[1,2]Aly H. Abdeldaim

[2]Gøran J. Nilsen

[1]Lucy Clark

1]School of Chemistry, University of Birmingham, Edgbaston, Birmingham, B15
2TT, UK

2]ISIS Neutron and Muon Source, Didcot, Oxfordshire, OX11 0QX,UK

4]Diamond Light Source, Didcot, Oxfordshire, OX11 0DE, UK

3]Deutsches Elektronen-Synchrotron DESY, Hamburg, D-22607, Germany

5]London Centre for Nanotechnology and Department of Physics and Astronomy,
University College London, London, WC1E 6BT, UK

6]Felix Bloch Institute for Solid-State Physics, University of Leipzig,
Leipzig, 04103, Germany

# Kitaev Interactions Through an Extended Superexchange Pathway in the
$j_{\mathsf{eff}}=1/2$ Ru3+ Honeycomb Magnet, RuP3SiO11

<EMAIL_ADDRESS>Hlynur Gretarsson  Sarah J. Day  M. Duc Le  Gavin
B. G. Stenning  Pascal Manuel  Robin S. Perry  Alexander A. Tsirlin
<EMAIL_ADDRESS><EMAIL_ADDRESS>[  [  [  [  [  [

###### Abstract

Magnetic materials are composed of the simple building blocks of magnetic
moments on a crystal lattice that interact via short-range magnetic exchange
interactions. Yet from these simple building blocks emerges a remarkable
diversity of magnetic states. Some of these, such as ferromagnetism, are
familiar in our everyday lives, while others reveal the deep quantum
mechanical origins of magnetism. A prime example of the latter are quantum
spin liquid (QSL) states in which—unlike in a ferromagnet where magnetic
moments are driven by their exchange interactions to adopt long-range
order—magnetic moments remain disordered at low temperatures but are
simultaneously correlated over long length scales through quantum
entanglement. Realising such states holds significant promise for advancing
our understanding of quantum many-body physics as well as our ability to
design materials for next-generation quantum technologies. A particularly
promising theoretical model of a QSL is the Kitaev model, composed of unusual
bond-dependent exchange interactions between magnetic moments on a honeycomb
lattice. However, the Kitaev QSL is extremely challenging to realise
experimentally as it is unstable to competing exchange interactions and
crystal lattice perturbations that inevitably arise in real materials. This
makes it essential to understand the relationship between the structure and
interactions that may give rise to Kitaev interactions in new candidate
materials. Here we show that the material requirements for the Kitaev QSL
survive for an extended pseudo-edge-sharing superexchange pathway of Ru3+
$4d^{5}$ octahedra within the honeycomb layers of the inorganic framework
solid, RuP3SiO11. Through materials synthesis and structural characterisation,
resonant inelastic X-ray and neutron scattering experiments, we confirm the
requisite $j_{\mathsf{eff}}=\frac{1}{2}$ state of Ru3+ in RuP3SiO11 and
resolve the hierarchy of exchange interactions that provide experimental
access to an otherwise unexplored region of the extended Kitaev phase diagram.
This demonstrates the importance of inorganic and hybrid framework materials
in the design of new Kitaev QSL candidates, as the relevant exchange
interactions in their more open crystal structures may be more readily tuned
than in their dense inorganic counterparts studied to date.

###### keywords:

Quantum materials, Kitaev quantum spin liquid, materials synthesis, neutron
scattering, X-ray scattering.

†Current address: Diamond Light Source, Didcot, Oxfordshire, OX11 0DE, UK.

##  1 Introduction

The pursuit of new magnetic materials which allow us to explore and
potentially exploit novel quantum states of matter serves as an important
demonstration of the symbiosis between condensed matter experiment and theory.
Traditionally, the impetus to search for novel phenomena in magnetic materials
stems directly from advances in theory—a prime example being the landmark
developments of the concepts of topology in condensed matter [1, 2] that are a
major motivation for modern experimental materials research [3]. But
increasingly, the design and synthesis of new magnetic materials provide an
equally important guide to push the boundaries of state-of-the-art theory.
Materials-led discoveries often highlight the critical role played by the
complexities of real magnetic systems—such as further near-neighbour exchange
interactions, lattice distortions and disorder—that go beyond idealised models
of magnetism but ultimately govern the magnetic properties we can observe and
exploit [4]. Quantum spin liquids (QSLs) present particularly compelling
challenges for both experiment and theory and are highly sought by both
communities for their manifestation of long-range quantum entanglement and
topological excitations that may provide alternative routes to quantum
computing [5]. Experimentally, QSLs are widely sought in magnetic materials
where the underlying crystal structure introduces a geometric frustration that
prevents the simultaneous energy minimisation of every pair-wise magnetic
exchange interaction within the system [6]. However, from a theoretical
perspective, this geometric magnetic frustration often compromises the
analytical and numerical tractability of the exchange Hamiltonian of model
systems that are predicted to host QSL ground states, such that the true
nature of the QSL states predicted for many archetypal models of geometrically
frustrated magnetism is still outstanding [7, 8]. For instance, the character
of the QSL describing the ground state of the $S=\frac{1}{2}$ Heisenberg
kagome antiferromagnet is still widely debated [9, 10] and while several
materials candidates of this model have been identified [11], the presence of
disorder within their crystal structures makes it challenging to determine
unambiguously the most relevant model to describe the correct magnetic ground
state [12, 13, 14]. Accordingly, magnetic models with exactly solvable QSL
ground states that can be extended to account for the complexities of
candidate materials are highly attractive.

In this vein, the Kitaev model serves as an important and rare example of an
exactly solvable model of frustrated magnetism [15]. It is most widely
formulated in terms of a honeycomb lattice of ferromagnetically coupled
magnetic moments (see Figure 1a), whose interactions are frustrated by an
easy-axis exchange anisotropy orthogonal to the bond that connects nearest-
neighbour moments [16]. The resulting exchange Hamiltonian is commonly
represented in a cubic Cartesian reference frame as [8, 17]

$\mathcal{H}=\sum_{\langle
i,j\rangle\in{\gamma}}K^{\gamma}_{ij}S^{\gamma}_{i}S^{\gamma}_{j},$ (1)

where the summation of the spin operators, $S$, covers the $\langle
i,j\rangle$ pairs of nearest-neighbour spins along some [$\alpha$, $\beta$,
$\gamma$] $=[(y,x,z),(z,x,y),(x,y,z)]$ bond direction, and $K$ is the bond-
dependent Kitaev exchange interaction (see Figure 1a). In the quantum limit,
the Kitaev model is exactly solvable by fractionalizing the spin degrees of
freedom into Majorana fermions and gauge fluxes, yielding a QSL ground state
with characteristic topological excitations [15].

Initially, it was unclear how the bond-dependent exchange interactions of the
Kitaev model could be realised experimentally in a magnetic material until the
breakthrough development of the Jackeli-Khaliullin mechanism, a superexchange
theory that sets out the structural and electronic criteria required for the
formation of dominant Kitaev-like interactions in real materials [18]. The
Jackeli-Khaliullin mechanism demonstrates that the Kitaev model can be
realised by placing spin-orbit entangled $j_{\mathsf{eff}}=\frac{1}{2}$
moments on a two-dimensional honeycomb lattice via an edge-sharing octahedral
connectivity. The anisotropy of this spin-orbit entangled state ensures that
exchange interactions between neighbouring $j_{\mathsf{eff}}=\frac{1}{2}$
moments are extremely sensitive to their bonding geometry, giving rise to the
frustrated bond-dependent exchange interaction through a 90° metal-ligand-
metal edge-sharing geometry. This arrangement ensures that the electron
hopping processes responsible for the isotropic nearest-neighbour Heisenberg
interaction, $J$, exactly cancel out, leaving only the anisotropic Kitaev
interaction, $K$, as the dominant energy scale. While this theory was
originally developed for low-spin $d^{5}$ electronic configurations, more
recent work has extended the Jackeli-Khaliullin mechanism to include $d^{7}$
[19, 20] and $f$-electron [21] systems with strong spin-orbit coupling.

The requirements of the Jackeli-Khaliullin mechanism, however, pose
considerable synthetic challenges for realising candidate materials for the
Kitaev QSL. For the originally proposed $d^{5}$ electronic configuration, only
five transition metal ions with sufficiently strong spin-orbit coupling
required for the requisite $j_{\mathsf{eff}}=\frac{1}{2}$ state are available
for study, with, typically, chemically inaccessible oxidation states. This is
reflected in the very limited pool of candidate materials studied in the
context of the Kitaev model [22], which includes only a handful of Ru3+ and
Ir4+-containing materials, with the more synthetically challenging Re2+, Os3+,
and Rh4+-based materials remaining largely unexplored [23, 24]. Additionally,
the ideal octahedral crystal field environment and edge-sharing bonding
geometry are often broken by the structural symmetry of real materials [16],
and the extended orbital wavefunctions of their $4d$ and $5d$ transition metal
ions lead to appreciable direct nearest-neighbour exchange as well as further
neighbour interactions. Therefore, additional terms in the exchange
Hamiltonian are generally inevitable in candidate systems, which in turn are
more accurately described by an extended Kitaev model,
$\mathcal{H}_{JK\Gamma\Gamma^{\prime}}$. This includes an isotropic Heisenberg
exchange interaction, $J$, an off-diagonal bond-dependent frustrated
interaction, $\Gamma$, and an additional interaction that arises from trigonal
distortions to the perfect octahedral symmetry in the crystal field of
candidate materials, $\Gamma^{\prime}$ (see Methods). The vital role that
these additional exchange interactions play in driving candidate Kitaev
materials away from the QSL ground state is exemplified in one of the most-
studied candidate systems to date, $\alpha$-RuCl3 [25, 26, 22]. Despite
hosting all prerequisites for dominant Kitaev interactions, the ground state
of $\alpha$-RuCl3 is magnetically ordered [27, 28]. This is likely because in
addition to a dominant ferromagnetic Kitaev interaction, significant
contributions from all other terms in the extended
$\mathcal{H}_{JK\Gamma\Gamma^{\prime}}$ model along with further-neighbour
interactions are present in the exchange Hamiltonian of $\alpha$-RuCl3 [17,
29]. However, the proximity of the ground state of $\alpha$-RuCl3 to the
Kitaev QSL is still intensely debated [17, 29], and it and all other candidate
materials highlight the experimental challenges in attaining the criteria
described by the Jackeli-Khaliullin mechanism as well as the fragility of the
Kitaev QSL [22].

Figure 1: Mapping the crystal structure of RuP3SiO11 (RPSO) to the Kitaev
model on a honeycomb network. a The Kitaev model on a honeycomb network
describes a system of magnetic moments (vertices) with anisotropic bond-
dependent exchange interactions on cubic (x, y, z) axes resulting in magnetic
frustration. The basis vectors of the coordinate system used to define the
magnetic Hamiltonian are indicated by arrows. Such a model may be realised in
materials in which $j_{\mathsf{eff}}=\frac{1}{2}$ moments are connected on a
honeycomb network via octahedral edge-sharing. b In the $R\bar{3}c$ crystal
structure of RPSO, honeycomb layers of Ru3+ ions are stacked by pyrophosphate
(P2O${}_{7}^{4–}$) and pyrosilicate (Si2O${}_{7}^{6–}$) units in an ABC
stacking sequence along the $c$-axis. c The honeycomb rings of trigonally
distorted RuO6 octahedra viewed down the $c$-axis with a d pseudo-edge-sharing
connectivity through two phosphate (PO${}_{4}^{3–}$) units.

Given the experimental complexities associated with the existing set of Kitaev
materials, recent theoretical studies highlight the importance of extending
the pool of candidates to encompass framework materials with more open and
extended crystal structures than the dense inorganic materials studied to date
[22]. In particular, ab initio studies have predicted that the Jackeli-
Khaliullin mechanism will still apply via an extended pseudo-edge-sharing
superexchange pathway in a metal-organic framework solid [30, 31]. At the same
time, this approach should also increase the typical nearest-neighbour
distances within the honeycomb layers of candidate materials, thus reducing
the direct orbital overlap of $4d$ and $5d$ transition metal ions that give
rise to the non-Kitaev exchange interactions in the extended
$\mathcal{H}_{JK\Gamma\Gamma^{\prime}}$ model [31, 16]. In this way, framework
materials may provide a route to tuning the ratio of exchange interactions in
the extended Kitaev model, providing experimental access to new regions of the
magnetic phase diagram, including, possibly, the Kitaev QSL ground state.
Motivated by this hypothesis, we have identified the inorganic framework
material, RuP3SiO11 (RPSO), as an alternative candidate for exploring the
Kitaev QSL beyond $\alpha$-RuCl3. Through comprehensive structural
characterisation and resonant inelastic X-ray scattering, we show that RPSO
consists of well-separated honeycomb layers of pseudo-edge-sharing
$j_{\mathsf{eff}}=\frac{1}{2}$ Ru3+ ions. Magnetic susceptibility, specific
heat, and neutron diffraction measurements of the low-temperature magnetic
properties of RPSO confirm it adopts a magnetic ground state below
$T_{\mathsf{N}}=1.3$ K that is distinct from $\alpha$-RuCl3 due to its unique
exchange Hamiltonian, which we show contains a dominant anisotropic Kitaev
interaction through analysis of inelastic neutron scattering data. Above a
critical field $H_{\mathsf{C}}=3.55$ T, we show that $T_{\mathsf{N}}$ is
suppressed and RPSO enters a field-induced phase, highlighting its more
readily tuneable exchange interactions in comparison to $\alpha$-RuCl3 with
$H_{\mathsf{C}}\approx 8$ T [32].

##  2 Results

###  2.1 Crystal Structure of RPSO: Two-dimensional honeycomb layers with
pseudo-edge-sharing Ru3+ octahedra

To determine the crystal structure of RPSO, we have performed Rietveld
analysis of high-resolution synchrotron X-ray and neutron powder diffraction
data. This analysis confirms that RPSO adopts a trigonal $R\bar{3}c$ structure
[33] over the measured temperature range of $0.08~{}-~{}300$ K (see Methods).
The structure of RPSO is composed of quasi-two-dimensional buckled honeycomb
layers of octahedrally coordinated Ru3+ ions (see Figure 1b,c). Each corner of
the octahedron is occupied by an O2- anion from a phosphate (PO${}_{4}^{3–}$)
tetrahedron. The latter connect neighbouring Ru3+ ions within the honeycomb
layers in a pseudo-edge-sharing fashion (see Figure 1d). Each phosphate group
within the honeycomb layers is part of a larger pyrophosphate
(P2O${}_{7}^{4–}$) linker that connects the honeycomb layers along the
$c$-axis of the crystal structure in an ABC stacking sequence. In the centre
of each honeycomb ring is a pyrosilicate (Si2O${}_{7}^{6–}$) group which
stabilises the open inorganic framework structure (see Figure 1c). The
arrangement of the pyrophosphate groups within the crystal structure of RPSO
results in a subtle trigonal distortion of the Ru3+ octahedral crystal field,
with two distinct Ru-O bond lengths within each octahedron of $2.027(1)$ Å and
$2.049(1)$ Å at $300$ K.

Inspection of the crystal structure of RPSO allows us to infer the possible
magnetic exchange pathways between neighbouring Ru3+ ions. The leading
nearest-neighbour exchange interaction within the honeycomb layers, $J$,
occurs through the two equivalent Ru-O-P-O-Ru pathways of the two phosphate
groups generating the pseudo-edge-sharing connectivity of neighbouring
octahedra (see Figure 1d). The next-nearest-neighbour exchange interaction
within the honeycomb layers does not have an obvious superexchange pathway,
while the further-neighbour coupling, $J_{3}$, requires a much longer pathway
comprised of one silicate and two phosphate groups. Between the honeycomb
layers, an interlayer exchange pathway, $J_{\perp}$, runs along the
pyrophosphate linkers. Compared with $\alpha$-RuCl3, the more open framework
structure extends the intra- and interlayer exchange pathways. In RPSO, the
Ru-Ru distance across the nearest-neighbour exchange pathway is $4.800(1)$ Å
compared with $\approx 3.46$ Å in $\alpha$-RuCl3 [26, 27, 28]. The interlayer
Ru-Ru distance is also extended in RPSO to $7.172(1)$ Å from $\approx 6.01$ Å
in $\alpha$-RuCl3 [26, 27, 28]. At the same time, the nearest-neighbour
superexchange angles are comparable in both systems at $94.36(3)$°and $\approx
93.1$° in RPSO and $\alpha$-RuCl3 [26, 27, 28], respectively. Thus, we
hypothesise that the local coordination environment and connectivity of the
Ru3+ ions in RPSO should preserve the requirements for anisotropic Kitaev
exchange interactions, while the more open framework in comparison to
$\alpha$-RuCl3 should tune the hierarchy of exchange interactions within the
extended $\mathcal{H}_{JK\Gamma\Gamma^{\prime}}$ model and enhance the two-
dimensionality of the honeycomb layers.

###  2.2 Local and Collective Magnetic Properties of RPSO:
$j_{\mathsf{eff}}=1/2$ Ru3+ moment and Néel ground state

An essential ingredient to produce the anisotropic Kitaev exchange coupling on
the honeycomb network of RPSO is a $j_{\mathsf{eff}}=\frac{1}{2}$ state for
its Ru3+ ions. In the case of the $4d^{5}$ configuration of Ru3+, a
$j_{\mathsf{eff}}=\frac{1}{2}$ state emerges when a low-spin octahedral
crystal field splitting, $\Delta_{\mathsf{O}}$, creates a single electron hole
in the threefold degenerate $t_{2g}$ manifold with electron spin
$s={\frac{1}{2}}$ and an effective orbital angular momentum
$l_{\mathsf{eff}}=-1$. The spin-orbit coupling interaction, $\lambda$, mixes
these levels, creating the required $j_{\mathsf{eff}}=\frac{1}{2}$ state (see
Figure 2a). To examine the relevance of this spin-orbit entangled
$j_{\mathsf{eff}}=\frac{1}{2}$ state in RPSO, we measured its resonant
inelastic X-ray scattering (RIXS) spectrum at the Ru $L_{3}$-edge at $25$ K
(see Methods). The measured spectrum along with the calculated electronic
transitions are shown in Figure 2b. A single sharp transition is observed
below 1 eV, which bears a strong resemblance to the
$j_{\mathsf{eff}}=\frac{1}{2}$ $\rightarrow\frac{3}{2}$ excitation of
$\alpha$-RuCl3 [34], both in terms of its width—with a $\textrm{FWHM}\approx
150$ meV—and its energy, $E\approx 250$ meV. Moreover, unlike in the related
Ir4+ honeycomb compounds, (Na,Li)2IrO3 [35], no trigonal crystal field
splitting is observed within the instrumental resolution. Taken together, this
provides strong experimental evidence for the existence of a
$j_{\mathsf{eff}}=\frac{1}{2}$ state in RPSO.

Above 1 eV in the RIXS spectrum of RPSO (see Figure 2b), a series of peaks are
observed originating from excitations into the $e_{g}$ manifold of its
octahedral crystal field. Although the overall spectral form is similar to
$\alpha$-RuCl3, these excitation bands are shifted by about $250$ meV higher
in energy and are also noticeably sharper. The energy shift relative to
$\alpha$-RuCl3 is due to the larger octahedral crystal field splitting
expected for O2- compared to Cl-, while the peak sharpness likely stems from a
smaller overlap within the electron-hole continuum, i.e., a larger optical
gap. Additional insight can be gained by comparing the measured spectrum with
full atomic multiplet calculations (see Methods). In particular, the Hund’s
rule coupling, $J_{\mathsf{H}}=340$ meV, and crystal field splitting,
$\Delta_{\mathsf{O}}=2.4$ eV, determined previously for $\alpha$-RuCl3 by RIXS
[34] must be increased to $J_{\mathsf{H}}=460$ meV and
$\Delta_{\mathsf{O}}=2.8$ eV for RPSO to capture the sharper $e_{\textrm{g}}$
spectral features as well as their shift to higher energies. Such an increase
in $J_{\mathsf{H}}$ can arise from a weaker screening effect in more
insulating compounds, for example, as has been observed in K2RuCl6 [36] and
RuX3 (X $=$ Cl, Br, I) [37]. Despite the differences in $\Delta_{\mathsf{O}}$
and $J_{\mathsf{H}}$, the spin-orbit coupling in RPSO, $\lambda=130$ meV, is
comparable to that reported for $\alpha$-RuCl3, $\lambda=150$ meV [34], which
highlights the significant role of spin-orbit coupling in the physics of RPSO.

Figure 2: Confirming the $j_{\mathsf{eff}}=\frac{1}{2}$ state of Ru3+ in RPSO
by resonant inelastic X-ray scattering (RIXS). a In an octahedral crystal
field environment, the $d$-orbitals of a Ru3+ 4d5 ion are split into a
$t_{2g}$ manifold with a single electron hole and an empty $e_{g}$ manifold
separated by the octahedral crystal field splitting parameter,
$\Delta_{\mathsf{O}}$. In the presence of spin-orbit coupling, $\lambda$, the
$t_{2g}$ manifold can be further split, creating the possibility of a spin-
orbit-entangled $j_{\mathsf{eff}}=\frac{1}{2}$ moment for Ru3+. If such
moments are coupled by edge-sharing octahedral connectivity, then anisotropic
Kitaev-type exchange interactions may dominate between them. b RIXS data
collected for RPSO at $25$ K (IRIXS, DESY) confirm the
$j_{\mathsf{eff}}=\frac{1}{2}$ state of the Ru3+ ions in this system through
the single sharp transition observed below $1$ eV. The RIXS spectrum can be
successfully simulated using full atomic multiplet calculations, from which
estimates for the octahedral crystal field splitting, $\Delta_{\mathsf{O}}$,
Hund’s coupling, $J_{\mathsf{H}}$, and spin-orbit coupling, $\lambda$, of RPSO
can be extracted.

Having established the relevance of the $j_{\mathsf{eff}}=\frac{1}{2}$ state
to the local magnetic properties of RPSO, we now turn to the collective
magnetic properties and elucidation of its magnetic ground state. Curie-Weiss
fitting the inverse magnetic susceptibility of RPSO (see Figure 3a, Methods)
between $100~{}-~{}300$ K yields a Weiss constant,
$\theta_{\mathsf{CW}}=-5.70(1)$ K, a Curie constant, $C=0.33(1)$ emu K mol-1,
and a temperature-independent term,
$\chi_{\textrm{0}}=2.90(1)\times~{}10^{-4}$ emu mol-1. This implies dominant
antiferromagnetic interactions between neighbouring Ru3+ moments in RPSO, and
an effective moment $\mu_{\mathsf{eff}}=1.63(1)~{}\mu_{\mathsf{B}}$ per Ru3+
ion consistent with the $j_{\mathsf{eff}}=\frac{1}{2}$ state confirmed by
RIXS. The temperature-independent term, $\chi_{\textrm{0}}$, most likely stems
from the Pauli paramagnetism of the small Ru and RuO2 phases identified in the
sample by powder diffraction (see Methods). Efforts to improve the Curie-Weiss
analysis by incorporating temperature-dependent magnetic moments—often a more
suitable approach for strongly spin-orbit coupled systems [38]—proved
inconclusive and most likely requires single-crystal magnetometry data. Below
$15$ K, the development of short-range magnetic correlations is evidenced in
the magnetic susceptibility data through the deviation from Curie-Weiss
behaviour followed by a cusp just above $1$ K that indicates the onset of
long-range magnetic order. This behaviour is also observed in the low-
temperature specific heat data of RPSO (see Figure 3b, Methods)—where the
lattice contribution to the total heat capacity will be minimal—in which a
broad peak near 15 K is followed by a sharp $\lambda$-type anomaly at
$T_{\mathsf{N}}=1.3$ K.

Two similar features have been observed previously in specific heat
measurements of polycrystalline samples of $\alpha$-RuCl3 [39]. These have
been attributed to the onset of two magnetically ordered regimes that arise
due to the presence of stacking faults between the van der Waals layers of
$\alpha$-RuCl3 [27, 39]. In the case of RPSO, the pyrophosphate and
pyrosilicate linkers that pillar the honeycomb layers will increase the energy
barrier to stacking fault formation compared to the weaker intermolecular
forces between the layers of $\alpha$-RuCl3, and indeed, there is no evidence
of stacking faults from the peak shape and intensity of high-resolution
synchrotron powder X-ray diffraction data (see Supporting Information).
Alternatively, Monte-Carlo simulations have indicated that such a two-step
magnetic entropy release for the extended
$\mathcal{H}_{JK\Gamma\Gamma^{\prime}}$ model is evidence of spin
fractionalisation [40]. The temperatures at which these features appear,
however, are inconsistent with the energy scales of the
$\mathcal{H}_{JK\Gamma\Gamma^{\prime}}$ model in RPSO obtained through
analysis of inelastic neutron scattering data (see Section 2.3). Thus, we
hypothesise that the broad feature near 15 K in the specific heat of RPSO
simply reflects the onset of the correlated paramagnetic regime, rather than a
distinct long-range ordered magnetic phase.

To determine the nature of the magnetic ground state of RPSO below
$T_{\mathsf{N}}=1.3$ K, neutron magnetic scattering was isolated by
subtracting neutron powder diffraction data collected above and below
$T_{\mathsf{N}}$ ($\Delta T=0.08$ K $-~{}2.5$ K, see Methods). This
temperature subtraction (see Figure 3c) reveals three magnetic Bragg peaks at
$d$-spacings corresponding to the (012), (104), and (018) reflections of the
$R\bar{3}c$ crystal structure, which indicates a ground state magnetic
structure described by the commensurate propagation vector,
$\textbf{{k}}=(0,0,0)$. Symmetry analysis (see Methods) results in four
maximal magnetic space groups with irreducible representations that are
compatible with this propagation vector and the underlying crystal structure
of RPSO: $m\Gamma_{1+}$, $m\Gamma_{1-}$, $m\Gamma_{2+}$ and $m\Gamma_{2-}$ in
Miller-Love notation [41]. Here, $m\Gamma_{2+}$ corresponds to a
ferromagnetically ordered structure, while the $m\Gamma_{2-}$, $m\Gamma_{1+}$
and $m\Gamma_{1-}$ modes describe C, A and G-type Néel ordered states,
respectively. We also considered the magnetic subgroups defined by the
irreducible representations $m\Gamma_{3+}$ and $m\Gamma_{3-}$. Refining each
of these models against the temperature-subtracted data reveals that only the
model corresponding to the G-type Néel order ($m\Gamma_{1-}$ representation)
is consistent with the observed magnetic diffraction (see Figure 3c). Thus,
the magnetic ground state of RPSO is composed of antiferromagnetic honeycomb
layers of Ru3+ ions that couple antiferromagnetically along the $c$-axis
(G-type Néel order, see Figure 3d). The ordered magnetic moment obtained from
this magnetic structure refinement,
$\mu_{\mathsf{ord}}=0.35(1)~{}\mu_{\mathsf{B}}$ per Ru3+, is reduced from the
full $j_{\mathsf{eff}}=\frac{1}{2}$ ordered moment of $1~{}\mu_{\mathsf{B}}$
and the calculated ordered moment for the $S=\frac{1}{2}$ honeycomb Heisenberg
antiferromagnet with $\mu_{\mathsf{ord}}~{}\approx~{}0.55~{}\mu_{\mathsf{B}}$
[42]. Such a reduced ordered moment is characteristic of low-dimensional
magnetic materials and implies the presence of frustrated exchange
interactions. It is also consistent with the observed ordered moment in
$\alpha$-RuCl3, which is typically reported in the range of
$0.3~{}-~{}0.7~{}\mu_{\mathsf{B}}$ per Ru3+ ion [43, 27].

The significance of the G-type Néel ordered ground state of RPSO is that it
provides experimental access to an otherwise unexplored region of the magnetic
phase diagram of the extended $\mathcal{H}_{JK\Gamma\Gamma^{\prime}}$ Kitaev
model. Indeed, all other experimental realisations of Ru3+ and Ir4+ honeycomb
magnets studied in the context of the Kitaev QSL and that undergo a magnetic
phase transition to a long-range ordered ground state adopt either zig-zag or
incommensurate spin spiral magnetic structures at low temperatures [22]. This
implies that RPSO has a unique hierarchy of exchange interactions within the
$\mathcal{H}_{JK\Gamma\Gamma^{\prime}}$ model that gives rise to its distinct
magnetic ground state. The delicate balance of competing interactions that
yields the rich magnetic phase diagram for this model means that candidate
materials may be tuned from one phase to another by an external perturbation,
such as applied pressure or magnetic field [44]. For instance, in the case of
$\alpha$-RuCl3, the zig-zig order within the magnetic ground is suppressed
upon application of a critical field $H_{\mathsf{C}}\approx 8$ T, which has
been attributed to the formation of a field-induced quantum critical phase
[32, 45, 46].

To explore the tunability of the magnetic ground state of RPSO, we have
measured the field dependence of its magnetic susceptibility $\chi(T;H)$ and
isothermal magnetisation $M(H;T)$ below $T_{\mathsf{N}}$ (see Methods). Upon
increasing applied field strength, the cusp in the magnetic susceptibility of
RPSO—that indicates the onset of the long-range Néel order—broadens and shifts
lower in temperature up to an applied field of $3.5$ T, above which
$T_{\mathsf{N}}$ is no longer observable down to $0.5$ K (see Supporting
Information). Below $T_{\mathsf{N}}$, the isothermal magnetisation increases
linearly with the applied field up to $2$ T, beyond which the rate of increase
in magnetisation becomes steeper before reaching a critical field,
$H_{\mathsf{C}}=3.55$ T. Such a field dependence of the magnetisation is
typical of a field-induced phase transition, which can be more easily observed
in the field derivative of the magnetisation (see Figure 4a). The resulting
field-temperature phase diagram (see Figure 4b) reveals the reduction in
$T_{\mathsf{N}}$ upon increasing the applied field until a critical
paramagnetic phase is reached above $H_{\mathsf{C}}=3.55$ T. Understanding the
nature of this field-induced phase of RPSO—whether it is a simple field-
polarised or a quantum paramagnetic state—requires further investigation of
single-crystal samples, but the field-temperature phase diagram is again
reminiscent of $\alpha$-RuCl3, albeit with a lower critical field, which stems
from the weaker exchange interactions in RPSO.

Figure 3: Elucidating Néel order in RPSO below
$\boldsymbol{T_{\mathsf{N}}=1.3}$ K. a The magnetic susceptibility data
($\chi_{\mathsf{m}}$, left axis) measured for RPSO between $0.4~{}-~{}300$ K
in an applied field of $0.1$ T after zero-field cooling shows a cusp just
above $1$ K, indicating the onset of a magnetic phase transition. Curie-Weiss
fitting the inverse magnetic susceptibility ($\chi_{\mathsf{m}}^{–1}$, right
axis) between $100~{}-~{}300$ K gives a Weiss constant,
$\theta_{\mathsf{CW}}=-5.70(1)$ K, a Curie constant, $C=0.33(1)$ emu K mol-1,
and a temperature-independent term, $\chi_{0}=2.90(1)\times 10^{–4}$ emu
mol-1. b The low-temperature zero-field specific heat ($C_{p}$) of RPSO
confirms the magnetic phase transition at $T_{\mathsf{N}}=1.3$ K with a sharp
$\lambda$-type anomaly at this temperature. The broader feature at $T\approx
15$ K likely reflects the development of short-range magnetic correlations. c
Temperature-subtracted powder neutron diffraction data (WISH, ISIS) reveal
magnetic Bragg peaks below $T_{\mathsf{N}}$ that can be indexed with a
$\textbf{{k}}=(0,0,0)$ propagation vector. Of the four magnetic models
compatible with this k and the symmetry of the crystal structure, the best
magnetic Rietveld fit (shown) is obtained with the $m\Gamma_{1-}$ irreducible
representation ($\chi^{2}=1.07$ and $R_{\mathsf{mag}}=10.2$%). This model
corresponds to the magnetic space group
$R\bar{3}^{{}^{\prime}}c^{{}^{\prime}}$ and a d G-type antiferromagnetic order
of the Ru3+ moments in the honeycomb layers of RPSO with an ordered moment,
$\mu_{\mathsf{ord}}=0.35(1)~{}\mu_{\mathsf{B}}$. Figure 4: Supressing Néel
order in RPSO upon application of a magnetic field. a The field derivative of
the magnetisation isotherms shows the development of the critical field below
$T_{\mathsf{N}}\approx 1.3$ K. b The temperature-field phase diagram of RPSO
mapped through magnetic susceptibility and isothermal magnetisation
measurements reveals the suppression of the zero-field $T_{\mathsf{N}}=1.3$ K
upon application of a magnetic field. Above a critical field of
$H_{\mathsf{C}}=3.55$ T, $T_{\mathsf{N}}$ appears completely suppressed within
the temperature range of the measurement.

###  2.3 Exchange Hamiltonian of RPSO: Kitaev interactions from inelastic
neutron scattering

To understand the origin of the G-type Néel ordered ground state of RPSO, we
need to establish the interactions within the underlying exchange Hamiltonian
(see Methods). Experimentally, this can be achieved through the analysis of
the dynamical structure factor of RPSO, $S\text{(}Q,\Delta
E=\hbar\omega\text{)}_{\textrm{exp}}$, measured by inelastic neutron
scattering (INS, see Methods). Figure 5a shows the $S\text{(}Q,\Delta
E=\hbar\omega\text{)}_{\textrm{exp}}$ measured for RPSO below $T_{\mathsf{N}}$
at $T=0.08$ K with an incident neutron energy $E_{\mathsf{i}}=1.78$ meV. The
magnetic INS spectrum is gapped, with a gap $\Delta\approx 0.1$ meV, and two
clear bands of magnetic scattering intensity are observed with bandwidths
$0.1~{}-~{}0.4$ meV and $0.42~{}-~{}0.82$ meV, respectively. The sharply
dispersing feature in Figure 5a above $0.8$ meV is the roton excitation of
superfluid 4He arising from the He exchange gas loaded in the sample can to
cool the sample (see Methods). The intensity seen at low-$\lvert Q\rvert$ and
small-$\Delta E$ in Figure 5a is also spurious, as verified by the observation
that its $\lvert Q\rvert$-dependence varies with $E_{\mathsf{i}}$. Overall,
the form of the magnetic excitation spectrum of RPSO below
$T_{\mathsf{N}}$—particularly that the spectrum is gapped—indicates a strongly
anisotropic exchange Hamiltonian.

To establish the relevance of $\mathcal{H}_{JK\Gamma\Gamma^{\prime}}$ in RPSO,
we performed a grid search using linear spin wave theory (LSWT), in which we
compared $S\text{(}Q,\Delta E=\hbar\omega\text{)}_{\textrm{exp}}$ to solutions
of the extended Kitaev $\mathcal{H}_{JK\Gamma\Gamma^{\prime}}$ model within
the region of the phase diagram where the experimentally observed Néel ground
state is stable (see Methods). The resulting sets of exchange parameters that
gave the best fit to the data in the initial grid search were then further
optimised using a simulated annealing algorithm (see Methods). This
optimisation of the $\mathcal{H}_{JK\Gamma\Gamma^{\prime}}$ model yields two
regions of the phase diagram for which the fits to the experimental data are
indistinguishable, which have either a dominant ferromagnetic ($K<0$) or
antiferromagnetic ($K>0$) Kitaev exchange coupling. However, these two regions
are physically equivalent and are related by the self-duality of the extended
Kitaev $\mathcal{H}_{JK\Gamma\Gamma^{\prime}}$ model when defined in the cubic
axes of the Kitaev framework and are introduced by a global $\pi$ rotation
about the crystallographic $c$-axis (see Supporting Information) [47, 17].
Thus, we find five solutions to the $\mathcal{H}_{JK\Gamma\Gamma^{\prime}}$
model that give the lowest $\chi^{2}$ against the experimental data and that
are well separated in $\chi^{2}$ from the next set of solutions (see Table 1,
Supporting Information). Figure 5b shows the $S\text{(}Q,\Delta
E=\hbar\omega\text{)}_{\textrm{calc}}$ simulated by LSWT for the set of
exchange parameters in Solution 1, and Figures 5c–f show representative
comparisons of this solution to several $Q$-integrated cuts to the data.
Within the ferromagnetic $K$ region, all five of the lowest $\chi^{2}$ LSWT
solutions of the $\mathcal{H}_{JK\Gamma\Gamma^{\prime}}$ model unveil two
dominant anisotropic exchange interactions for RPSO: a ferromagnetic Kitaev
exchange interaction $K$, which is approximately equal in strength to an
antiferromagnetic $\Gamma$. In addition to these two bond-dependent,
frustrated exchange interactions, we also find an antiferromagnetic isotropic
nearest-neighbour exchange interaction, $J~{}\approx-2/3~{}K$, and a
ferromagnetic $\Gamma^{\prime}$ that is consistently smaller than the other
three parameters. Notably, the latter aligns with our analysis of the RIXS
spectrum and supporting DFT calculations111The trigonal field splitting of the
$t_{2g}$ manifold is obtained as $\Delta_{\mathsf{O}}=0.06$ eV in density-
functional-theory calculations. For full details of density-functional theory
calculations, see Supporting Information., which all show that the trigonal
splitting of the octahedral crystal field in RPSO is small.

Further efforts to distinguish between the optimised parameters of Solutions 1
– 5 using experimental data did not yield a single, unique solution. When
comparing the predicted critical field between the Néel ordered ground state
and field-induced phase with the experimental $H_{\mathsf{C}}=3.55$ T, all
solution sets consistently yield $H_{\mathsf{C,calc}}=3.8~{}-~{}4.1$ T.
Additionally, calculating the mean-field Weiss constant [38] results in
$\theta_{\mathsf{CW}}\approx-1$ K for all solutions, which makes a direct
comparison between the Curie-Weiss analysis of the magnetic susceptibility
data and LSWT challenging. Thus, while the calculated
$\mathcal{H}_{JK\Gamma\Gamma^{\prime}}$ model successfully captures many of
the prominent features in $S(Q,\Delta E=\hbar\omega)_{\textrm{exp}}$ (see
Figure 5), the observed discrepancy with the experimental intensities may
suggest further exchange interactions are required for a full microscopic
description of RPSO. Indeed, the Ru-Ru bond symmetry of further near-neighbour
couplings in RPSO allows for anisotropic exchange tensors, which can add up to
an exchange Hamiltonian with at least 18 independent parameters that cannot be
fit reasonably to the current powder-averaged INS data. On the other hand,
this fitting discrepancy may also stem from the semi-classical formalism of
LSWT, which underestimates both quasi-particle interactions [48, 49] and other
quantum effects [50], and thus creates a bottleneck for the complete analysis
of INS data of low-dimensional, frustrated quantum magnets. Multi-magnon
interactions and the resulting reduction in the single magnon lifetime
generally result in a characteristic broadening of spectral lines. These
challenges have also been highlighted in the context of $\alpha$-RuCl3 [51].
Hence, in the case of RPSO, where the two dominant exchange interactions are
strongly anisotropic—combined with the small ordered moment and quantum
spin—we expect that the mismatch in fitting intensity arises from the
limitations inherent to LSWT. A definitive determination of the exchange
Hamiltonian, therefore, demands future developments in experiment and theory,
including single-crystal inelastic neutron scattering measurements with
neutron polarisation analysis and a more rigorous treatment of the resulting
excitations, including magnon-interactions and other quantum effects.

###  2.4 Discussion and Outlook

While the energy scales of the exchange parameters determined for RPSO are at
least an order of magnitude smaller than those observed in $\alpha$-RuCl3 [17,
16, 52], considering their relative magnitudes allows for a useful comparison
of the two systems. Table 1 summarises a representative set of
$\mathcal{H}_{JK\Gamma\Gamma^{\prime}}$ parameters normalised to the
ferromagnetic Kitaev exchange for $\alpha$-RuCl3 and RPSO. We note that Table
1 is not exhaustive, as the full microscopic description of $\alpha$-RuCl3 is
still heavily debated with at least $20$ estimates of the relevant exchange
parameters (e.g., see Table 1 in [17, 29]). However, overall, the absolute
relative magnitudes of the $\mathcal{H}_{JK\Gamma\Gamma^{\prime}}$ parameters
appear consistent across both systems, with dominant $K$ and $\Gamma$ and
smaller $J$ and $\Gamma^{\prime}$. Interestingly, the relative magnitudes of
the nearest-neighbour Heisenberg exchange, $J$, are similar in both RPSO and
$\alpha$-RuCl3. This is surprising given that one might expect the strength of
this interaction to be relatively reduced by the more open framework structure
of RPSO [31] and perhaps highlights the more complex influence of the multi-
atom ligand facilitating the pseudo-edge-sharing superexchange on the relative
magnitude of $J$. Otherwise, in the analysis of $S\text{(}Q,\Delta
E=\hbar\omega\text{)}_{\textrm{exp}}$ for RPSO, the further-near-neighbour
Heisenberg exchange interaction, $J_{3}$, appears to be negligible, which is
consistent with DFT (see Supporting Information). This is in stark contrast to
$\alpha$-RuCl3 and other Kitaev-related materials where $J_{3}$ is at least
$\lvert K\rvert/4$ [53, 16, 29, 17], and thus appears to be an important
consequence of the more open framework structure of RPSO.

Collectively, the exchange parameters determined here for RPSO align with the
classically calculated phase diagram of the extended
$\mathcal{H}_{JK\Gamma\Gamma^{\prime}}$ Kitaev model [54, 29], forming a line
of solutions that closely border the predicted stripy antiferromagnetic and
the experimentally observed incommensurate spin-spiral [22] magnetic ground
states. To the best of our knowledge, this makes RPSO the first material
realisation of the $\mathcal{H}_{JK\Gamma\Gamma^{\prime}}$ model to fall
within the Néel ordered region of the phase diagram and the second Ru3+-based
material relevant to this model beyond the $\alpha$-Ru$X_{3}$ family ($X=$ Cl,
Br, I) [55]. Perhaps most importantly, RPSO also serves as an experimental
proof-of-concept that anisotropic Kitaev interactions can be transmitted
through a complex extended pseudo-edge-sharing superexchange pathway,
corroborating the ab initio predictions of Yamada et al. [30, 31].
Consequently, this work significantly broadens the pool of materials suitable
for exploring the Kitaev QSL. Coupled with other recent ab initio studies
mapping the Kitaev model to more synthetically accessible $d^{7}$ systems [20,
19], this opens the door to an extensive experimental exploration of the
$\mathcal{H}_{JK\Gamma\Gamma^{\prime}}$ phase diagram in framework materials.
Moving forward, it will also be important to investigate why the frustrated
bond-dependent XY-type interaction, $\Gamma$, also appears to play a dominant
role in the exchange Hamiltonian of a multi-atom superexchange pathway. This
might be fruitful for future theoretical and experimental investigations as
further framework materials are identified as candidate Kitaev systems.
Nevertheless, in the case of RPSO, the extended superexchange pathway appears
to have significantly diminished the influence of the further-near-neighbour
Heisenberg interaction, $J_{3}$, which is a significant energy scale in all
other Kitaev-related materials studied to date [22, 29]. This—in addition to
its relatively weaker exchange interactions—makes RPSO an ideal system in
which to examine the effect of external perturbations, such as applied strain,
pressure, and magnetic field, that may ultimately lead us to the Kitaev QSL in
this inorganic framework solid.

Table 1: Exchange parameters (in meV) of the lowest $\chi^{2}$ solutions of the $\mathcal{H}_{JK\Gamma\Gamma^{\prime}}$ model obtained for RPSO in this work by fitting inelastic neutron scattering data to linear spin wave theory. For comparison, the range of $-K$ normalised exchange parameters reported for $\alpha$-RuCl3 [17, 52, 16] have also been included. RPSO | $J$ | $K$ | $\Gamma$ | $\Gamma^{\prime}$ | $\chi^{2}$
---|---|---|---|---|---
Solution 1 | $0.32$ | $-0.54$ | $0.44$ | $-0.12$ | $67.7$
Solution 2 | $0.23$ | $-0.31$ | $0.38$ | $-0.08$ | $68.9$
Solution 3 | $0.16$ | $-0.15$ | $0.33$ | $-0.04$ | $69.7$
Solution 4 | $0.25$ | $-0.39$ | $0.41$ | $-0.08$ | $69.7$
Solution 5 | $0.37$ | $-0.65$ | $0.49$ | $-0.14$ | $70.5$
Normalised | $J$ | $K$ | $\Gamma$ | $\Gamma^{\prime}$ | $J_{3}$
RPSO | $0.60-1.10$ | $-1$ | $0.75-2.20$ | $-0.22--0.27$ | –
$\alpha$-RuCl3 [17, 16, 52] | $-0.30--0.53$ | $-1$ | $0.41-0.85$ | $0.22-0.52$ | $0.25-0.50$
Figure 5: Determining the exchange Hamiltonian of RPSO through inelastic
neutron scattering a Experimental dynamical structure factor,
$S\text{(}Q,\Delta E=\hbar\omega\text{)}_{\textrm{exp}}$, measured at 0.08 K
with neutrons of incident energy $E_{\mathsf{i}}=1.78$ meV (LET, ISIS). b
Powder-averaged and energy-convoluted $S\text{(}Q,\Delta
E=\hbar\omega\text{)}_{\textrm{calc}}$ simulated using linear spin wave theory
for the exchange Hamiltonian, $\mathcal{H}_{JK\Gamma\Gamma^{\prime}}$, with
$J=0.32$ meV, $K=-0.54$ meV, $\Gamma=0.44$ meV, and $\Gamma^{\prime}=-0.12$
meV (Solution 1, Table 1). c-f $\Delta E$-integrated cuts (blue circles)
fitted to $S\text{(}Q,\Delta E=\hbar\omega\text{)}_{\textrm{calc}}$ (red
lines).

##  3 Methods

###  3.1 Synthesis of RPSO

Polycrystalline samples of RuP3SiO11 (RPSO) were prepared via a three-step
synthesis. In the first step, a precursor of RPSO, H2RuP3O10, was synthesised
via a modified procedure reported in the literature [33]. In a typical
reaction, RuCl${}_{3}\cdot$xH2O (Sigma-Aldrich, 99.98%) and H3PO4 (Sigma-
Aldrich, 85 wt.% in H2O, 99.99% trace metals basis) were combined in a round
bottom flask in a $1:4$ molar ratio, respectively, and heated in air at 350 K
for 6 hours under constant stirring. The resulting viscous brown solution was
transported to an alumina crucible and heated at 1 K min–1 to 650 K under
flowing Ar gas. After 7 days of heating, the furnace was cooled to room
temperature at a rate of 1 K min–1. This resulted in a light brown, water
insoluble, and glass-like amorphous product, which has been reported as
H2RuP3O10 [56]. In the second step, a water-washed pressed-powder pellet of
H2RuP3O10 was heated for 7 days in an evacuated sealed quartz ampoule refilled
with 425 mbar of Ar gas to form a pale orange powder of Ru(PO3)3. Temperature
control at this stage of the reaction is crucial, as Ru(PO3)3 can adopt three
different polymorphs [56] of which the monoclinic cyclo-hexaphosphate
(Ru2P6O18)—prepared by slow heating (0.5 K min–1) to 823 K—reacts to form
RPSO. Finally, in the third step, a stoichiometric mixture of Ru(PO3)3 and
SiO2 (Alfa Aesar, 99.99%) was ground for 30 minutes, pressed into a pellet and
sealed in an evacuated quartz ampoule refilled with 425 mbar of Ar gas without
a crucible. This mixture was heated at 3 K min–1 to 1233 K for 3 days before
cooling to room temperature. The resulting final product of RPSO is yellow
coloured, water insoluble and air stable for at least a month.

###  3.2 Crystal and Magnetic Structure Determination

High-resolution synchrotron powder X-ray diffraction data were collected on
the I11 beamline at the Diamond Light Source using the MAC detector. The
sample was densely packed in a 0.5 mm borosilicate capillary by sonication and
diffraction data were measured at 300 K with an incident wavelength of
$\lambda=0.826015(5)$ Å. Neutron powder diffraction data were collected on the
time-of-flight diffractometer WISH at the ISIS Neutron and Muon Source. 1 g of
RPSO powder was packed in a cylindrical copper can and diffraction data were
measured at 0.08 K and 2.5 K using a dilution refrigerator. Crystal structure
Rietveld refinements to 300 K (X-ray), 2.5 K and 0.08 K (neutron) diffraction
data were performed using the GSASII [57] software package (Fig. 6, Supporting
Information). For the Rietveld analysis of data collected at 300 K, lattice
parameters, atomic positions and isotropic thermal parameters were varied
within the $R\bar{3}c$ structural model, yielding the refined structure
summarised in Table 2 (Supporting Information). Additional phases of RuO2 and
Ru metal were found to account for approximately 4.6% of the sample mass. For
the Rietveld fit to 2.5 K data, lattice parameters, atomic positions and
isotropic thermal parameters were varied, with additional Al and Cu phases
modelled using the Le Bail method to account for the scattering from the
dilution fridge insert and sample can used (Table 3), Supporting Information).
At 0.08 K, all structural and instrumental parameters were fixed to their
values obtained from the 2.5 K fit, with the exception of the lattice
parameters, which contracted isotropically upon cooling. Representational
analysis and magnetic structure refinements performed against temperature-
subtracted ($\Delta T=0.08$ K $-~{}2.5$ K) neutron diffraction data utilised
the MAXMAGN [58] and FullProf suite [59] softwares, respectively. The Ru3+
form factor used was interpolated from data presented in [60].

###  3.3 Magnetic Susceptibility and Specific Heat Measurements

Temperature-dependent magnetic susceptibility data were measured in DC mode
for a 28 mg sample of RPSO in a Quantum Design MPMS3 with a low-temperature
3He insert. Data were collected in an applied field of 0.1 T between
$0.4~{}-~{}300$ K in both zero-field cooled and field-cooled protocols. The
sample was contained within a gelatine capsule packed tightly with Teflon tape
and held in a plastic straw. Additional temperature-dependent magnetic
susceptibility measurements were carried out within the temperature range of
$0.4~{}-~{}1.78$ K. These measurements were taken in $1$ T steps of applied
field between $1~{}-~{}5$ T and additionally at $3.25$ T, $3.5$ T, $3.75$ T
and $4.25$ T. Isothermal magnetisation data were measured over the temperature
range of $0.40~{}-~{}1.8$ K, with measurements taken in increments of $0.06$
K, over an applied magnetic field range of $-7~{}-~{}7$ T. Specific heat data
were measured on a Quantum Design Physical Property Measurement System (PPMS).
2.6 mg of RPSO was pressed into a pellet, mounted onto a puck using Apiezon
N-grease and 300 data points were recorded in zero applied magnetic field
between $2~{}-~{}300$ K using a cryostat and between $0.1~{}-~{}4$ K using a
dilution refrigerator. The measurement at each temperature point was repeated
twice to obtain an average.

###  3.4 Resonant Inelastic X-ray Scattering

Resonant inelastic X-ray scattering (RIXS) data were collected on the IRIXS
instrument at beamline P01 of PETRA III synchrotron radiation source at DESY.
RPSO powder was pressed into a pellet and mounted on a copper sample holder.
The position of the zero-energy-loss spectral line was determined by measuring
non-resonant spectra from glue deposited next to the sample. Data were
collected at 25 K at the Ru $L_{3}$-edge, where the overall energy resolution
(FWHM) of the IRIXS spectrometer is 75 meV. To model the measured RIXS
spectrum of RPSO, full atomic multiplet calculations were performed using the
Quanty code [61] assuming a $4d^{5}$ configuration for Ru3+. Spin-orbit
coupling ($\lambda$), Hund’s rule coupling ($J_{\mathsf{H}}$), and octahedral
crystal field splitting ($\Delta_{\textrm{O}}$) were included in the model
Hamiltonian for RPSO, similar to the fitting approach developed for recent
measurements of $\alpha$-RuX3 (X = Cl, Br, I) [62, 37]. These three parameters
were then refined to best fit the experimental data.

###  3.5 Defining the Exchange Hamiltonian

The extended Kitaev model, $\mathcal{H}_{JK\Gamma\Gamma^{\prime}}$, used
widely to describe the relevant exchange Hamiltonian of candidate Kitaev
materials is defined as

$\displaystyle\mathcal{H}_{JK\Gamma\Gamma^{\prime}}=$
$\displaystyle\sum_{\langle
ij\rangle\in\gamma}[J\mathbf{S}_{i}\cdot\mathbf{S}_{j}+KS_{i}^{\gamma}S_{j}^{\gamma}+\Gamma(S_{i}^{\alpha}S_{j}^{\beta}+S_{i}^{\beta}S_{j}^{\alpha})$
$\displaystyle+\Gamma^{\prime}(S_{i}^{\alpha}S_{j}^{\gamma}+S_{i}^{\gamma}S_{j}^{\alpha})\pm\Gamma_{ij}^{\prime}(S_{i}^{\beta}S_{j}^{\gamma}+S_{i}^{\gamma}S_{j}^{\beta})]+\sum_{\langle\langle\langle
ij\rangle\rangle\rangle}J_{3}\mathbf{S}_{i}\cdot\mathbf{S}_{j}.$ (2)

Here, $J$ is the isotropic Heisenberg exchange interaction, $\Gamma$—which is
symmetry allowed in the Kitaev model—is an off-diagonal bond-dependent
frustrated XY-type interaction that depends on a complex combination of metal-
metal and metal-ligand assisted electron hoppings, $\Gamma’$ is dependent on
the local symmetry at the metal ion site and arises from trigonal distortions
to the perfect octahedral symmetry in the crystal field of candidate materials
[63], and $J_{3}$ is the third-nearest-neighbour Heisenberg exchange
interaction. The $\pm$ sign emphasises that the bond-dependent sign structure
of $\Gamma^{\prime}$ varies depending on the frame of reference used to define
the model [29].

The standard parameterisation of the extended Kitaev model in Eqn. 2 assumes
$C_{2}$ symmetry at the Ru-Ru bond centre. In RPSO, the $\bar{1}$ point
symmetry at the Ru-Ru bond centre allows for two more parameters in the
exchange tensor, $\xi$ and $\zeta$. The resulting nearest-neighbour exchange
tensor, $\mathcal{J}_{a}^{\gamma}$, of the exchange Hamiltonian is thus
populated with six nonzero parameters such that

$\mathcal{J}_{a}^{x}=\begin{pmatrix}J+K&-\Gamma^{\prime}-\zeta&-\Gamma^{\prime}+\zeta\\\
-\Gamma^{\prime}-\zeta&J-\xi&\Gamma\\\
-\Gamma^{\prime}+\zeta&\Gamma&J+\xi\end{pmatrix},~{}\mathcal{J}_{a}^{y}=\begin{pmatrix}J+\xi&-\Gamma^{\prime}+\zeta&\Gamma\\\
-\Gamma^{\prime}+\zeta&J+K&-\Gamma^{\prime}-\zeta\\\
\Gamma&-\Gamma^{\prime}-\zeta&J-\xi\end{pmatrix},$

$\mathcal{J}_{a}^{z}=\begin{pmatrix}J-\xi&\Gamma&-\Gamma^{\prime}-\zeta\\\
\Gamma&J+\xi&-\Gamma^{\prime}+\zeta\\\
-\Gamma^{\prime}-\zeta&-\Gamma^{\prime}+\zeta&J+K\end{pmatrix},$

for spins that are projected on axes with $\gamma$ defining the local basis
vectors, given in the crystallographic coordinates:

$\begin{pmatrix}\textbf{x}\\\ \textbf{y}\\\
\textbf{z}\end{pmatrix}=\begin{pmatrix}0.224&0.112&-0.029\\\
-0.112&-0.224&-0.029\\\ -0.112&0.112&-0.029\end{pmatrix}.$

The exchange tensors furthermore transform with the threefold rotation axis of
the $R\bar{3}c$ space group between bonds in the $ab$-plane, and the $c$-glide
operation between adjacent planes

$\mathcal{J}_{b}^{x}=\begin{pmatrix}J+K&-\Gamma^{\prime}+\zeta&-\Gamma^{\prime}-\zeta\\\
-\Gamma^{\prime}+\zeta&J+\xi&\Gamma\\\
-\Gamma^{\prime}-\zeta&\Gamma&J-\xi\end{pmatrix},~{}\mathcal{J}_{b}^{y}=\begin{pmatrix}J-\xi&-\Gamma^{\prime}-\zeta&\Gamma\\\
-\Gamma^{\prime}-\zeta&J+K&-\Gamma^{\prime}+\zeta\\\
\Gamma&-\Gamma^{\prime}+\zeta&J+\xi\end{pmatrix},$

$\mathcal{J}_{b}^{z}=\begin{pmatrix}J+\xi&\Gamma&-\Gamma^{\prime}+\zeta\\\
\Gamma&J-\xi&-\Gamma^{\prime}-\zeta\\\
-\Gamma^{\prime}+\zeta&-\Gamma^{\prime}-\zeta&J+K\end{pmatrix}.$

Alternatively, $\mathcal{H}_{JK\Gamma\Gamma^{\prime}}$ can be rewritten in a
basis aligned with the crystallographic axes [44, 17] of the $R\bar{3}c$ space
group such that

$\displaystyle\begin{split}\mathcal{H}_{XXZ}={}&\sum_{\langle
ij\rangle}\\{J_{ij}^{xy}(S_{i}^{x}S_{j}^{x}+S_{i}^{y}S_{j}^{y})+J_{ij}^{z}S_{i}^{z}S_{j}^{z}\\\
&-2J_{i,j}^{\pm\pm}[(S_{i}^{x}S_{j}^{x}+S_{i}^{y}S_{j}^{y})\tilde{c}_{\alpha}-(S_{i}^{x}S_{j}^{y}+S_{i}^{y}S_{j}^{x})\tilde{s}_{\alpha}]\\\
&-J_{i,j}^{z\pm}[(S_{i}^{x}S_{j}^{z}+S_{i}^{z}S_{j}^{x})\tilde{c}_{\alpha}+(S_{i}^{y}S_{j}^{z}+S_{i}^{z}S_{j}^{y})\tilde{s}_{\alpha}\\}].\end{split}$
(3)

Here, $\tilde{c}_{\alpha}=cos(\phi_{\alpha})$,
$\tilde{s}_{\alpha}=sin(\phi_{\alpha})$, and the phase
$\phi_{\alpha}=[2\pi/3,-2\pi/3,0]$ for the $[X,Y,Z]$ bonds, respectively. We
emphasise that $\mathcal{H}_{XXZ}$ and $\mathcal{H}_{JK\Gamma\Gamma^{\prime}}$
are equivalent descriptions of the exchange Hamiltonian that are defined using
different Cartesian bases. The $\mathcal{H}_{JK\Gamma\Gamma^{\prime}}$ and
$\mathcal{H}_{XXZ}$ parameters are related by [17]

$\displaystyle\begin{split}J^{xy}=&~{}J+\frac{1}{3}(K-\Gamma-2\Gamma^{\prime}),\\\
J^{z}=&~{}J+\frac{1}{3}(K+2\Gamma+4\Gamma^{\prime}),\\\
2J_{i,j}^{\pm\pm}=&~{}-\frac{1}{3}(K+2\Gamma-2\Gamma^{\prime}),\\\
\sqrt{2}J_{i,j}^{z\pm}=&~{}\frac{2}{3}(K-\Gamma+\Gamma^{\prime}).\end{split}$
(4)

###  3.6 Inelastic Neutron Scattering

Inelastic neutron scattering (INS) data were collected on the direct geometry
time-of-flight cold neutron multi-chopper spectrometer, LET, at the ISIS
Neutron and Muon Source. 1 g of RPSO was packed and sealed with He exchange
gas in an annular copper can mounted in a dilution refrigerator. Data were
collected at 0.08 K, 2 K, 3 K, 4 K, 5 K, 10 K, and 15 K with incident neutron
energies, $E_{\mathsf{i}}=$ 1.78 meV, 3.14 meV and 7 meV. The choppers were
operated on high-flux mode with the resolution disk and pulse-removal disk
choppers spinning at frequencies of 200 Hz and 100 Hz, respectively, yielding
an elastic line resolution of approximately 0.03 meV at $E_{\mathsf{i}}=$ 1.78
meV.

To analyse the data, linear spin wave theory (LSWT) simulations were performed
on the SpinW MATLAB package [64]. All computations were powder averaged and
convoluted with the energy-dependent instrumental resolution. To extract the
relevant exchange Hamiltonian of RPSO from the data, eight constant momentum
transfer $|Q|$ cuts of the INS data collected at $0.08$ K were used in the
exchange parameter grid search, integrated over $\delta Q$ = 0.1 Å, and
centred about $0.25~{}-~{}0.95$ Å in 0.1 Å steps. For each cut, the amplitude
was fitted and a constant background term was subtracted. Given the large
parameter space of the extended Kitaev model, we made the starting assumption
that $\zeta$, $\xi$, and any further neighbour couplings in RPSO are
comparatively smaller than the four leading parameters, $J$, $K$, $\Gamma$ and
$\Gamma^{\prime}$ in the $\mathcal{H}_{JK\Gamma\Gamma^{\prime}}$ model. This
is justified by the relative magnitudes of the hopping integrals contributing
to these parameters estimated by DFT (see Section A.3, Supporting Information)
and the open framework structure of RPSO which will likely weaken couplings
beyond nearest neighbours. In the initial grid search, the four exchange
parameters in the $\mathcal{H}_{JK\Gamma\Gamma^{\prime}}$ model were varied in
14 linearly spaced steps spanning two parameter spaces, one in which
$-0.6<J<0.6$ meV, $-1<K<0$ meV, $-1<\Gamma<1$ meV, and $-1<\Gamma^{\prime}<1$
meV, and another with the same $J$, $\Gamma$, and $\Gamma^{\prime}$ but with
$0<K<1$ meV. Together these represent parameter spaces with ferromagnetic and
antiferromagnetic Kitaev exchange coupling, $K$, respectively. This approach
yielded multiple almost equivalent solutions (see Figure 12, Supporting
Information) with the lowest $\chi^{2}$, all with either dominant anisotropic
Kitaev-type ferromagnetic $K$ and antiferromagnetic $\Gamma$ interactions (see
Table 1), or antiferromagnetic $K$ for the symmetry-related dual set.

From this initial grid search, a series of local optimisations were performed
around the lowest $\chi^{2}$ solutions of the
$\mathcal{H}_{JK\Gamma\Gamma^{\prime}}$ model in an attempt to identify a
unique set of exchange parameters for RPSO. These local optimisations were
carried out on grid search solutions with $\chi^{2}<30$, and a simulated
annealing algorithm was used to fit model exchange Hamiltonians to 60 $\Delta
E$ cuts spanning the full experimental INS spectrum at $0.08$ K. Three models
were tested in this local optimisation (1) the
$\mathcal{H}_{JK\Gamma\Gamma^{\prime}}$ model, (2) the
$\mathcal{H}_{JK\Gamma\Gamma^{\prime}}$ model with the additional $\xi$ and
$\zeta$ exchange couplings allowed by the symmetry of the Ru-Ru bond centre in
RPSO and (3) a 9-parameter model, including $J$, $K$, $\Gamma$, $\Gamma’$,
$\xi$, $\zeta$ and the isotropic further-neighbour couplings $J_{\textrm{2}}$,
$J_{\textrm{3}}$, and $J_{\perp}$. However, including parameters beyond the
$\mathcal{H}_{JK\Gamma\Gamma^{\prime}}$ model did not yield improved fits.
While this does not conclusively dismiss the relevance of the additional
parameters tested, it underscores the limitation of the powder-averaged
datasets currently available, for which there is a risk of over-parameterising
when adding further parameters beyond the
$\mathcal{H}_{JK\Gamma\Gamma^{\prime}}$ model. Further details and results of
the INS fitting procedure can be found in Supporting Information.

## Supporting information

Supporting Information contains further details of the crystal structure
refinement of RPSO, field-dependent magnetic susceptibility and isothermal
magnetisation measurements, supporting density-functional theory calculations
and inelastic neutron scattering analysis.

## Acknowledgments

This work was supported by the UKRI Science and Technology Facilities Council
through the award of an ISIS Facility Development and Utilisation PhD
Studentship to AHA as well as access to beamtime at the Diamond Light Source
and ISIS Neutron and Muon Source. We also acknowledge DESY—a member of the
Helmholtz Association HGF—for access to beamtime. Work at the University of
Birmingham was supported by the UKRI Engineering and Physical Sciences
Research Council grant EP/V028774/1. Work at the University of Leipzig was
supported by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) grant TRR 360 – 492547816. The authors thank Dr Matthew Coak
(University of Birmingham) for providing helpful feedback on a manuscript
draft.

## Data availability

The data that support the findings of this study are available via the
following:

  * •

https://syncandshare.desy.de/index.php/s/xZ2Jwrzw9mpDkxN (IRIXS, DESY),

  * •

https://doi.org/10.5286/ISIS.E.RB2210080 (WISH, ISIS),

  * •

https://doi.org/10.5286/ISIS.E.RB2210081 (LET, ISIS).

Any requests for further data, analysis or code should be made to the
corresponding authors.

## Author contributions

LC and GJN conceived and supervised all aspects of the study. AHA devised and
performed the synthesis of RPSO with support from RSP. AHA collected and
analysed the magnetometry data. AHA collected heat capacity with support from
GBGS. AHA collected and analysed synchrotron X-ray diffraction, neutron
diffraction and neutron spectroscopy datasets with support from SJD, PM and
GJN, respectively. AHA, MDL and GJN devised the analysis procedure for the
neutron spectroscopy data. AAT performed and interpreted the density-
functional theory calculations. HG performed and analysed resonant inelastic
X-ray scattering measurements. AHA and LC wrote the manuscript with input from
all the authors.

## Competing interests

The authors declare no competing interests.

##  Appendix A Supporting Information

###  A.1 Crystal Structure Refinement

The $R\bar{3}c$ structural model of RuP3SiO11 was confirmed through Rietveld
analysis of high-resolution synchrotron X-ray and neutron powder diffraction
at temperatures of $300$ K, $2.5$ K, and $0.08$ K (see Figure 6). The
temperature-dependent analysis, detailed in Tables. 2-3, shows an isotropic
contraction of the unit cell parameters that are consistent with the isotypic
compounds, MoP3SiO11 [65] and FeP3SiO11 [66].

Figure 6: a Rietveld analysis of powder X-ray diffraction data (300 K, I11, Diamond) using the $R\bar{3}m$ structural model that describes the crystal structure of RuP3SiO11. b,c Rietveld analysis of powder neutron diffraction data (WISH, ISIS) confirming that RuP3SiO11 maintains the $R\bar{3}c$ structural model down to a base temperature of 0.08 K. Table 2: Structural parameters of RuP3SiO11 obtained by Rietveld refinement of the $R\bar{3}c$ model against powder X-ray diffraction data measured at 300 K on I11 (Diamond). The resulting unit cell parameters are $a=b=8.2410(1)$ Å and $c=39.230(1)$ Å with goodness-of-fit parameters $\chi^{2}=3.55$ and $R_{\mathsf{p}}=8.06\%$. Atom | Site | $x$ | $y$ | $z$ |  $U_{\mathsf{iso}}$ (Å2)
---|---|---|---|---|---
Ru | $12c$ | 0 | 0 | 0.15854(1) | 0.0057(1)
P | $36f$ | 0.3707(1) | 0.0377(1) | 0.11972(1) | 0.0003(2)
Si | $12c$ | 0 | 0 | 0.04017(4) | 0.0012(1)
O1 | $36f$ | 0.0440(3) | 0.1929(3) | 0.05235(5) | 0.0037(6)
O2 | $36f$ | 0.2233(3) | 0.0829(2) | 0.12706(5) | 0.0043(5)
O3 | $36f$ | 0.1374(3) | 0.2322(3) | 0.18896(5) | 0.0097(6)
O4 | $18e$ | 0.2101(3) | 0 | $\frac{3}{4}$ | 0.0027(4)
O5 | $6b$ | 0 | 0 | 0 | 0.0110(7)
Table 3: Structural parameters of RuP3SiO11 obtained by Rietveld refinement of the $R\bar{3}c$ model against powder neutron diffraction data measured at 2.5 K on WISH (ISIS). The resulting unit cell parameters are $a=b=8.2297(5)$ Å and $c=39.20(2)$ Å with goodness-of-fit parameters $\chi^{2}=3.23$ and $R_{\mathsf{p}}=5.97\%$. Atom | Site | $x$ | $y$ | $z$ |  $U_{\mathsf{iso}}$ (Å2)
---|---|---|---|---|---
Ru | $12c$ | 0 | 0 | 0.1583(2) | 0.02(1)
P | $36f$ | 0.3720(8) | 0.0381(8) | 0.1196(1) | 0.02(1)
Si | $12c$ | 0 | 0 | 0.0397 (3) | 0.001(2)
O1 | $36f$ | 0.0484(7) | 0.2042(6) | 0.0533(1) | 0.02(3)
O2 | $36f$ | 0.2273(7) | 0.0877(7) | 0.1273(1) | 0.01(2)
O3 | $36f$ | 0.1377(6) | 0.2316(6) | 0.1891(1) | 0.009(3)
O4 | $18e$ | 0.2080(8) | 0 | $\frac{3}{4}$ | 0.01(1)
O5 | $6b$ | 0 | 0 | 0 | 0.02(3)

###  A.2 Temperature-Field Magnetic Phase Diagram

As shown in Figure 7a, the magnetization isotherm of RPSO measured at 0.4 K,
(below $T_{\mathsf{N}}=1.3$ K), exhibits an anomaly at a critical field of
$H_{\mathsf{C}}=3.55$ T. The temperature dependence of this anomaly is
displayed in Figure 7b, which softens as the paramagnetic regime is
approached. A closer examination of the field and temperature dependence,
shown in Figure 7c, of $M/H$ reveals that beyond 0.1 T, the anomaly indicating
the onset of long-range magnetic order broadens, and its temperature
dependence decreases steadily with increasing applied field. Upon reaching
$3.5$ T, $T_{\mathsf{N}}$ is no longer discernible within the $0.5~{}–~{}1.7$
K temperature range of the measurement.

Figure 7: a The magnetisation isotherm of RPSO measured at $0.4$ K increases
linearly with the applied field up to $2$ T, beyond which the rate of increase
in magnetisation becomes steeper before reaching a critical field,
$H_{\mathsf{C}}=3.55$ T. a, inset The second field derivative of the
magnetisation highlighting $H_{\mathsf{C}}=3.55$ T. b The effect of this
critical field can be observed in the field dependence of $M/H$, where the
ordering temperature is suppressed above $H_{\mathsf{C}}$.

###  A.3 First-Principles Calculations

Ab initio methods were used to independently verify the validity of the
$j_{\mathsf{eff}}=\frac{1}{2}$ character of Ru3+ in RPSO as determined in the
main manuscript by RIXS, as well as to investigate the nature of the exchange
Hamiltonian. Density-functional theory (DFT) band structure calculations were
performed in the FPLO [67] and VASP [68, 69] codes using the experimentally
determined crystal structure and the Perdew-Burke-Ernzerhof (PBE) [70] flavour
of the exchange-correlation potential. Correlation effects in the Ru $4d$
shell were accounted for on the mean-field DFT+U level with the on-site
Coulomb repulsion $U=3$ eV and Hund’s coupling $J_{\mathsf{H}}=0.46$ eV, the
latter value determined by RIXS.

Figure 8: PBE density of states for a RPSO and b $\alpha$-RuCl3 (bottom). The
Fermi level ($E_{\mathsf{f}}$) is at zero energy. The inset shows a comparison
of PBE and PBE$+$SO density of states around $E_{\mathsf{f}}$, with the arrows
labeling the regions of $j_{\mathsf{eff}}=\frac{3}{2}$ and $\frac{1}{2}$
states.

To validate the experimental observation of the $j_{\mathsf{eff}}=\frac{1}{2}$
state of Ru3+ in RPSO and to directly compare it with $\alpha$-RuCl3, we have
performed electronic band structure calculations for both systems at the
scalar relativistic level (see Figure 8). To ensure the validity of this
comparison, we employed the widely accepted low-temperature rhombohedral
structure for band-structure calculations of $\alpha$-RuCl3 [71]. In this
$R\bar{3}$ structure, the octahedral crystal field is trigonally distorted as
in RPSO, yielding the same local 3-fold rotational symmetry at the Ru3+ sites
of each system. Calculating the density of states allows the orbital energies
of each system to be determined from their Wannier projections, and the
corresponding octahedral crystal field splitting, $\Delta_{\mathsf{O}}$, for
RPSO was calculated to be 2.52 eV, compared with 2.42 eV for $\alpha$-RuCl3.
The increased $\Delta_{\mathsf{O}}$ for the oxide crystal field of RPSO
compared with the chloride crystal field of $\alpha$-RuCl3 is in keeping with
the RIXS data analysis and reflects the greater ionic character of O2– versus
Cl–. The trigonal crystal field splitting, on the other hand, is equivalent in
magnitude but opposite in sign in both systems with
$\lvert\Delta_{\mathsf{t}}\rvert$ = 0.06 eV. Overall, the calculated orbital
energies indicate that RPSO meets the same prerequisite requirements for the
formation of the $j_{\mathsf{eff}}=\frac{1}{2}$ state for Ru3+ as in
$\alpha$-RuCl3, which is fully consistent with the RIXS data analysis in the
main manuscript.

This conclusion is verified by calculating the PBE$+$SO density of states of
RPSO. The resulting density of states reveals a clear splitting of the
$t_{2g}$ band into lower and higher lying $j_{\mathsf{eff}}=\frac{1}{2}$ and
$j_{\mathsf{eff}}=\frac{3}{2}$ manifolds, respectively, as has also been seen
in the PBE+SO density of states of Ir4+ compounds [72]. Spin-polarised
calculations (PBE$+$U$+$SO) further reveal that the magnetic moments of Ru3+
comprise $\mu^{x}_{\mathsf{spin}}=0.19~{}{\mu_{\mathsf{B}}}$ of the spin
contribution and $\mu^{x}_{\mathsf{orb}}=0.38~{}{\mu_{\mathsf{B}}}$ of the
orbital contribution when directed along $a$. Similarly, along the $c$ axis,
the magnetic moments are $\mu^{z}_{\mathsf{spin}}=0.20~{}{\mu_{\mathsf{B}}}$
and $\mu^{z}_{\mathsf{orb}}=0.47~{}{\mu_{\mathsf{B}}}$. The almost isotropic
nature of the moment and the nearly $1:2$ ratio of $\mu_{\mathsf{spin}}$ and
$\mu_{\mathsf{orb}}$ are both consistent with a $j_{\mathsf{eff}}=\frac{1}{2}$
state in RPSO.

We also expanded our DFT calculations to include the magnetic exchange
interactions between the $j_{\mathsf{eff}}=\frac{1}{2}$ states of the Ru3+
ions in RPSO by analysing the hopping integrals $t$ within superexchange
theory [63, 73]. Using $U=3$ eV, as well as $J_{\mathsf{H}}=0.46$ eV and
$\lambda=0.13$ eV from RIXS, we find $(J,K,\Gamma,\Gamma^{\prime})$ =
$(-0.16,-0.65,0.26,-0.42)$ meV along with $\zeta=0.03$ meV and $\xi=0.02$ meV.
These $K$ and $\Gamma$ values are compatible with the LSWT estimates, whereas
$\Gamma^{\prime}$ seems overestimated, and $J$ is weakly ferromagnetic, in
contrast to the weakly antiferromagnetic $J$ required for stabilising the Néel
order. These discrepancies may be caused by the overall weakness of the
exchange couplings and by the residual trigonal crystal-field splitting, which
was not taken into account within the superexchange theory. Nevertheless,
these estimates support our description of RPSO in terms of the
$\mathcal{H}_{JK\Gamma\Gamma^{\prime}}$ model and justify the neglect of
$\zeta$ and $\xi$ as minor terms in the spin Hamiltonian. We also estimated
the interlayer coupling of $J_{\perp}=0.18$ meV, which is somewhat weaker than
the leading in-plane terms. Finally, our calculations confirm that further-
neighbor in-plane couplings should be as small as $J_{2}\approx 0.003$ meV and
$J_{3}\approx 0.0006$ meV and thus truly negligible in this material.

###  A.4 Inelastic Neutron Scattering Data Analysis

As shown in Figure 9, at an incident energy of $E_{\mathsf{i}}=3.14$ meV, the
experimental dynamical structure factor shows only magnetic scattering. Above
$T_{\mathsf{N}}$, the $\Delta=0.1$ meV gap closes, and the spectra are peaked
at low $\lvert Q\rvert$ and show scattering up to $\Delta E\approx 0.8$ meV.
This behavior is observed up to $15$ K and is shown in Figure 9, which
includes the spectrum and representative cuts at selected temperatures.

As detailed in the main manuscript, a grid search was used to explore and
compare the $\mathcal{H}_{JK\Gamma\Gamma^{\prime}}$ parameter space to the
experimental dynamical structure factor at $0.08$ K. The resulting five-
dimensional space, encompassing the fitted parameter sets of
$(J,K,\Gamma,\Gamma^{\prime})$ and their corresponding goodness-of-fit
$\chi^{2}$ to the data, is shown in Figures 10 and 11 for $K>0$ and $K<0$,
respectively. Here, missing spaces in the four-dimensional spaces are
solutions that are incompatible with the magnetic structure. A summary of the
number of the lowest $\chi^{2}$ parameter sets — subsequently used for the
simulated annealing optimisation (see Methods) — is presented in Figure 12.
The best fitting optimised solutions for $K<0$ are presented in Table 1. We
note that the mismatch in $\chi^{2}$ values between the grid search and the
simulated annealing optimisation stems from the different number of cuts used
for fitting — 8 in the former and 60 in the latter.

Figure 9: a The experimental dynamical structure factor, $S\text{(}Q,\Delta
E=\hbar\omega\text{)}_{\textrm{exp}}$ of RPSO measured at 0.08 K and
$E_{\mathsf{i}}=3.14$ meV (LET, ISIS) shows a clear excitation gap of
approximately 0.1 meV and two branches of magnetic excitations extending up to
0.8 meV. b Above $T_{\mathsf{N}}$ at $2$ K, the excitation gap closes, and c-d
$\Delta$E\- and $|Q|$-integrated cuts show that structured magnetic scattering
remains up to $10$ K. Figure 10: Fit quality, represented by $\chi^{2}$, for
the $\mathcal{H}_{JK\Gamma\Gamma^{\prime}}$ model fitted to 8 $\Delta E$ cuts
with $K>0$. Each four-dimensional space illustrates the variation of
$-0.6<J<0.6$ meV, $0<K<1$ meV, and $-1<\Gamma<1$ meV, in 14 linearly spaced
steps for a specific $\Gamma^{\prime}$ value, with the intensity reflecting
the corresponding $\chi^{2}$ values. Missing regions represent solutions that
are incompatible with the Néel ordered ground state. Figure 11: Fit quality,
represented by $\chi^{2}$, for the $\mathcal{H}_{JK\Gamma\Gamma^{\prime}}$
model fitted to 8 $\Delta E$ cuts with $K<0$. Each four-dimensional space
illustrates the variation of $-0.6<J<0.6$ meV, $-1<K<0$ meV, and $-1<\Gamma<1$
meV, in 14 linearly spaced steps for a specific $\Gamma^{\prime}$ value, with
the intensity reflecting the corresponding $\chi^{2}$ values. Missing regions
represent solutions that are incompatible with the Néel ordered ground state.
Figure 12: The number solutions as a function of $\chi^{2}$ for the parameter
sets obtained in the $\mathcal{H}_{JK\Gamma\Gamma^{\prime}}$ grid search for
$K>0$ and $K<0$.

For low $\chi^{2}$ solutions with $K>0$, we find a connection to the optimised
$K<0$ parameter sets through the self-duality of the
$\mathcal{H}_{JK\Gamma\Gamma^{\prime}}$ model when defined using the set of
rotated cubic axes. Within this frame of reference, a physically identical set
of parameters can be obtained by applying a $\pi$ rotation to the honeycomb
plane about the crystallographic $c$-axis [17]

${\begin{pmatrix}[c]J\\\ K\\\ \Gamma\\\
\Gamma^{\prime}\end{pmatrix}_{\mathsf{dual}}}=\begin{pmatrix}[r]1&\frac{4}{9}&-\frac{4}{9}&\frac{4}{9}\\\
0&-\frac{1}{3}&\frac{4}{3}&-\frac{4}{3}\\\
0&\frac{4}{9}&\frac{5}{9}&\frac{4}{9}\\\
0&-\frac{2}{9}&\frac{2}{9}&\frac{7}{9}\end{pmatrix}\begin{pmatrix}[c]J\\\ K\\\
\Gamma\\\ \Gamma^{\prime}\end{pmatrix}.$

This transformation is evident for the representative optimised
$(J,K,\Gamma,\Gamma^{\prime})$ parameter sets, $(-0.16,0.86,-0.01,0.12)$ meV
and $(-0.11,0.56,0.12,0.09)$ meV, with $\chi^{2}=68.8$ and $72.4$
respectively. Applying the transformation matrix to these solutions yields
$(0.28,-0.46,0.43,-0.1)_{\mathsf{dual}}$ meV and
$(0.12,-0.15,0.36,-0.02)_{\mathsf{dual}}$ meV, corresponding approximately to
Solutions 1 and 3 in Table 1.

Table 4: Exchange parameters (in meV) of the five lowest $\chi^{2}$ solutions of the $\mathcal{H}_{XXZ}$ model obtained for RPSO in this work by fitting inelastic neutron scattering data to linear spin wave theory. RPSO | $J^{xy}$ | $J^{z}$ | $J^{\pm\pm}$ | $J^{z\pm}$ | $\chi^{2}$
---|---|---|---|---|---
Solution 1 | $0.08$ | $0.28$ | $-0.10$ | $-0.52$ | $67.7$
Solution 2 | $0.05$ | $0.28$ | $-0.10$ | $-0.36$ | $68.9$
Solution 3 | $0.02$ | $0.28$ | $-0.10$ | $-0.25$ | $69.7$
Solution 4 | $0.07$ | $0.27$ | $-0.10$ | $-0.42$ | $69.7$
Solution 5 | $0.08$ | $0.29$ | $-0.10$ | $-0.61$ | $70.5$

A recent trend in the literature is a growing preference for the
crystallographic frame to formulate the exchange Hamiltonian of the extended
Kitaev model [74, 17]. In this framework, the exchange Hamiltonian is
expressed as $\mathcal{H}_{XXZ}$ (Equation 3), and the optimised solutions in
Table 1 can be rewritten using Equations 4, resulting in the parameters
presented in Table 4. Using this notation, only $J^{z\pm}$ varies across the
the optimised solutions, while $(J^{xy},J^{z},J^{\pm\pm})$ are consistently
around $(0.05,0.28,-0.1)$ meV.

## References

  *   * Haldane [1983] Haldane, F.D.M.: Nonlinear field theory of large-spin Heisenberg antiferromagnets: Semiclassically quantized solitons of the one-dimensional easy-axis Néel state. Phys. Rev. Lett. 50, 1153–1156 (1983)
  * Kosterlitz and Thouless [1973] Kosterlitz, J.M., Thouless, D.J.: Ordering, metastability and phase transitions in two-dimensional systems. J. Phys. C 6, 1181–1203 (1973)
  * Bernevig et al. [2022] Bernevig, B.A., Felser, C., Beidenkopf, H.: Progress and prospects in magnetic topological materials. Nature 603, 41–51 (2022)
  * Balz et al. [2016] Balz, C., Lake, B., Reuther, J., Luetkens, H., Schönemann, R., Herrmannsdörfer, T., Singh, Y., Nazmul Islam, A.T.M., Wheeler, E.M., Rodriguez-Rivera, J.A., Guidi, T., Simeoni, G.G., Baines, C., Ryll, H.: Physical realization of a quantum spin liquid based on a complex frustration mechanism. Nat. Phys. 12, 942–949 (2016)
  * Broholm et al. [2020] Broholm, C., Cava, R.J., Kivelson, S.A., Nocera, D.G., Norman, M.R., Senthil, T.: Quantum spin liquids. Science 367, 0668 (2020)
  * Clark and Abdeldaim [2021] Clark, L., Abdeldaim, A.H.: Quantum spin liquids from a materials perspective. Annu. Rev. Mater. Sci. 51, 495–519 (2021)
  * Savary and Balents [2016] Savary, L., Balents, L.: Quantum spin liquids: A review. Rep. Prog. Phys. 80, 016502 (2016)
  * Hermanns et al. [2018] Hermanns, M., Kimchi, I., Knolle, J.: Physics of the Kitaev model: Fractionalization, dynamic correlations, and material connections. Annu. Rev. Condens. Matter Phys. 9, 17–33 (2018)
  * Ran et al. [2007] Ran, Y., Hermele, M., Lee, P.A., Wen, X.-G.: Projected-wave-function study of the spin-$1/2$ Heisenberg model on the kagomé lattice. Phys. Rev. Lett. 98, 117205 (2007)
  * Yan et al. [2011] Yan, S., Huse, D.A., White, S.R.: Spin-liquid ground state of the $S=1/2$ kagome Heisenberg antiferromagnet. Science 332, 1173–1176 (2011)
  * Norman [2016] Norman, M.R.: Colloquium: Herbertsmithite and the search for the quantum spin liquid. Rev. Mod. Phys. 88, 041002 (2016)
  * Han et al. [2012] Han, T.-H., Helton, J.S., Chu, S., Nocera, D.G., Rodriguez-Rivera, J.A., Broholm, C., Lee, Y.S.: Fractionalized excitations in the spin-liquid state of a kagome-lattice antiferromagnet. Nature 492, 406–410 (2012)
  * Fu et al. [2015] Fu, M., Imai, T., Han, T.-H., Lee, Y.S.: Evidence for a gapped spin-liquid ground state in a kagome Heisenberg antiferromagnet. Science 350, 655–658 (2015)
  * Khuntia et al. [2020] Khuntia, P., Velazquez, M., Barthélemy, Q., Bert, F., Kermarrec, E., Legros, A., Bernu, B., Messio, L., Zorko, A., Mendels, P.: Gapless ground state in the archetypal quantum kagome antiferromagnet ZnCu3(OH)6Cl2. Nat. Phys. 16(4), 469–474 (2020)
  * Kitaev [2006] Kitaev, A.: Anyons in an exactly solved model and beyond. Ann. Phys. 321, 2–111 (2006)
  * Winter et al. [2017] Winter, S.M., Tsirlin, A.A., Daghofer, M., Brink, J., Singh, Y., Gegenwart, P., Valentí, R.: Models and materials for generalized Kitaev magnetism. J. Phys. Condens. Matter. 29, 493002 (2017)
  * Maksimov and Chernyshev [2020] Maksimov, P.A., Chernyshev, A.L.: Rethinking $\alpha$-RuCl3. Phys. Rev. Res. 2, 033011 (2020)
  * Jackeli and Khaliullin [2009] Jackeli, G., Khaliullin, G.: Mott insulators in the strong spin-orbit coupling limit: From Heisenberg to a quantum compass and Kitaev models. Phys. Rev. Lett. 102, 017205 (2009)
  * Stavropoulos et al. [2019] Stavropoulos, P.P., Pereira, D., Kee, H.-Y.: Microscopic mechanism for a higher-spin Kitaev model. Phys. Rev. Lett. 123, 037203 (2019)
  * Liu and Khaliullin [2018] Liu, H., Khaliullin, G.: Pseudospin exchange interactions in $d^{7}$ cobalt compounds: Possible realization of the Kitaev model. Phys. Rev. B 97, 014407 (2018)
  * Motome et al. [2020] Motome, Y., Sano, R., Jang, S., Sugita, Y., Kato, Y.: Materials design of Kitaev spin liquids beyond the Jackeli–Khaliullin mechanism. J. Phys. Condens. Matter. 32, 404001 (2020)
  * Takagi et al. [2019] Takagi, H., Takayama, T., Jackeli, G., Khaliullin, G., Nagler, S.E.: Concept and realization of Kitaev quantum spin liquids. Nat. Rev. Phys. 1, 264–280 (2019)
  * Kataoka et al. [2020] Kataoka, K., Hirai, D., Yajima, T., Nishio-Hamane, D., Ishii, R., Choi, K.-Y., Wulferding, D., Lemmens, P., Kittaka, S., Sakakibara, T., Ishikawa, H., Matsuo, A., Kindo, K., Hiroi, Z.: Kitaev spin liquid candidate OsxCl3 comprised of honeycomb nano-domains. J. Phys. Soc. Japan 89, 114709 (2020)
  * Katukuri et al. [2015] Katukuri, V.M., Nishimoto, S., Rousochatzakis, I., Stoll, H., Brink, J., Hozoi, L.: Strong magnetic frustration and anti-site disorder causing spin-glass behavior in honeycomb Li2RhO3. Sci. Rep. 5, 14718 (2015)
  * Plumb et al. [2014] Plumb, K.W., Clancy, J.P., Sandilands, L.J., Shankar, V.V., Hu, Y.F., Burch, K.S., Kee, H.-Y., Kim, Y.-J.: $\alpha$-RuCl3: A spin-orbit assisted Mott insulator on a honeycomb lattice. Phys. Rev. B 90, 041112 (2014)
  * Banerjee et al. [2016] Banerjee, A., Bridges, C.A., Yan, J.-Q., Aczel, A.A., Li, L., Stone, M.B., Granroth, G.E., Lumsden, M.D., Yiu, Y., Knolle, J., Bhattacharjee, S., Kovrizhin, D.L., Moessner, R., Tennant, D.A., Mandrus, D.G., Nagler, S.E.: Proximate Kitaev quantum spin liquid behaviour in a honeycomb magnet. Nat. Mater. 15, 733–740 (2016)
  * Johnson et al. [2015] Johnson, R.D., Williams, S.C., Haghighirad, A.A., Singleton, J., Zapf, V., Manuel, P., Mazin, I.I., Li, Y., Jeschke, H.O., Valentí, R., Coldea, R.: Monoclinic crystal structure of $\alpha$-RuCl3 and the zigzag antiferromagnetic ground state. Phys. Rev. B 92, 235119 (2015)
  * Sears et al. [2015] Sears, J.A., Songvilay, M., Plumb, K.W., Clancy, J.P., Qiu, Y., Zhao, Y., Parshall, D., Kim, Y.-J.: Magnetic order in $\alpha$-RuCl3: A honeycomb-lattice quantum magnet with strong spin-orbit coupling. Phys. Rev. B 91, 144420 (2015)
  * Rousochatzakis et al. [2024] Rousochatzakis, I., Perkins, N., Luo, Q., Kee, H.-Y.: Beyond Kitaev physics in strong spin-orbit coupled magnets. Rep. Prog. Phys. (2024)
  * Yamada et al. [2017a] Yamada, M.G., Dwivedi, V., Hermanns, M.: Crystalline Kitaev spin liquids. Phys. Rev. B 96, 155107 (2017)
  * Yamada et al. [2017b] Yamada, M.G., Fujita, H., Oshikawa, M.: Designing Kitaev spin liquids in metal-organic frameworks. Phys. Rev. Lett. 119, 057202 (2017)
  * Banerjee et al. [2018] Banerjee, A., Lampen-Kelley, P., Knolle, J., Balz, C., Aczel, A.A., Winn, B., Liu, Y., Pajerowski, D., Yan, J., Bridges, C.A., Savici, A.T., Chakoumakos, B.C., Lumsden, M.D., Tennant, D.A., Moessner, R., Mandrus, D.G., Nagler, S.E.: Excitations in the field-induced quantum spin liquid state of $\alpha$-RuCl3. npj Quantum Mater. 3, 8 (2018)
  * Fukuoka et al. [1996] Fukuoka, H., Imoto, H., Saito, T.: Synthesis and crystal structure of a new ruthenium silicophosphate: RuP3SiO11. J. Solid State Chem. 121, 247–250 (1996)
  * Suzuki et al. [2021] Suzuki, H., Liu, H., Bertinshaw, J., Ueda, K., Kim, H., Laha, S., Weber, D., Yang, Z., Wang, L., Takahashi, H., Fürsich, K., Minola, M., Lotsch, B.V., Kim, B.J., Yavaş, H., Daghofer, M., Chaloupka, J., Khaliullin, G., Gretarsson, H., Keimer, B.: Proximate ferromagnetic state in the Kitaev model material $\alpha$-RuCl3. Nat. Commun. 12, 4512 (2021)
  * Gretarsson et al. [2013] Gretarsson, H., Clancy, J.P., Liu, X., Hill, J.P., Bozin, E., Singh, Y., Manni, S., Gegenwart, P., Kim, J., Said, A.H., Casa, D., Gog, T., Upton, M.H., Kim, H.-S., Yu, J., Katukuri, V.M., Hozoi, L., Brink, J., Kim, Y.-J.: Crystal-field splitting and correlation effect on the electronic structure of A2IrO3. Phys. Rev. Lett. 110, 076402 (2013)
  * Takahashi et al. [2021] Takahashi, H., Suzuki, H., Bertinshaw, J., Bette, S., Müuhle, C., Nuss, J., Dinnebier, R., Yaresko, A., Khaliullin, G., Gretarsson, H., Takayama, T., Takagi, H., Keimer, B.: Nonmagnetic ${J}=0$ state and spin-orbit excitations in K2RuCl6. Phys. Rev. Lett. 127, 227201 (2021)
  * Gretarsson et al. [2024] Gretarsson, H., Fujihara, H., Sato, F., Gotou, H., Imai, Y., Ohgushi, K., Keimer, B., Suzuki, H.: $J=1/2$ Pseudospins and $d$-$p$ Hybridization in the Kitaev Spin Liquid Candidates Ru$X_{3}$ ($X$ = Cl, Br, I) (2024)
  * Li et al. [2021] Li, Y., Winter, S.M., Kaib, D.A.S., Riedl, K., Valentí, R.: Modified Curie-Weiss law for ${j}_{\mathrm{eff}}$ magnets. Phys. Rev. B 103, 220408 (2021)
  * Kim et al. [2022] Kim, S., Yuan, B., Kim, Y.-J.: $\alpha$-RuCl3 and other Kitaev materials. APL Mater. 10, 080903 (2022)
  * Hickey and Trebst [2019] Hickey, C., Trebst, S.: Emergence of a field-driven U(1) spin liquid in the Kitaev honeycomb model. Nat. Commun. 10, 530 (2019)
  * Miller and Love [1967] Miller, S.C., Love, W.F.: Tables of Irreducible Representations of Space Groups and Co-representations of Magnetic Space Groups. Pruett Press, Colorado (1967)
  * Reger et al. [1989] Reger, J.D., Riera, J.A., Young, A.P.: Monte carlo simulations of the spin-1/2 Heisenberg antiferromagnet in two dimensions. J. Phys. Condens. Matter. 1, 1855–1865 (1989)
  * Cao et al. [2016] Cao, H.B., Banerjee, A., Yan, J.-Q., Bridges, C.A., Lumsden, M.D., Mandrus, D.G., Tennant, D.A., Chakoumakos, B.C., Nagler, S.E.: Low-temperature crystal and magnetic structure of $\alpha$-RuCl3. Phys. Rev. B 93, 134423 (2016)
  * Trebst and Hickey [2022] Trebst, S., Hickey, C.: Kitaev materials. Phys. Rep. 950, 1–37 (2022)
  * Baek et al. [2017] Baek, S.-H., Do, S.-H., Choi, K.-Y., Kwon, Y.S., Wolter, A.U.B., Nishimoto, S., Brink, J., Büchner, B.: Evidence for a field-induced quantum spin liquid in $\alpha$-RuCl3. Phys. Rev. Lett. 119, 037201 (2017)
  * Hentrich et al. [2020] Hentrich, R., Hong, X., Gillig, M., Caglieris, F., C̆ulo, M., Shahrokhvand, M., Zeitler, U., Roslova, M., Isaeva, A., Doert, T., Janssen, L., Vojta, M., Büchner, B., Hess, C.: High-field thermal transport properties of the Kitaev quantum magnet $\alpha$-RuCl3: Evidence for low-energy excitations beyond the critical field. Phys. Rev. B 102, 235155 (2020)
  * Chaloupka and Khaliullin [2015] Chaloupka, J., Khaliullin, G.: Hidden symmetries of the extended Kitaev-heisenberg model: Implications for the honeycomb-lattice iridates A2IrO3. Phys. Rev. B 92, 024413 (2015)
  * Bai et al. [2023] Bai, X., Zhang, S.-S., Zhang, H., Dun, Z., Phelan, W.A., Garlea, V.O., Mourigal, M., Batista, C.D.: Instabilities of heavy magnons in an anisotropic magnet. Nat. Commun. 14, 4199 (2023)
  * Kim et al. [2023] Kim, C., Kim, S., Park, P., Kim, T., Jeong, J., Ohira-Kawamura, S., Murai, N., Nakajima, K., Chernyshev, A.L., Mourigal, M., Kim, S.-J., Park, J.-G.: Bond-dependent anisotropy and magnon decay in cobalt-based Kitaev triangular antiferromagnet. Nat. Phys. 19, 1624–1629 (2023)
  * Taylor and Müller [1983] Taylor, J.H., Müller, G.: Limitations of spin-wave theory in ${T}=0$ spin dynamics. Phys. Rev. B 28, 1529–1533 (1983)
  * Winter et al. [2017] Winter, S.M., Riedl, K., Maksimov, P.A., Chernyshev, A.L., Honecker, A., Valentí, R.: Breakdown of magnons in a strongly spin-orbital coupled magnet. Nat. Commun. 8, 1152 (2017)
  * Liu et al. [2022] Liu, H., Chaloupka, J., Khaliullin, G.: Exchange interactions in ${d}^{5}$ Kitaev materials: From Na2IrO3 to $\alpha$-RuCl3. Phys. Rev. B 105, 214411 (2022)
  * Kim et al. [2020] Kim, J., Chaloupka, J., Singh, Y., Kim, J.W., Kim, B.J., Casa, D., Said, A., Huang, X., Gog, T.: Dynamic spin correlations in the honeycomb lattice Na2IrO3 measured by resonant inelastic X-ray scattering. Phys. Rev. X 10, 021034 (2020)
  * Rau and Kee [2014] Rau, J.G., Kee, H.-Y.: Trigonal distortion in the honeycomb iridates: Proximity of zigzag and spiral phases in Na2IrO3 (2014)
  * Kaib et al. [2022] Kaib, D.A.S., Riedl, K., Razpopov, A., Li, Y., Backes, S., Mazin, I.I., Valentí, R.: Electronic and magnetic properties of the RuX3 (X = Cl, Br, I) family: two siblings—and a cousin? npj Quantum Materials 7(1), 75 (2022)
  * Fukuoka et al. [1995] Fukuoka, H., Imoto, H., Saito, T.: New polymorphs of RuIIIP3O9: Cyclo-hexaphosphate Ru2P6O18 and metaphosphate Ru(PO3)3 with a novel structure. J. Solid State Chem. 119, 107–114 (1995)
  * Toby and Von Dreele [2013] Toby, B.H., Von Dreele, R.B.: GSAS-II: The Genesis of a Modern Open-source All Purpose Crystallography Software Package. J. Appl. Crystallogr. 46, 544–549 (2013)
  * Perez-Mato et al. [2015] Perez-Mato, J.M., Gallego, S.V., Tasci, E.S., Elcoro, L., Flor, G., Aroyo, M.I.: Symmetry-based computational tools for magnetic crystallography. Annu. Rev. Mater. Res. 45, 217–248 (2015)
  * Rodríguez-Carvajal [1993] Rodríguez-Carvajal, J.: Recent advances in magnetic structure determination by neutron powder diffraction. Phys. B: Condens. Matter. 192, 55–69 (1993)
  * Cromer and Waber [1965] Cromer, D.T., Waber, J.T.: Scattering factors computed from relativistic Dirac–Slater wave functions. Acta Crystallogr. 18, 104–109 (1965)
  * Haverkort [2016] Haverkort, M.W.: Quanty for core level spectroscopy - excitons, resonances and band excitations in time and frequency domain. J. Phys. Conf. Ser. 712, 012001 (2016)
  * Yang et al. [2023] Yang, Z., Wang, L., Zhao, D., Luo, M., Laha, S., Güth, A., Taniguchi, T., Watanabe, K., Lotsch, B.V., Smet, J.H., Minola, M., Gretarsson, H., Keimer, B.: Resonant inelastic x-ray scattering from electronic excitations in $\alpha$-RuCl3 nanolayers. Phys. Rev. B 108, 041406 (2023)
  * Winter et al. [2016] Winter, S.M., Li, Y., Jeschke, H.O., Valentí, R.: Challenges in design of Kitaev materials: Magnetic interactions from competing energy scales. Phys. Rev. B 93, 214431 (2016)
  * Toth and Lake [2015] Toth, S., Lake, B.: Linear spin wave theory for single-Q incommensurate magnetic structures. J. Phys. Condens. Matter. 27, 166002 (2015)
  * Badrtdinov et al. [2021] Badrtdinov, D.I., Ding, L., Ritter, C., Hembacher, J., Ahmed, N., Skourski, Y., Tsirlin, A.A.: MoP3SiO11 honeycomb antiferromagnet with disconnected octahedra. Phys. Rev. B 104, 094428 (2021)
  * Elbouaanani et al. [1999] Elbouaanani, L.K., Malaman, B., Gérardin, R.: Synthesis, crystal structure, and magnetic properties of FeP3SiO11: First example of iron(III)silicophosphate. J. Solid. State Chem. 147, 565–572 (1999)
  * Koepernik and Eschrig [1999] Koepernik, K., Eschrig, H.: Full-potential nonorthogonal local-orbital minimum-basis band-structure scheme. Phys. Rev. B 59, 1743–1757 (1999)
  * Kresse and Furthmüller [1996] Kresse, G., Furthmüller, J.: Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set. Computational Materials Science 6, 15–50 (1996)
  * Kresse and Furthmüller [1996] Kresse, G., Furthmüller, J.: Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B 54, 11169–11186 (1996)
  * Perdew et al. [1996] Perdew, J.P., Burke, K., Ernzerhof, M.: Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865–3868 (1996)
  * Zhang et al. [2024] Zhang, H., McGuire, M.A., May, A.F., Chao, H.-Y., Zheng, Q., Chi, M., Sales, B.C., Mandrus, D.G., Nagler, S.E., Miao, H., Ye, F., Yan, J.: Stacking disorder and thermal transport properties of $\alpha$-RuCl3. Phys. Rev. Mater. 8, 014402 (2024)
  * Bertinshaw et al. [2019] Bertinshaw, J., Kim, Y.K., Khaliullin, G., Kim, B.J.: Square lattice iridates. Ann. Rev. Condens. Matter Phys. 10, 315–336 (2019)
  * Rau et al. [2014] Rau, J.G., Lee, E.K.-H., Kee, H.-Y.: Generic spin model for the honeycomb iridates beyond the Kitaev limit. Phys. Rev. Lett. 112, 077204 (2014)
  * Halloran et al. [2023] Halloran, T., Desrochers, F., Zhang, E.Z., Chen, T., Chern, L.E., Xu, Z., Winn, B., Graves-Brook, M., Stone, M.B., Kolesnikov, A.I., Qiu, Y., Zhong, R., Cava, R., Kim, Y.B., Broholm, C.: Geometrical frustration versus Kitaev interactions in BaCo2(AsO4)2. Proc. Natl. Acad. Sci. U.S.A. 120, 2215509119 (2023)