energy. A more realistic treatment to find the ground state $L$ and $S$ manifold is called Hund’s rules [40, 93, 13]. This specifies that (1): $S$ will be maximised, and (2): given (1) $L$ will be maximised. The third rule includes spin-orbit coupling which is treated separately here. In the ${\text{Co}}^{2+}$example, this results in an $L=3$, $S=3/2$, 28-fold degenerate ground state, ignoring spin-orbit coupling.

The next leading order perturbation considered is a relativistic correction to the energy levels, spin-orbit coupling (SOC). SOC adds a contribution to the energy

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

(1.5)

where the spin-orbit coupling constant $\lambda$ is positive for shells less than half full, and negative for shells more than half full. If a total angular momentum operator is defined

$$\widehat{𝑱}=\widehat{𝑳}+\widehat{𝑺}$$

(1.6)

with quantum number $J$, defined similarly to $L$ and $S$. $\widehat{𝑱}$ commutes with the spin-orbit Hamiltonian, so $J$ is a good quantum number, while $L$ and $S$ are not. This splits the $L,S$ manifolds of states into $J=|L-S|,|L-S|+1,\mathrm{\dots },L+S$ multiplets, each with degeneracy $2J+1$. In the ${\text{Co}}^{2+}$example, and the ground state is $J=L+S=9/2$ with additional multiplets at $J=7/2,5/2,3/2$ in increasing order of energy.

The total magnetic moment operator can be written in terms of this new total angular momentum operator as

$$\widehat{𝝁}={\mu }_B{g}_J\widehat{𝑱}$$

(1.7)

where $g_J$ is the Landé g-factor [58] which evaluates as

$$g_J\approx \frac{3}{2}+\frac{S(S+1)-L(L+1)}{J(J+1)}$$

(1.8)

Note that the time-reversal symmetry of the Hamiltonian enforces over any degenerate multiplet of states. The moment $\widehat{𝝁}$ couples to a magnetic field by the Zeeman energy

$${\widehat{\mathscr{H}}}_{\text{Zeeman}}=-\widehat{𝝁}\cdot 𝑩$$

(1.9)

which is sufficient to break the symmetry and produce a non-zero moment, even for an infinitesimally small field $B$.

This gives the free-ion ground state, however in crystals the electrons in the ion will feel an electrostatic potential which is not spherically symmetric. In general, the continuous spherical symmetry will be broken into discrete symmetries, depending on the point group symmetry of the ion. This will typically lift the degeneracy of the multiplets of states. In some cases, this can be treated as a perturbation on the spin-orbit ground state. However, in all the systems of interest in this thesis, it at least comparably strong to the SOC. The additional contribution from the crystal environment is called the crystal-electric field (CEF), and its energy contribution can be written in terms of Stevens operators $O_k^q$ [1]

$${\widehat{\mathscr{H}}}_{\text{CEF}}=\sum _{k\in \{2,4,6\}}\sum _{q=-k}^k{B}_k^q{O}_k^q$$

(1.10)

where the coefficients $B_k^q$ are constrained by the point group symmetry of the environment. For example, in a cubic octahedral $O_h$ environment the only non-zero terms are $B_4^0,B_4^4,B_6^0,B_6^0$ under the constraints $B_4^4=5B_4^0$, $B_6^4=-21B_6^0$ [1, 74]. This means that the octahedral crystal field (OCF) CEF Hamiltonian can be written as

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

(1.11)

These Stevens operators are caused by an energy dependence on the orbitals involved in the state. This comes from the interactions between these orbitals and the crystal environment, for example an octahedral environment of ${\text{O}}^{2-}$ions as is seen in all the materials of interest in this thesis. The electronic states whose wave-functions have large overlap with the oxygen ions develop a higher energy than the electrons with a smaller overlap, which lifts the degeneracy of the free ion multiplet. Because this effect is dependent only on the orbital degree of freedom, the Stevens operators can be written in terms of only polynomials of the orbital angular momentum operators ${\widehat{L}}_z,{\widehat{L}}_{+},{\widehat{L}}_{-}$. For example

	$O_4^0$	$={\widehat{L}}_z^4-(30L(L+1)-25){\widehat{L}}_z^2+(3L^2{(L+1)}^2-6L(L+1))𝕀$		(1.12)
	$O_4^4$	$={\displaystyle \frac{1}{2}}({\widehat{L}}_{+}^4+{\widehat{L}}_{-}^4)$		(1.13)

where $𝕀$ is the identity operator. These operators are straightforward to compute [86] and are tabulated up to $k=6$ [1, 3]. In all of the systems investigated in this thesis, the most straightforward approach to including both SOC and the CEF is numerically, computing the Hamiltonian and diagonalising to give the energy levels. Frequently, an effective angular momentum $j_{\text{eff}}$ will be defined for the resulting ground state multiplet. This is defined such that the degeneracy of this multiplet is $2j_{\text{eff}}+1$. The g-factor of this multiplet can be defined in terms of the moment operator by taking the limit of a small magnetic field (here along a nominal $𝒛$ axis)

(1.14)

where the expectation value is the expectation of the ground state, once a small field has been applied along the $𝒛$ axis. This means that the magnetic moment has the same form as the free-ion case

$${\widehat{𝝁}}_z={\mu }_{B}g{\widehat{𝒋}}_{\text{eff}}^z$$

(1.15)

Given some ground state multiplet with non-zero $g$ a magnetic field $𝑩=B\widehat{𝒛}$ applied parallel to the $\widehat{𝒛}$ axis will be split by the Zeeman Hamiltonian (1.9)

	${\widehat{\mathscr{H}}}_{\text{Zeeman}}$	$=-\widehat{𝝁}\cdot 𝑩$		(1.16)
		$=-{\mu }_{B}gB{\widehat{j}}_{\text{eff}}^z$		(1.17)

This splits the multiplet into $2j_{\text{eff}}+1$ separate levels described by the ${\widehat{j}}_{\text{eff}}^z$ eigenvalue $m\in \{-j_{\text{eff}},-j_{\text{eff}}+1,\mathrm{\dots },j_{\text{eff}}\}$. These levels have a magnetic moment ${\mu }_z={\mu }_{B}gm$, and energy $E_m=-{\mu }_{B}gmB$.

Note that the $g$-factor is not isotropic in general. This extension can be included by defining a $g$-factor tensor $𝔾$ such that

$$\widehat{𝝁}={\mu }_B𝔾{\widehat{𝒋}}_{\text{eff}}$$

(1.18)

which is used frequently later in the thesis.

Given a collection of isolated (non-interacting) magnetic moments, the behaviour in finite temperature can be described by statistical mechanics. For the sake of brevity, $j_{\text{eff}}$ here is written as $j$. Given the energy splittings above, the partition function [12] can be written

	$Z$	$={\displaystyle \sum _m}\exp (m\beta g{\mu }_{B}B)$		(1.19a)
		$={\displaystyle \sum _m}\exp (mx)$		(1.19b)

where $x≔\beta g{\mu }_{B}B$ and $\beta =\frac{1}{k_{B}T}$. From this, the expectation value of $m$ can be written

		$={\displaystyle \frac{1}{Z}}{\displaystyle \sum _m}m\exp (mx)$		(1.20a)
		$={\displaystyle \frac{1}{Z}}{\displaystyle \frac{\partial Z}{\partial x}}={\displaystyle \frac{\partial \ln Z}{\partial x}}$		(1.20b)
		$=j{B}_j(jx)$		(1.20c)

where $B_j(jx)$ is the Brillouin function [56]. This shows that in the large-field, low-temperature limit $\beta B\to \mathrm{\infty }$, the expectation value . In the inverse high temperature case, $\beta B\to 0$ this expectation tends towards zero. The expectation value of the moment is then . This expression itself gives the behaviour of the magnetic moment in a small field and finite temperature. One commonly measured observable is the temperature dependent magnetic susceptibility $\chi$, defined as

	$\chi$			(1.21)
		$={\displaystyle \frac{n{\mu }_0{\mu }_{\text{eff}}^2}{3k_{B}T}}$		(1.22)

for number density $n$ and effective moment ${\mu }_{\text{eff}}^2=g^2{\mu }_B^{2}j(j+1)$. This is a Curie relationship [13].

In real materials, there are additional multiplets at higher energies. The most straightforward way of computing their effect on the temperature dependent magnetisation is to numerically compute the Hamiltonian, including SOC, CEF, and Zeeman terms, and to then diagonalise and compute the expectation value of the moment operator over all of the energy levels. Fitting this to experimental data can be used to determine the values of spin-orbit coupling and crystal-electric-field parameters.

In a crystal at low temperatures, the interactions between magnetic ions become important to understand how the material orders magnetically. For the sake of convention, the effective angular momentum operators are written ${\widehat{𝑺}}_i$. The canonical minimal example of an interaction term between spins $i,j$ is the Heisenberg interaction, which can originate as an exchange energy from the Coulomb repulsion of the electrons, when combined with the Pauli exclusion principle. The Heisenberg Hamiltonian can be written

$$\widehat{\mathscr{H}}=\frac{1}{2}\sum _{i,j}{J}_{ij}{\widehat{𝑺}}_i\cdot {\widehat{𝑺}}_j$$

(1.23)

where $J_{ij}$ is the exchange term between spins $i,j$, and the half is needed as each bond is summed over twice. Terms with prefer spins $i,j$ to be parallel i.e. ferromagnetic; $J_{ij}>0$ prefers spins to be antiparallel i.e. antiferromagnetic.

The Heisenberg Hamiltonian is completely isotropic, it only favours a relative orientation of spins. In general, interactions can come from higher order processes such as superexchange and can be anisotropic, depending on the symmetry of the bond. At second-order perturbation theory these will come about as bilinear terms in the spin operators, and can be written in the most general form as

(1.24)

where the sum over indexes each bond only once, and $\alpha ,\beta \in \{x,y,z\}$. This means that the bilinear interaction on each bond is given as a tensor $J_{ij}^{\alpha \beta }$. This can capture common anisotropic interactions such as XXZ

$${\widehat{\mathscr{H}}}_{\text{XXZ}}^{ij}=J_{ij}\left({\widehat{S}}_i^x{\widehat{S}}_j^x+{\widehat{S}}_i^y{\widehat{S}}_j^y+\mathrm{\Delta }{\widehat{S}}_i^z{\widehat{S}}_j^z\right)$$

(1.25)

which is sufficient to prefer a particular axis. If $\mathrm{\Delta }>1$, this is called an easy-axis or Ising-type XXZ interaction, otherwise if this is an easy-plane interaction.

Another important type of interaction which appears in this thesis, is Dzyaloshinskii–Moriya interaction (DMI), or antisymmetric exchange. This has the form

$${\widehat{\mathscr{H}}}_{\text{DMI}}^{ij}={𝑫}_{ij}\cdot \left({\widehat{𝑺}}_i\times {\widehat{𝑺}}_j\right)$$

(1.26)

and can also be written as an antisymmetric component to the $J_{ij}^{\alpha \beta }$ interaction tensor. This type of interaction is antisymmetric under swapping of the two spins. This means that if DMI is present, the interaction tensor has to be specified on a directed bond.

A particular model of interest is the Kitaev model [55] on a honeycomb lattice with

(1.27)

where indexes over nearest-neighbour honeycomb bonds and $\gamma \in \{x,y,z\}$ is dependent on the bond direction. This model is discussed more in a later section.

The origin of the anisotropy seen in each of these cases is the anisotropy of the electronic orbitals in the crystal. In order to couple this to the spin degrees of freedom, the spin must be coupled to the orbital levels. This means that spin-orbit coupling is a requirement for anisotropic interactions.

In an ideal paramagnetic phase, in the low-field limit, every level in the degenerate multiplet with degeneracy $2j+1$ will be populated equally. This will result in an entropy of

$$S=k_B\ln (2j+1)$$

(1.28)

per particle. As the temperature of the system approaches zero, the effect of interactions becomes more dominant, and in conventional magnetic materials the system will undergo a symmetry-breaking phase transition from the high-temperature paramagnetic state to a low-temperature ordered state with zero entropy at $T=0$ (in the limit of large systems).

For example, in the ferromagnetic Heisenberg system, time-reversal symmetry will be broken at the ordering temperature to produce a ferromagnet. This will have magnetisation as the order parameter for the transition.

In general, finding the quantum mechanical ground state for a system of many interacting spins and some specified interaction Hamiltonian is not straightforward or necessarily even feasible. An often fruitful approach is to consider the mean-field approximation, where spins are treated as classical vectors of fixed length $S$, replacing the operators $S^x,S^y,S^z$ with their expectation values.

In a crystal, the locations of the individual spins can be described by a Bravais lattice. For example, consider the square lattice, with only nearest-neighbour Heisenberg-type antiferromagnetic couplings. The energy on every bond can be minimised by setting every adjacent bond antiparallel, as is seen in Figure 1.1. This is known as Néel order [73], and breaks both time-reversal and translational symmetry. There is a new, magnetic unit cell, twice as large as the structural unit cell. An equivalent Néel order state will occur on any such bipartite model with only nearest-neighbour antiferromagnetic Heisenberg interactions.

In a more general case, rather than attempting to specify the spin orientation on every site on a crystal, magnetic orders can be described more succinctly in reciprocal space. This means that the spin on lattice vector $𝒑$ can be written in terms of Fourier components:

$${𝑺}_{𝒑}=\sum _{𝒒}{\widetilde{𝑺}}_{𝒒}{e}^{2\pi i𝒒\cdot 𝒑}$$

(1.29)

While in general, there are as many $𝒒$ vectors for which ${\widetilde{𝑺}}_{𝒒}$ has to be specified as there are sites in the crystal, magnetic structures order periodically, so only a small number of these are non-zero. In many cases, only one ${\widetilde{𝑺}}_{𝒒}$ is non-zero and the $𝒒$ where this occurs is referred to as the propagation vector of the magnetic order. In the square lattice Néel order example above, ${𝑺}_{𝒑}=S\widehat{𝒛}\cos \pi (p_x+p_y)$ so the system has $𝒒={(1/2,1/2)}^T$. There is no need in general for this $𝒒$ to be rational; if it is irrational the magnetic order is described as incommensurate. All the materials investigated in this thesis order as commensurate magnetic structures, so the magnetic unit cell is an integer multiple of the primitive structural unit cell.

An interesting feature in many ordered systems is the presence of field-induced phase transitions. This occurs because the magnetic field couples to the Hamiltonian of the system through the Zeeman energy [112]

$${\widehat{\mathscr{H}}}_{\text{Zeeman}}=-{\mu }_B\sum _{i\mu \nu }{𝔾}^{\mu \nu }{B}^{\mu }{S}_i^{\nu }$$

(1.30)

where the g-factor tensor $𝔾$ allows for anisotropic g-factors, with arbitrary principal axes.

The most straightforward example of a field-induced transition is the transition to polarisation in a two-sublattice Heisenberg antiferromagnet. This has the Hamiltonian

(1.31)

where the magnetic field is chosen to apply along the $𝒛$ axis, and selects nearest-neighbour bonds. Treating the spins as classical vectors, this can be rewritten in terms of sublattice magnetisations ${𝑺}_A$, ${𝑺}_B$ as

$$\mathscr{H}=nJ{𝑺}_A\cdot {𝑺}_B-g{\mu }_{B}B(S_A^z+S_B^z)$$

(1.32)

with $n$ as the number of nearest-neighbours for each site. Taking the sublattice magnetisations ${𝑺}_A$, ${𝑺}_B$ as unit vectors, and defining parameters $𝑨=\frac{1}{2}\left({𝑺}_A-{𝑺}_B\right)$, $𝑭=\frac{1}{2}\left({𝑺}_A+{𝑺}_B\right)$ with the constraints $𝑨\cdot 𝑭=0$, ${𝑨}^2+{𝑭}^2=1$, this can be rewritten as

$$\mathscr{H}=nJ-2nJ{𝑨}^2-2g{\mu }_{B}B{F}_z$$

(1.33)

In this system with isotropic interactions, the ordered $𝑨$ vector can be chosen to be perpendicular to the field, in this example parallel to the $𝒙$ direction. As the magnetic field increases, so does $F_z$, and the spins cant towards the field. If the field is large enough, $A_x\ll 1$ and $F_z=\sqrt{1-A_x^2}\approx 1-\frac{1}{2}{A}_x^2-\frac{1}{8}{A}_x^4-𝒪(A_x^6)$ so the energy can be written as

$$\mathscr{H}\approx {\mathscr{H}}_0-(2nJ-g{\mu }_{B}B)A_x^2+\frac{1}{4}g{\mu }_{B}B{A}_x^4+𝒪(A_x^6)$$

(1.34)

where ${\mathscr{H}}_0$ is a constant. This Hamiltonian can be understood in terms of Landau theory [57] to undergo a second order phase transition at $g{\mu }_B{B}_{\text{P}}=2nJ$ to a high-symmetry polarised $A_x=0,F_z=1$ phase for $B>B_{\text{P}}$. This causes an anomaly in the differential susceptibility $-{\displaystyle \frac{{\partial }^2\mathscr{H}}{\partial B^2}}$, which can be experimentally detected straightforwardly.

In the presence of magnetic anisotropy, there can be more complex magnetic phase transitions. Two simple examples include uniaxial anisotropy: easy-axis XXZ exchange anisotropy

$${\mathscr{H}}_{\text{XXZ}}=nJ(S_A^x{S}_B^x+S_A^y{S}_B^y+\mathrm{\Delta }{S}_A^z{S}_B^z)-g{\mu }_{B}B(S_A^z+S_B^z)$$

(1.35)

and single-ion easy axis $-D{S}_z^2$ anisotropy

$${\mathscr{H}}_{\text{D}}=nJ{𝑺}_A\cdot {𝑺}_B-D(S_A^z{}_{}{}^2+S_B^z{}_{}{}^2)-g{\mu }_{B}B(S_A^z+S_B^z)$$

(1.36)

In both cases at low field the system orders into an $A_z$ antiferromagnetic order along the $𝒛$ axis, which has a lower energy than $A_x$ or $A_y$. Figure 1.2 shows the phase diagrams for the two cases, with magnetic field applied along the unique $𝒛$ axis. As field is increases, the ability to cant towards field lowers the energy of the $(A_x,F_z)$ phase. At some point the energy of this phase is lower than the $A_z$ state and the system undergoes a first-order magnetic phase transition called a spin-flop [14, 72, 73]. In the case of $-D{S}_z^2$ type anisotropy, if $D$ is sufficiently large, the system will undergo a spin-flip transition from $A_z$ to $F_z$ directly, without undergoing a spin-flop transition. Both spin-flop [113] and spin-flip [50] phase transitions have been realised in many materials experimentally.

More complex magnetic models can undergo a large variety of phase transitions in field. When these transitions are experimentally observed, this can reveal what model is governing the material’s behaviour, and allow our understanding of fundamental theory to be tested. In frustrated magnetic systems, introduced next section, magnetic fields can induce transitions to many kinds of unconventional magnetically ordered phases. A large number of interesting field-induced phase transitions have been observed experimentally such as in $\alpha$-RuCl₃ [108, 8, 44], polymorphs of Li₂IrO₃ [18, 85], triangular lattice antiferromagnet 2$H$-AgNiO₂ [19], and Ising-chain CoNb₂O₆ [20].

In many materials, it is not possible to minimise the energy on every bond simultaneously; when this is the case, the system is described as frustrated. One way this can happen is through geometric frustration, the canonical example of which is antiferromagnetic Heisenberg interactions on the triangular lattice. In these frustrated systems, there are large numbers of degenerate classical ground states [102], giving the potential for exotic quantum phases to occur. In the triangular lattice case, it was originally conjectured to form a resonating valence bond (RVB) state at low temperature [4]. This RVB state is a kind of quantum spin liquid (QSL), involving pairs of $S=1/2$ spins coupling as spin singlets, fluctuating in a liquid-like fashion with long-range entanglement [96, 7]. In the end, it was shown that these triangular lattices order with spins at a $120\mathrm{°}$ angle from their neighbours as predicted by a mean-field approximation, compromising to minimise total energy [15]. However, this exotic RVB-QSL phase is predicted to occur in other highly frustrated models [110, 39].

An alternate approach to achieving exotic magnetic phases is with frustration introduced by bond-dependent interactions. Such is the example of the Kitaev model [55] introduced earlier. The honeycomb lattice is bipartite, so antiferromagnetic Heisenberg interactions will stabilise a Néel order, and there is no geometric frustration. Instead, the Kitaev model introduces frustration by having mutually orthogonal Ising axes on each of the three bonds which meet at a site on the honeycomb lattice. This means that there is no way to minimise the energy on all three bonds simultaneously. This model is important because it has an exactly solvable quantum spin liquid ground state, thereby proving that a QSL ground state can exist in 2D and beyond, and revealing exotic excitations including quasiparticles with generalised (anyonic) statistics. Theoretical work to understand how these interactions may be realised experimentally first predicted the honeycomb iridates as candidates [41, 16, 82, 17]. An extensive search has been made for experimental realisation of a Kitaev quantum spin liquid, including Na₂IrO₃ [62, 18], $\alpha$-Li₂IrO₃ [104], and $\alpha$-RuCl₃ [79, 44]. Generalisations of this model to include Heisenberg interactions [16, 92] and symmetric off-diagonal exchange $\mathrm{\Gamma }$ gives the nearest-neighbour Hamiltonian

(1.37)

Each bond has an associated Ising axis $\gamma \in \{x,y,z\}$ and $\alpha \beta (\gamma )$ refer to the orthogonal plane to this axis. This shows an incredibly complex phase diagram[81]. This includes a finite volume of phase space stabilising a QSL around the exactly solvable pure Kitaev model, as well as many regions of unconventional magnetic order, some of which have been experimentally realised in the candidate materials searched [8, 9]. This provided part of the motivation to search for possible realisations of models with frustrated bond-dependent interactions and unconventional magnetic order in the $f$-electron ${\text{Na}}_2{\text{PrO}}_3$systems explored later in this thesis.

The Kitaev model, and its exact solutions, can be generalised onto three-dimensional tricoordinated lattices of spin-$1/2$ magnetic ions such as the hyperhoneycomb and harmonic honeycomb family of lattices [65, 53]. Some of these lattices have be realised as polymorphs of ${\text{Li}}_2{\text{IrO}}_3$[68, 10, 98, 11], but they seem to be experimentally rare. Much like the honeycomb case, this exactly solvable pure Kitaev model can be extended into a $J,K,\mathrm{\Gamma }$ model, which is predicted to show a rich phase diagram of exotic magnetic phases depending on the precise balance between the three terms in the model [59].

This thesis reports studies on three materials: the honeycomb Kitaev-candidate $\alpha {\text{-Na}}_2{\text{PrO}}_3$[42], its hyperhoneycomb polymorph $\beta {\text{-Na}}_2{\text{PrO}}_3$[43], and two honeycomb transition-metal oxides ${\text{FeTiO}}_3$and ${\text{CoTiO}}_3$. The three materials are linked by their tricoordinated lattices of magnetic ions, and their anisotropic interactions driven by strong spin-orbit-coupling [46].