# Magnetism and spin dynamics of an $S$ = 3/2 frustrated trillium lattice
antiferromagnet K2CrTi(PO4)3

J. Khatua  Department of Physics, Indian Institute of Technology Madras,
Chennai 600036, India  Department of Physics, Sungkyunkwan University, Suwon
16419, Republic of Korea  Suheon Lee  Center for Artificial Low Dimensional
Electronic Systems, Institute for Basic Science, Pohang 37673, Republic of
Korea  Gyungbin Ban  Department of Physics, Sungkyunkwan University, Suwon
16419, Republic of Korea  Marc Uhlarz  Dresden High Magnetic Field Laboratory
(HLD-EMFL), Helmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden, Germany
Kwang-Yong Choi<EMAIL_ADDRESS>Department of Physics, Sungkyunkwan
University, Suwon 16419, Republic of Korea  P. Khuntia<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology Madras, Chennai 600036,
India  Quantum Centre of Excellence for Diamond and Emergent Materials, Indian
Institute of Technology Madras, Chennai 600036, India.

###### Abstract

Competing magnetic interactions, frustration-driven quantum fluctuations, and
spin-correlation offer an ideal route for the experimental realization of
emergent quantum phenomena and exotic quasi-particle excitations in three-
dimensional frustrated magnets. In this context, trillium lattice, wherein
magnetic ions decorate a three-dimensional chiral network of corner-shared
equilateral triangular motifs, provides a viable ground. Herein, we present
the crystal structure, magnetic susceptibility, specific heat, electron spin-
resonance (ESR), muon spin-relaxation ($\mu$SR) results on the polycrystalline
samples of K2CrTi(PO4)3 wherein the Cr3+ ions form a perfect trillium lattice
without any detectable anti-site disorder. The Curie-Weiss fit of the magnetic
susceptibility data above 100 K yields a Curie-Weiss temperature $\theta_{\rm
CW}$ = $-$ 23 K, which indicates the presence of dominant antiferromagnetic
interactions between $S$ = 3/2 moments of Cr3+ ions. The specific heat
measurements reveal the occurrence of two consecutive phase transitions, at
temperatures $T_{\rm L}$ = 4.3 K and $T_{\rm H}$ = 8 K, corresponding to two
different magnetic phases. Additionally, it unveils the existence of short-
range spin correlations above the ordering temperature $T_{\rm H}$. The power-
law behavior of ESR linewidth suggests the persistence of short-range spin
correlations over a relatively wide critical region $(T-T_{\rm H}$)/$T_{\rm
H}>$ 0.25 in agreement with the specific heat results. The $\mu$SR results
provide concrete evidence of two different phases corresponding to two
transitions, coupled with the critical slowing down of spin fluctuations above
$T_{\rm L}$ and persistent spin dynamics below $T_{\rm L}$, consistent with
the thermodynamic results. Moreover, the $\mu$SR results reveal the
coexistence of static and dynamic local magnetic fields below $T_{\rm L}$,
signifying the presence of complex magnetic phases owing to the entwining of
spin correlations and competing magnetic interactions in this three-
dimensional frustrated magnet.

††preprint: APS

##  I Introduction

Frustrated quantum materials, where competing interactions between localized
spin moments and frustration-induced strong quantum fluctuations prevent
classical Néel order, are highlighted as promising contenders for the
discovery of emergent physical phenomena such as quantum spin liquid (QSL),
and spin ice with exotic excitations that goes beyond conventional symmetry
breaking paradigms in condensed matter Balents (2010); Mermin and Wagner
(1966); Khuntia (2019); Khatua  _et al._ (2023).
A QSL state is a highly entangled state, in which frustration-induced strong
quantum fluctuations defy long-range magnetic order even at absolute zero
temperatures, despite strong exchange coupling between magnetic moments in the
host spin-lattice Zhou  _et al._ (2017); Wen  _et al._ (2019); Savary and
Balents (2016); Broholm  _et al._ (2020a); Anderson (1973). Instead, two spins
in the corresponding lattice form resonating spin-singlet pairs or long-range
entangled states with exotic fractionalized excitations such as spinons
coupled to emergent gauge fields Wen (1991). Experimental realization of
fractional quantum numbers, their identification and interaction are of
profound importance in elucidating the underlying mechanism of some of the
groundbreaking phenomena such as high-temperature superconductivity,
fractional Hall effect, topological spin-textures, and monopoles in quantum
condensed matter Baskaran  _et al._ (1987); Anderson (1987); Jiang and
Kivelson (2021); Fradkin  _et al._ (2015); Castelnovo  _et al._ (2012); Nayak
_et al._ (2008); Phillips  _et al._ (2020); Göbel  _et al._ (2021) that could
have far-reaching ramifications in both fundamental physics and innovative
quantum technologies Nayak  _et al._ (2008). The realization of QSL state is
well established in one-dimensional magnets, however, its identification in
higher dimensional spin system remains a long-standing challenge in modern
condensed matter physics.
In the quest of this long-sought goal, significant effort has been devoted to
the exploration of two-dimensional (2D) frustrated lattices Khatua  _et al._
(2023), which has led to the identification of a few QSL candidates in 2D
systems Jeon  _et al._ (2023); Arh  _et al._ (2022). Meanwhile, three-
dimensional (3D) frustrated spin-lattices such as hyperkagome and pyrochlore
lattice are found to host QSL as well. Despite the fact that quantum
fluctuations are less pronounced in 3D lattices, certain frustrated 3D
lattices, including transition metal-based hyperkagome lattices PbCuTe2O6
Khuntia  _et al._ (2016); Chillal  _et al._ (2020) and Na4Ir3O8 Shockley  _et
al._ (2015); Dally  _et al._ (2014) along with pyrochlore lattice NaCaNi2F7
Plumb  _et al._ (2019); Krizan and Cava (2015); Cai  _et al._ (2018) as well
as rare-earth-based pyrochlore lattice Ce2Zr2O7 Gao  _et al._ (2019), shows
signatures of QSL state driven by frustration-induced strong quantum
fluctuations Khatua  _et al._ (2022a). However, the experimental realization
of QSL in 3D spin lattice remains scarce, as most of the candidate materials
show spin-freezing or magnetic ordering owing to perturbing terms in the spin
Hamiltonian, lattice imperfections, and exchange anisotropy. In this respect,
it is pertinent to investigate promising frustrated magnets on 3D spin
lattices wherein competition between emergent degrees of freedom offers an
alternate route to stabilize exotic quantum and topological states Khatua  _et
al._ (2023); Broholm  _et al._ (2020b); Kivelson and Sondhi (2023); Lancaster
(2023); Khatua  _et al._ (2022b)
To date, hexagonal-based layered frustrated magnets have been studied
rigorously for their 2D QSL and certain cubic lattice systems for their 3D
counterparts Nguyen and Cava (2021). Recently, there is growing interest in
non-centrosymmetric frustrated quantum materials which crystallize in the
cubic space group $P2_{1}3$ Bradley and Cracknell (2010); Kakihana  _et al._
(2017); Redpath and Hopkinson (2010). The lack of inversion symmetry in such
non-centrosymmetric materials gives rise to interesting physical phenomena
owing to anisotropic Dzyaloshinskii-Moriya (DM) interactions Chizhikov and
Dmitrienko (2013); Bradley and Cracknell (2010); Dzyaloshinskii (1964); Ding
_et al._ (2021); Ramakrishnan  _et al._ (2019); Izyumov (1984). For example,
non-centrosymmetric cubic quantum materials are known to host a wide range of
quantum phenomena including magnetic skyrmions in insulator Cu2OSe2O3 Qian
_et al._ (2018), and in itinerant magnets MnSi Mühlbauer  _et al._ (2009), and
FeGe Yu  _et al._ (2011), and topological Hall effect in MnGe Kanazawa  _et
al._ (2011). Apart from quantum phenomena induced by DM interactions, the spin
frustration induced quantum phenomena in such 3D non-centrosymmetric chiral
magnets have garnered significant attention in the quest to achieve distinct
quantum phenomena. It is observed that in such quantum materials that
crystallizes in the cubic structure ($P2_{1}3$) the magnetic ions form a
trillium lattice i.e., a 3D chiral network of corner-shared equilateral
triangular motifs which provides the origin of spin frustration and may host a
myriads of frustration-induced physical phenomena such as QSL. It is a well
known that a geometrically frustrated lattice has the potential to harbor
frustration-induced new quantum phenomena, as observed in intermetallic
compounds. Notable examples include the presence of a dynamic state above the
transition temperature in EuPt(Si/Ge) Higa  _et al._ (2021), a spin-ice-like
state in CeIrSi Kneidinger  _et al._ (2019), and the occurrence of pressure-
induced quantum phase transitions in MnSi Pfleiderer  _et al._ (1997).
Nevertheless, oxide-based materials having a single trillium lattice structure
are extremely rare. Recently, the trillium lattice, composed of corner-sharing
equilateral triangles with six nearest neighbors, has emerged as an
alternative route to further explore the influence of quantum fluctuations in
the 3D QSL state and to realize unique quantum phenomena Hopkinson and Kee
(2006); Redpath and Hopkinson (2010). Remarkably, a recent study on the
langbeinite family member K2Ni2(SO4)3, which crystallizes in a cubic ($P$213)
crystal structure, reveals that Ni2+ ($S$ = 1) ions form a coupled trillium
lattice Živković  _et al._ (2021). This material demonstrates several magnetic
sublattices with competing magnetic interactions and exhibits a 3D QSL state
induced by an applied magnetic field Živković  _et al._ (2021). More recently,
continuum spin excitations driven by strong quantum fluctuations are revealed
by inelastic neutron scattering measurement on single crystals of K2Ni2(SO4)3
Yao  _et al._ (2023). Furthermore, despite possessing a high spin moment, the
signature of a spin-liquid state has been observed in the trillium compounds
KSrFe2(PO4)3 (Fe3+; $S$ = 5/2) Boya  _et al._ (2022) and Na[Mn(HCOO)3] (Mn2+;
$S$ = 5/2) Bulled  _et al._ (2022), that invokes further investigation of the
ground state and associated quasi-particle excitations in analogous trillium
lattice antiferromagnets. In addition, the spin correlations, interplay
between competing interactions and the topology of the electronic band
structure in non-centrosymmetric magnets stabilizing in 3D spin lattice make
them an exciting class of materials with potential technological applications
Huai and Tran (2023); Hall  _et al._ (2023).
Herein, we focus on a new phosphate langbeinite K2CrTi(PO4)3 (henceforth
KCTPO), wherein Cr3+ ions with $S$ = 3/2 form a trillium lattice without any
detectable anti-site disorder between constituent atoms. Magnetization data
reveals the presence of dominant antiferromagnetic interactions between $S$ =
3/2 moments of Cr3+ ions. Specific heat measurements demonstrate two magnetic
transitions, namely at $T_{\rm L}$ = 4.3 K and $T_{\rm H}$ = 8 K, as well as
the development of short-range spin correlations above $T_{\rm H}$, which is
supported by critical-like behavior of the ESR line width. Furthermore,
$\mu$SR data uncovers an intriguing evolution of static and dynamic spin
correlations, notably below $T_{\rm L}$, and a critical slowing down of spin
dynamics above $T_{\rm L}$. Our experimental findings reveal a complex
magnetic landscape with coexisting states, successive phase transitions, and
dynamic behavior of Cr3+ moments decorating a frustrated 3D trillium spin-
lattice over a wide temperature regime.

Figure 1: (a) Rietveld refinement pattern of the room temperature powder x-ray
diffraction data of K2CrTi(PO4)3. The orange circle, black line, olive
vertical bars, and blue line show the experimentally observed points, the
result of Rietveld fitting, expected Bragg reflection positions, and the
difference between observed and calculated intensities, respectively. (b)
Schematic depicting the one unit cell of K2CrTi(PO4)3. The Cr3+ ions form
distorted CrO6 octahedra (blue), and the P5+ ions form PO4 tetrahedra (pink).
These are connected through a shared oxygen ion, resulting in the formation of
a Cr-O-P-O-Cr super-exchange bridge with nearest CrO6 octahedra. (c) The
nearest-neighbor Cr3+ ions (solid blue line; 6.10 Å) arrange themselves in a
trillium lattice, featuring six nearest neighbors. Trillium lattice is
composed of three motifs of equilateral triangle of Cr3+ ions with possible
nearest-neighbor superexchange routes. The second (8.32 Å), third (9.79 Å),
and fourth (10.06 Å) nearest neighbors are shown by the dashed orange, red,
and olive lines, respectively. Other atoms are omitted to ensure a clearer
view of the trillium lattice and potential nearest-neighbor exchange paths.

.

##  II Experimental details

Polycrystalline samples of KCTPO were prepared by a conventional solid-state
reaction method. The appropriate stoichiometric amounts of K2CO3 (Alfa Aesar,
99.997 %), Cr2O3 (Alfa Aesar, 99.97 %), TiO2 (Alfa Aesar, 99.995 %) and
(NH4)2HPO4 (Alfa Aesar, 98 %) were mixed. Prior to use, the reagent K2CO3 was
preheated in air at 100∘C to eliminate moisture. The stoichiometric mixture
was pelletized, and the pellet was sintered at 300∘C for 4 hrs. This sintering
procedure was repeated at several intermediate temperatures before annealing
the sample at 800∘C for 48 hrs to achieve a single-phase composition. Powder
x-ray diffraction (XRD) data were collected using a smartLAB Rigaku x-ray
diffractometer with Cu Kα radiation ($\lambda$ = 1.54 Å) at room temperature.
Magnetization measurements were performed using the VSM option of Physical
Properties Measurement System (PPMS, Quantum Design) in the temperature range
of 2 K $\leq$ T $\leq$ 300 K and in magnetic fields up to 7 T. Specific heat
measurements were performed using PPMS by thermal relaxation method in the
temperature range of 1.9 K $\leq$ T $\leq$ 250 K and in magnetic fields up to
7 T. High-field magnetization measurement was conducted at Dresden High
Magnetic Field Laboratory, sweeping a magnetic field up to 30 T at 4 K using a
nondestructive pulsed magnet. The obtained data were scaled to the isothermal
magnetization taken with PPMS at 4 K.
X-band ($\nu$ = 9.5 GHz) electron spin resonance (ESR) measurements were
performed using a Bruker EMXplus-9.5/12/P/L spectrometer with a continuous He
flow cryostat in the temperature range of 4 K $\leq$ $T$ $\leq$ 300 K. The
$\mu^{+}$SR experiments in zero field and in weak transverse magnetic field
(32 G) were performed at the surface muon beamline M20 at TRIUMF in the
temperature range 1.93 K $\leq$ $T$ $\leq$ 80 K. Powder samples (approximately
0.5 g) were placed into a thin envelope composed of Mylar tape coated with
aluminum ($\sim$ 50 $\mu$m thick), which was mounted on a Cu fork sample
stick. A standard 4He flow cryostat was employed to achieve the base
temperature of 1.93 K for TRIUMF/M20. The obtained $\mu$SR data was analyzed
using the musrfit software package Suter and Wojek (2012).

##  III results

###  III.1 Rietveld refinement and crystal structure

In order to ensure phase purity and determine atomic parameters of the
polycrystalline samples of KCTPO, the Rietveld refinement of powder XRD data
was performed using GSAS software Toby (2001). For the Rietveld refinement,
the initial atomic parameters were obtained from references Isasi and Daidouh
(2000); Boudjada and Perret (1977); Norberg (2002). The Rietveld refinement
reveals that KCTPO crystallizes in a cubic space group ($P2_{1}3$) and the
resulting refinement pattern is shown in Fig. 1 (a). No secondary phase was
found, confirming the successful synthesis of KCTPO in its single phase. The
observed sharp and well-defined XRD peaks imply that high-quality
polycrystalline samples have been used in this study. The obtained lattice
parameters as well as goodness factors from the Rietveld refinement of powder
XRD are tabulated in table 1. Our analysis shows the absence of any detectable
anti-site disorders between the atoms that constitute KCTPO.
Figure 1 (b) depicts the one unit cell of KCTPO, where Cr3+ ions form
distorted CrO6 octahedra with nearest-neighbor O2- that are separated from
adjacent CrO6 octahedra by PO4 tetrahedra. The bonding between the CrO6
octahedra and TiO4 tetrahedra through a common oxygen ion is expected to
construct a 3D Cr-O-P-O-Cr superexchange pathway between two nearest-neighbor
Cr3+ ions (see table 2). Most interestingly, the nearest-neighbor Cr3+ ions
(6.10 Å) form a trillium lattice, i.e., a 3D network of corner-shared
equilateral triangular lattice with coordination number six (Fig. 1 (c)),
potentially serving as a platform for hosting a frustrated spin-lattice in 3D.
From a structural point of view, KCTPO belongs to the langbeinite family
K2$M_{2}$(SO4)3 ($M$ = Mg, Co, Ni,..), which crystallizes in the cubic crystal
structure (space structure $P$213). Two crystallographic 4$a$ sites occupy
divalent magnetic ions that form a double trillium lattice in the sulfate
langbeinites without any anti-site disorder Živković  _et al._ (2021). Among
various phosphate langbeinites, the recently reported phosphate langbeinite
KSrFe2(PO4)3 shows a similar configuration, featuring a double trillium
lattice with two magnetic sites of Fe3+ ions and K/Sr anti-site disorder Boya
_et al._ (2022). Unlike the aforementioned double trillium lattice-based
langbeinites, KCTPO, a phosphate langbeinite, accommodates a single magnetic
site (4$a$) hosting Cr3+ ions, alongside another 4$a$ site occupied by non-
magnetic Ti4+ ions. This characteristic positions KCTPO as a considerably
simpler 3D lattice, suited for the experimental realization of theoretically
proposed physical phenomena intrinsic to the trillium lattice Isakov  _et al._
(2008); Kakihana  _et al._ (2017).

Figure 2: (a) Temperature dependence of magnetic susceptibility, $\chi(T)$, of
K2CrTi(PO4)3 in several magnetic fields. The inset depicts the temperature
dependence of inverse magnetic susceptibility at $\mu_{0}H$ = 1 T. The red
line represents the Curie-Weiss fits to the high-temperature inverse
susceptibility data. (b) Temperature dependence of zero-field cooled (ZFC) and
field-cooled (FC) magnetic susceptibility in the temperature range 2 K $\leq$
$T$ $\leq$ 5 K in magnetic fields 0.005 T $\leq$ $\mu_{0}H$ $\leq$ 0.05 T. The
dotted vertical line is at temperature $T_{\rm L}$ = 4.3 K, below which ZFC
and FC bifurcation begins in a magnetic field of $\mu_{0}H$ = 0.005 T. (c)
Magnetization as a function of an external magnetic field up to 30 T at 4 K.
The inset shows the isotherm magnetization up to 9 T at several temperatures.

.

Table 1:  Rietveld refinement of x-ray diffraction data at 300 K yielded structural parameters of K2CrTi(PO4)3. (Space group: $P$213, $a=b=c$ = 9.796 Å, $\alpha=\beta=\gamma=90^{\circ}$ and $\chi^{2}$ = 2.86, Rwp = 5.29 %, Rp = 3.56 %, and Rexp = 1.84%) Atom | Wyckoff position | x | y | z | Occ.
---|---|---|---|---|---
Cr | 4$a$ | 0.666 | 0.666 | 0.666 | 1
K1 | 4$a$ | 0.960 | 0.960 | 0.960 | 1
K2 | 4$a$ | 0.186 | 0.186 | 0.186 | 1
Ti | 4$a$ | 0.391 | 0.391 | 0.391 | 1
P | 12$b$ | 0.474 | 0.703 | 0.368 | 1
O1 | 12$b$ | 0.609 | 0.860 | 0.737 | 1
O2 | 12$b$ | 0.463 | 0.314 | 0.232 | 1
O3 | 12$b$ | 0.506 | 0.573 | 0.313 | 1
O4 | 12$b$ | 0.585 | 0.721 | 0.457 | 1
Table 2:  Bond lengths and angles between atoms that result in distinct antiferromagnetic interactions between Cr3+ spins.  Bond length (Å) | Bond angle (∘) |
---|---|---
Cr-O1 = 2.101 |  $\angle$ Cr-O1-P = 153.510 |
Cr-O4 = 2.262 |  $\angle$ O1-P-O4 = 120.635 |
P-O4 = 1.405 |  $\angle$ P-O4-Cr = 143.804 |
P-O1 = 1.273 |  $\angle$Cr-Cr-Cr = 60 |
Cr-P = 3.494 |  $\angle$ Cr-P-Cr = 128.355 |
O1-O4 = 2.32 |  $\angle$ Cr-O1 -Cr = 154.692 |

###  III.2 Magnetic susceptibility

Figure 2 (a) depicts the temperature dependence of magnetic susceptibility
$\chi(T)$ in several magnetic fields up to 7 T. Upon lowering temperature, a
sharp increase of $\chi(T)$ was observed below 10 K, tending to saturate in a
magnetic field $\mu_{0}H$ = 1 T. This behavior indicates the presence of long-
range magnetic order in KCTPO consistent with the specific heat result
presented in this work. However, this behavior is suppressed significantly in
magnetic fields $\mu_{0}H\geq 3$ T. A minor bump around 50 K that appeared in
all magnetic fields is attributed to residual oxygen trapped in the
polycrystalline sample that was wrapped with teflon during measurement Khan
_et al._ (2019). This signal is not an intrinsic characteristic of the
investigated compound in this study.
In order to determine the dominant magnetic interaction between the $S$ = 3/2
moments of Cr3+ ions, the linear region (100 K $\leq$ $T$ $\leq$ 300 K ) of
the 1/$\chi(T)$ data in a magnetic field $\mu_{0}H$ = 1 T was fitted by the
Curie-Weiss (CW) law, i.e., $\chi=\chi_{0}$ \+ $C$/($T$ $-$ $\theta_{\rm
CW}$). Here, $\chi_{0}$ is the sum of temperature-independent core diamagnetic
susceptibility ($\chi_{\rm core}$) and Van Vleck paramagnetic susceptibility
($\chi_{\rm VV}$), $C$ is the Curie constant used to estimate the effective
magnetic moment ($\mu_{\rm eff}$ = $\sqrt{8C}$ $\mu_{\rm B}$) and $\theta_{\rm
CW}$ is the Curie-Weiss temperature associated with the exchange interaction
between magnetic moments of Cr3+ ions. The corresponding CW fit (red line in
the inset of Fig. 2 (a)) yields $\chi_{0}$ = $-$7.42 $\times$ 10-4 cm3/mol,
$C$ = 1.88 cm3 K/mol, and $\theta_{\rm CW}$ = $-$23 $\pm$ 0.15 K. The
estimated effective magnetic moment $\mu_{\rm eff}$ = 3.87 $\mu_{\rm B}$, is
close to the value g$\sqrt{S(S+1)}$ = 3.87 $\mu_{\rm B}$ expected for $S$ =
3/2 moments of free Cr3+ ions. The negative $\theta_{\rm CW}$ indicates the
presence of dominant antiferromagnetic interactions between the $S$ = 3/2 spin
of Cr3+ ions. As depicted in the inset of Fig. 2 (a), the Curie-Weiss fit
begins to diverge below 80 K, signaling the onset of antiferromagnetic spin
correlations upon lowering the temperature Lee  _et al._ (2022).
Interestingly, the material shows anomalies at low temperatures associated
with magnetic phase transitions.
To further understand the effect of magnetic fields on the long-range magnetic
ordered state in KCTPO, the zero-field-cooled (ZFC) and field-cooled (FC)
$\chi$ measurements were performed in weak magnetic fields $\mu_{0}H\geq
0.005$ T. Figure 2 (b) reveals the presence of a clear bifurcation between ZFC
and FC $\chi$ below the temperature ($T_{\rm L}$) = 4.3 K in a magnetic field
of $\mu_{0}H=0.005$ T, which implies the presence of a ferromagnetic component
below $T_{\rm L}$. Nevertheless, as the magnetic field strength increases, the
deviation between ZFC and FC magnetic susceptibility gradually diminishes,
almost converging at a magnetic field $\mu_{0}H$ = 0.05 T.
In order to shed more insights concerning the nature of magnetically ordered
phases at different temperature regimes, isotherm magnetization measurements
were performed at several temperatures as shown in the inset of Fig. 2 (c).
Several features can be observed in the isotherm magnetization curves such as
a weak $S$-shaped curvature in low-fields at 2 K, indicating the existence of
a ferromagnetic component consistent with the bifurcation between ZFC and FC
susceptibility below $T_{\rm L}$ Zhu  _et al._ (2021); Bukowski  _et al._
(2022); Lai  _et al._ (2021). Notably, the magnetization at 2 K does not reach
saturation in a high magnetic field (9 T); instead, the linear behavior of the
magnetization curves implies the presence of a canted antiferromagnetic state
below $T_{\rm L}$. A similar scenario, attributed to the canted
antiferromagnetic phase, has also been observed in several frustrated magnets
Cao  _et al._ (2013); Verrier  _et al._ (2020). As the temperature increases,
the $S$-shaped curvature is gradually suppressed due to the quenching of the
ferromagnetic component, finally disappearing. The resulting magnetization
curve becomes linear at 10 K, indicating the crossover to an additional
antiferromagnetic phase. These observations imply that KCTPO exhibits a canted
antiferromagnetic state below $T_{\rm L}$, above which it hosts an additional
antiferromagnetic state corroborated by the specific heat results described
below. To find the saturation magnetic moment in high magnetic fields,
magnetization measurements were performed at 4 K using a pulsed field magnet
up to 30 T.
Figure 2 (c) represents the high-field magnetization data at 4 K, calibrated
using the VSM-SQUID data at 4 K. It is worth to note that a weak ferromagnetic
component persists in low fields $\mu_{0}H$ $\leq$ 0.012 T and beyond this
magnetic-field, the magnetization continues to increase linearly until
$\mu_{0}H$ $\leq$ 14 T and reaches a saturation magnetic moment of 3.04
$\mu_{\rm B}$ in 24 T. We recall that the saturation field is comparable to
the CW temperature discussed above. The obtained saturation magnetic moment of
3.04 $\mu_{\rm B}$ is close to the expected theoretical value of 3.0 $\mu_{\rm
B}$ per Cr3+ ions ($S$ = 3/2). The absence of a fractional “1/3” magnetization
plateau indicates either the classical behavior of Cr3+ spins or the presence
of perturbation terms such as magnetic anisotropy and longer-range
interactions Schmidt and Thalmeier (2017); Kimura  _et al._ (2022).

Figure 3: (a)Temperature dependence of specific heat (Cp) of K2CrTi(PO4)3 in
zero-magnetic field. The solid red line represents the lattice contributions
obtained by combining one Debye and three Einstein functions as described in
text. The inset shows a closer view of the low-temperature specific heat. (b)
Temperature dependence of $C_{p}(T)$ in several magnetic fields in the
temperature range 2 K $\leq$ $T$ $\leq$ 20 K. The dashed vertical lines
indicate two anomalies: one at temperature $T_{\rm L}$ = 4.30 K and another at
temperature $T_{\rm H}$ = 8 K in zero-field. (c) Temperature dependence of
$C_{\rm mag}(T)$ in several magnetic fields in semi-log scale. Below $T_{\rm
L}$ = 4.3 K, the solid lines tentatively represent $\sim T^{n}$ power-law
behavior of magnetic specific heat, where $n$ varies from 1.3 to 1 in an
applied magnetic field of 7 T. The dashed orange line represents the $C_{\rm
mag}(T)$ $\sim$ $T^{3}$ behavior typical for conventional antiferromagnets.
(d) Temperature dependence of entropy change in several magnetic fields with
the horizontal pink line indicating the expected entropy of $R$ln(4) for $S$ =
3/2 spin of Cr3+ ions.

.

###  III.3 Specific heat

Specific heat is an excellent probe to track magnetic order and associated
low-lying excitations in frustrated magnets. In order to discern the evidence
of magnetic long-range order, temperature-dependent specific heat measurements
were performed in several magnetic fields. Figure 3 (a) shows the temperature
dependence of specific heat in the whole measured temperature range in zero-
magnetic field. As shown in Fig. 3 (b), below $T$ = 10 K, the specific heat
exhibits two anomalies: one at the temperature $T_{\rm L}$ = 4.3 K and another
at the temperature $T_{\rm H}$ = 8 K in zero-magnetic field, suggesting the
occurrence of successive magnetic phase transitions in KCTPO. It is worth to
note that the low-temperature anomaly occurring at $T_{L}$ in specific heat
coincides with the temperature below which the splitting of ZFC and FC $\chi$
(Fig. 2 (b)) as well as the appearance of a ferromagnetic component in
magnetization are observed (Fig. 2 (c)). This coincidence suggests that the
anomaly at $T_{L}$ is most likely related to the transition temperature of the
canted antiferromagnetic phase in KCTPO. On the other hand, the anomaly at
$T_{\rm H}$ = 8 K is attributed to the transition temperature of additional
antiferromagnetic phase of KCTPO. We recall that the two successive magnetic
phase transitions are also observed in the two coupled trillium lattice
K2Ni2(SO4)3 Živković  _et al._ (2021).
Figure 3 (b) displays the specific heat in several magnetic fields in the low-
temperature regime. The salient observation is that the low-temperature
anomaly at $T_{\rm L}$ shifts towards higher temperatures upon the application
of a magnetic field, eventually vanishing at a magnetic field of $\mu_{0}H$ =
3 T. The increase in $T_{\rm L}$ with increasing magnetic field can be
ascribed to either strong quantum fluctuations or the presence of competing
anisotropic interactions. While the applied magnetic field increases, the
anomaly at $T_{\rm H}$ widens, shifts to slightly lower temperatures, and a
clear $\lambda$-like anomaly is observed in $\mu_{0}H$ = 7 T, suggesting the
presence of a typical antiferromagnetic second-order phase transition at
$T_{\rm H}$ in KCTPO. Moreover, in the temperature range 8 K $\leq$ $T$ $\leq$
20 K, the specific heat exhibits field dependency, indicating that specific
heat is of magnetic origin in this temperature range.
In order to estimate the magnetic entropy and understand the low-temperature
magnetic properties relevant to this trillium lattice antiferromagnet, one
must subtract the lattice specific heat ($C_{\rm latt}(T)$) from the total
specific heat ($C_{\rm p}(T)$). To extract the lattice contributions, we used
a combination of one Debye and three Einstein terms, i.e.,

$C_{\rm
latt}(T)=C_{D}[9k_{B}\left(\frac{T}{\theta_{D}}\right)^{3}\int_{0}^{\theta_{D}/T}\frac{x^{4}e^{x}}{(e^{x}-1)^{2}}dx]$
$+\sum_{i=1}^{3}C_{E_{i}}[3R\left(\frac{\theta_{E_{i}}}{T}\right)^{2}\frac{\text{exp}(\frac{\theta_{E_{i}}}{T})}{(\text{exp}(\frac{\theta_{E_{i}}}{T})-1)^{2}}],$
(1)

where $\theta_{D}$ is the Debye temperature, $\theta_{E_{i}s}$ are the
Einstein temperatures of the three modes, R and kB are the molar gas constant
and Boltzmann constant, respectively. The corresponding lattice fit, yielding
$\theta_{D}$ = 223 K, $\theta_{E_{1}}$ = 376 K, $\theta_{E_{2}}$ = 615 K, and
$\theta_{E_{3}}$ = 1344 K, is represented by the solid red line on top of the
experimental data point in Fig. 3 (a). In order to minimize the fitting
parameters, the coefficients were set at a fixed ratio of $C_{\rm D}$ :
$C_{E_{1}}$ : $C_{E_{2}}$ : $C_{E_{3}}$ = 1 : 1 : 1.5 : 6, closely matching
with the ratio of the number of heavy atoms (K, Cr, Ti, P) to light atoms (O)
in KCTPO.

Figure 4: (a) Derivative of the ESR absorption spectra of K2CrTi(PO4)3 at
selected temperatures. The solid red lines indicate the fitting to the
Lorentzian line shape above 8 K. (b) Temperature dependence of g-factor in
semi-log scale. (c) Semi-log plot of the ESR linewidth $\Delta H_{\rm w}$ as a
function of the reduced temperature expressed as $T_{\rm rl}$ = ($T-$ $T_{\rm
L}$)/$T_{\rm L}$ on the bottom $x$-axis and $T_{\rm rh}$ = $(T-T_{\rm
H})/T_{\rm H}$ on the upper $x$-axis. The dashed lines are power-law fits.

.

After subtraction of the lattice contributions, the obtained temperature
dependence of magnetic specific heat $C_{\rm mag}$ ($T$) is shown in Fig. 3
(c). As the temperature decreases in a zero-magnetic field, $C_{\rm mag}$
increases and exhibits a broad maximum around temperature $T^{*}$ = 14 K,
followed by a weak kink at $T_{\rm H}$. The broad maximum is attributed to the
presence of short-range spin correlations Živković  _et al._ (2021). The
presence of a moderate frustration parameter $f$, roughly estimated through
the ratio $f=|\theta_{\rm CW}|/T_{\rm H}\approx 3$, could account for both the
short-range order and the suppression of magnetic order. The anomaly at
$T_{\rm H}$ is relatively weak compared to the anomaly at $T_{\rm L}$ similar
to the observed anomalies in K2Ni2(So4)3 Živković  _et al._ (2021).
Nevertheless, upon increasing the magnetic field, the canted antiferromagnetic
phase diminishes, leading to an increase in the anomaly observed at $T_{\rm
H}$ in $\mu_{0}H$ = 7 T. Most notably, as shown in Fig. 3 (c), the magnetic
specific heat follows $C_{\rm mag}~{}$$\sim$ $T^{n}$ ($n$ = 1.28) power-law in
zero field for temperatures below $T_{\rm L}$, which significantly deviates
from the $C_{\rm mag}(T)$ $\sim$ $T^{3}$ behavior typical of conventional
antiferromagnets. In the presence of a magnetic field up to 7 T, the average
value of $n$ was found to be 1.10. It is worth noting that in future low
temperature specific heat measurements are required to confirm whether the
power-law behavior persists even in the sub-Kelvin temperature range. In
contrast to KCTPO, the trillium lattice structure with two interpenetrating
sublattices of K2Ni2(SO4)3, well below the transition temperature, the
magnetic specific heat follows $C_{\rm mag}$ $\sim$ $T^{2}$ behavior Živković
_et al._ (2021).

Figure 5: (a) Time dependence of the muon asymmetry in K2CrTi(PO4)3 in a weak
transverse field ($B_{\rm TF}$ = 32 G) at a several temperatures. The solid
lines are fits obtained using Eq. 2. (b) Temperature dependence of muon spin
relaxation rate in transverse applied field on a double logarithmic scale. The
inset shows the fraction of the oscillating component as a function of
temperature. (c) Temperature dependence of muon spin relation rate due to
internal magnetic field in the investigated sample. The dashed vertical lines
indicate the position of characteristic transition temperatures at $T_{\rm L}$
= 4.3 K and $T_{\rm H}$ = 8 K.

.

To quantify the entropy release associated with phase transitions and spin
dynamics, it is crucial to account the magnetic-specific heat below 2 K. In
the absence of no more phase transition below 2 K, we extend the corresponding
power-law to zero temperature. Next, the entropy was obtained by integrating
$C_{\rm mag}/T$ with respect to temperature, as shown in Fig. 3 (d). The
obtained saturation entropy is found to be 11.16 J/mol$\cdot$K at 30 K that is
close the theoretically expected value 11.52 J/mol$\cdot$K for $S$ = 3/2
moments. The missing $\sim$ 3.12 % entropy is ascribed to either the existence
of short-range magnetic correlations well above the transition temperature or
an overestimation of the lattice contribution given the limitations of the
model employed here Khatua  _et al._ (2021); Murugan  _et al._ (2023).
Additionally, around the transition temperature $T_{H}$ of 8 K, approximately
76 % of the saturation entropy is released, suggesting that roughly 20 %
originates from short-range spin correlations that begin forming as high as at
$\sim 3T_{\rm H}$. However, the lack of a few percentage of entropy because of
short-range order persisting above the transition temperature is typical for
frustrated magnets Lee  _et al._ (2022); Khatua  _et al._ (2021). This
observation is corroborated by the ESR results presented in the following
subsection.

###  III.4 Electron spin resonance

To provide microscopic insights into the temperature evolution of spin
correlations, we conducted X-band ESR measurements down to 4 K Abragam and
Bleaney (1970). The obtained ESR spectra at several temperatures are shown in
Fig. 4 (a), which fits well with a derivative of a Lorentzian curve above 8 K.
This indicates that the ESR signal is exchange narrowed. At temperatures below
$T$ = 10 K, the spectra cannot be fitted by a single Lorentzian curve,
indicating the development of antiferromagnetic resonance mode and supporting
the conclusions drawn from thermodynamic results. The temperature dependence
of estimated $g$-factors from the fit is shown in Fig. 4 (b). Above $T$ = 100
K, the obtained $g$ value remains relatively constant at approximately $g$ =
1.972, slightly lower than the standard $g$ = 2 for a free Cr3+ ion. This
value is typical for a less-than-half-filled Cr3+ ion with a negligible spin-
orbit interaction in an octahedral ligand coordination, as observed in similar
compounds Murugan  _et al._ (2023). The $g$-factor starts to decrease with
temperature below 100 K, which corresponds to the deviation from the CW fit
(see the inset of Fig. 2 (a)). This is attributed to the persistent magnetic
correlations up to several $|\theta_{\rm CW}|$, often observed in frustrated
magnets Lee  _et al._ (2022).
The ESR linewidth is directly proportional to the spin correlations in
transition metal-based systems with minimum spin-orbit coupling. Figure 4 (c)
shows the estimated ESR linewidth ($\Delta H_{\rm w}$) of KCTPO as a function
of reduced temperatures $T_{rl}$ = $(T-T_{\rm L})/T_{\rm L}$ in the bottom
$x$-axis and $T_{rh}$ = $(T-T_{\rm H})/T_{\rm H}$ in the upper $x$-axis. Above
the transition temperature $T_{\rm H}$, the observed linewidth exhibits a
power-law behavior of $T_{\rm rl}/T_{\rm rh}\sim T^{-p}$ in two distinct
temperature ranges. For temperatures spanning from 10 K to 24 K, the power
exponent $p$ takes values of 0.097 and 0.072, while in the range of 30 K to
100 K, the values of $p$ are 0.035 and 0.032 for $T_{\rm rl}$ and $T_{\rm
rh}$, respectively Glamazda  _et al._ (2017); Murugan  _et al._ (2023); Lee
_et al._ (2016, 2022). The critical-like broadening in the temperature range
10 K $\leq$ $T$ $\leq$ 24 K, alongside the weak hump at $T^{*}$ in $C_{\rm
mag}$ and the small percentage of entropy released above 8 K, supports the
notion of the critical-like spin fluctuations. Conversely, the broadening
observed in the temperature range of 30 K $\leq$ $T$ $\leq$ 100 K is ascribed
to cooperative paramagnetic behavior between the $S=$ 3/2 spin of Cr3+ ions.
Additionally, a deviation from the Curie-Weiss fit has also been noted below
100 K (refer to Fig. 2 (a)). The presence of two stage broadening is also
observed in other high-spin based frustrated magnets alluding to the two-step
thermal evolution of magnetic correlations Glamazda  _et al._ (2017); Lee  _et
al._ (2022, 2016).

###  III.5 Muon spin relaxation

To capture the microscopic details of spin dynamics and the internal magnetic
field distribution, $\mu$SR measurements were performed on polycrystalline
samples of KCTPO in zero-field (ZF) and weak transverse field (wTF). When a
weak transverse field (here $B_{\rm TF}$ = 32 G) is applied perpendicular to
the initial direction of the muon spin polarization, the implanted muon spin
precesses around the applied transverse field in the paramagnetic state of the
sample, leading to an oscillatory signal with a frequency of
$\gamma_{\mu}$$B_{\rm TF}$/2$\pi$, where $\gamma_{\mu}$ = 2$\pi$ $\times$
135.5 MHz/T represents the muon gyromagnetic ratio Yaouanc and De Reotier
(2011). In addition, when a static magnetic field is present in the sample,
the muon spin polarization exhibits a non-oscillatory signal, as observed in
the material under investigation Nocerino  _et al._ (2023); Khatua  _et al._
(2021); Papadopoulos  _et al._ (2022). Figure 5 (a) shows the evolution of the
TF spectra at selected temperatures that were fitted according to

$A_{\rm wTF}(t)=A_{0}f\text{cos}(\gamma_{\mu}B_{\rm TF}t+\phi)e^{-\lambda_{\rm
TF}t}+(1-f)A_{0}e^{-\lambda_{\rm L}t},$ (2)

which combines an exponentially decaying oscillatory component, corresponding
to muon spins experiencing zero-static field, and an exponentially decaying
non-oscillatory component that accounts for the component of the static local
field parallel to the initial muon polarization. In Eq. 2, $A_{0}$ represents
the initial asymmetry at time zero, $\phi$ denotes the relative phase,
$\lambda_{\rm TF}$ stands for the muon spin relaxation rate in the applied
transverse field, and $f$ quantifies the fraction of the oscillatory component
while $\lambda_{\rm L}$ refers to the muon spin relaxation rate caused by the
internal magnetic fields. At short time scales, we observe a significantly
damped signal with decreasing temperature without any loss of initial
asymmetry, which is a characteristic signature of a magnetically ordered state
in KCTPO.
The estimated fraction of the oscillatory signal $f$, associated to the volume
fraction experiencing zero-static field (inset of Fig. 5 (b)), remains
constant to a value of 0.93 in the temperature range 4.3 K $\leq$ $T$ $\leq$
78 K. This temperature independent value of $f<1$ is attributed to the
presence of weak static local fields above 4.3 K which is also observed in ESR
results. However, it gradually decreases below $T_{\rm L}$= 4.3 K, indicating
a crossover to anther magnetically ordered state. It is noteworthy that $f$
does not sharply drop to zero below the transition temperature $T_{\rm L}$,
suggesting the existence of dynamic local fields in the ordered state. Upon
lowering the temperature, the transverse field muon spin relaxation rate
gradually increases (see Fig. 5 (b)), as expected in the extreme motional
narrowing limit of fluctuating magnetic moments. A sharp peak in $\lambda_{\rm
TF}$ at $T_{\rm L}$ indicates the critical slowing down of spin fluctuations
above the transition temperature as evident from unsaturated magnetic entropy.
The notable observation is the occurrence of the critical spin relaxation
below $T_{\rm H}$ Higa  _et al._ (2021). Conversely, the muon spin relaxation
rate caused by the static local field (Fig. 5 (c)) exhibits a weak kink at
$T_{\rm H}$, attributed to the antiferromagnetic phase transition similar to
that observed in specific heat. This is followed by a rapid rise towards
$T_{\rm L}$, indicating the enhancement of internal field fluctuations through
spin reconfiguration to the canted antiferromagnetic state. It is interesting
to note that, below $T_{\rm L}$, the $\lambda_{\rm TF}$ remains constant down
to 1.93 K without displaying a sharp fall, persistent spin fluctuations even
in the magnetically ordered state, similar to that observed in coupled
trillium lattice K2Ni2(SO4)3 Živković  _et al._ (2021).

Figure 6: (a) Time dependence of zero-field $\mu$SR spectra of K2CrTi(PO4)3 at
temperatures higher than the temperature of canted antiferromagnetic phase
transition. The solid lines represent the fitted curve by a stretched
exponential function. (b), (c) Temperature dependence of muon spin relaxation
rate and stretched exponent as a function of temperature in zero magnetic
field on a double logarithmic scale. The dashed vertical bars indicate the
position of characteristic transition temperatures at $T_{\rm L}$ = 4.30 K and
$T_{\rm H}$ = 8 K. (d) Time evolution of zero-field spectra at temperatures
below the canted antiferromagnet state. The solid lines are the fits of Eq. 3
to the data. (e) Temperature dependence of fast (left y-axis) and slow (right
y-axis) muon spin relaxation rate on a double logarithmic scale. (f) Mean
value of the Gaussian-broadened Gaussian distribution as a function of
temperature where the solid lines are the fitted curves with critical scaling
behavior of internal field distribution with critical exponent $\beta$ = 0.37.
Inset shows the temperature dependence of $R$ i.e, the ratio of the width of
distributions (W) to the distribution mean($\Delta_{0}$).

.

To validate the spin dynamics in the ordered states, ZF-$\mu$SR measurements
were performed in the wide temperature range. Figure 6 (d) shows the
ZF-$\mu$SR spectra at temperatures above $T_{\rm L}$ without any change of the
initial asymmetry. Furthermore, despite an anomaly in specific heat associated
to the magnetic phase transition, the absence of any oscillation or a strong
damped signal even in short-time scale, suggests that a dynamical behavior of
electronic spins associated with the magnetic phase similar to that observed
in trillium lattice antiferromagnet K2Ni2(SO4)3. The corresponding $\mu$SR
data (Fig. 6) can be well fitted to the phenomenological model $P(t)$ =
$A_{0}$ e${}^{-(\lambda_{\rm ZF}t)^{\beta_{\rm ZF}}}$, where $A_{0}$ is the
initial asymmetry, $\lambda_{\rm ZF}$ is the muon spin relaxation rate in
zero-field and $\beta_{\rm ZF}$ is the stretched exponent. The estimated
relaxation rate exhibits several features as shown in Fig. 6 (b). Upon
lowering the temperature from 78 K, the $\lambda_{\rm ZF}$ remains constant in
the temperature range 8 K $\leq$ $T$ $\leq$ 78 K. The essentially same
behavior is observed by $\lambda_{\rm TF}$ (see Fig. 5 (b)). Below 8 K, the
$\lambda_{\rm ZF}$ shows a rapid increase, ascribed to the presence of the
canted antiferromagnetic phase transition accompanied by a critical slowing
down of spin fluctuation consistent with our wTF-$\mu$SR results. The obtained
stretched exponent, measuring the distribution of electronic moment, is shown
Fig. 6 (c). At high-temperatures, $\beta_{\rm ZF}$ is close to one,
corresponding to fast fluctuation limit. As the temperature decreases, the
$\beta_{\rm ZF}$ value gradually decreases, notably dropping to 0.85 at
$T_{\rm H}\sim$ 8 K. Below 8 K, the $\beta_{\rm ZF}$ value gradually
decreases, but there is a sudden drop at $T_{\rm L}$ to a value of 0.6. It is
observed that the ZF-$\mu$SR data (for $T\geq 4.41$ K) cannot be accurately
fitted using a single exponential function ($\beta_{\rm ZF}$ = 1) or models
with multi-exponential components Lee  _et al._ (2022). The obtained
$\beta_{\rm ZF}<1$ is associated with the presence of a distribution of
relaxation times, as expected when various spins fluctuate in different
timescales due to competing magnetic interactions in KCTPO. The observed
$\beta_{\rm ZF}$ of 0.6 ($>1/3$) value, as seen in Fig. 6 (c), implies that
the splitting in ZFC and FC $\chi$ data at 4.3 K is due to canted
antiferromagnetic phase rather than spin freezing Campbell  _et al._ (1994).
Figure 6 (d) displays the time evolution of ZF-$\mu$SR spectra at several
temperatures below 4.5 K in the short-time scale. As the temperature decreases
below 4.5 K, the polarization initially exhibits a moderate departure from
stretched exponential behavior Yaouanc  _et al._ (2008). As it reaches around
1.9 K, it undergoes an abrupt transition into a strongly non-exponential
signal. The shorter time domain clearly shows a substantially damped signal,
making it challenging to simulate the muon polarization throughout the whole
temperature range using a stretched exponential relaxation function.
Furthermore, the absence of any peak in the first Fourier transform of the
corresponding spectra indicates that the damped signal can not be attributable
to coherent oscillations typically observed in long-range ordered magnets Koo
_et al._ (2018); Weinhold  _et al._ (2023); Papadopoulos  _et al._ (2022).
Instead, it suggests the existence of disordered static moments of Cr3+ ions
below 4.3 K. After attempting with several models Li  _et al._ (2021); Khuntia
_et al._ (2017); Zorko  _et al._ (2010), it turns out that the ZF-$\mu$SR data
can be modeled with the following polarization function

$P_{\rm ZF}=f_{\rm z}P_{\rm GDG}(t;\Delta_{0},W)e^{-\lambda_{s}t}+(1-f_{\rm
z})e^{-\lambda_{\rm f}t},$ (3)

which takes into account a phenomenological Gaussian-broadened Gaussian (GBG)
function with a slow exponential decay with relaxation rate $\lambda_{\rm s}$
and a fast exponential decay with relaxation rate $\lambda_{\rm f}$ Noakes and
Kalvius (1997); Do  _et al._ (2018). The parameter $f_{\rm z}$ defines the
fraction of Cr3+ moments that produce static local fields, while the remaining
part contributes to dynamic fields. The following GBG polarization function

$\begin{split}P_{\rm
GBG}(t)=&\frac{1}{3}+\frac{2}{3}(\frac{1}{1+R^{2}\Delta_{0}^{2}t^{2}})^{3/2}(1-\frac{\Delta_{0}^{2}t^{2}}{1+R^{2}\Delta_{0}^{2}t^{2}})\\\
&\rm exp[-\frac{\Delta_{0}^{2}t^{2}}{2(1+R^{2}\Delta_{0}^{2}t^{2})}]\\\
\end{split}$ (4)

is an extension of the Gaussian Kubo-Toyabe function with a Gaussian
distribution width $W$ and a mean value $\Delta_{0}$, where $R$ =
$W/\Delta_{0}$. This function is commonly employed when the internal field
distribution exceeds the Gaussian field distribution. It effectively captures
the existence of disordered static magnetic moments with short-range
correlations as observed in 3D hyperkagome compound such as Na4Ir3O8 which
exhibits ZFC and FC splitting in dc susceptibility similar to the compound
studied here Dally  _et al._ (2014). The extracted temperature dependence of
fast relaxation rate ($\lambda_{\rm f};$ left $y$-axis), representing a few
fraction of Cr3+ moments exhibiting dynamic behavior, and the slow relaxation
rate ($\lambda_{s};$ right $y$-axis) corresponds to the remaining fraction
Cr3+ moments displaying static behavior are shown in Fig. 6 (e). Both
$\lambda_{\rm f}$ and $\lambda_{s}$ start to increase below $T_{\rm L}$ and
attain a constant value at low temperatures indicating the coexistence of
static and dynamic components of Cr3+ moments in KCTPO.
In KCTPO, the origin of static moments can be associated to the canted
antiferomagnetic phase involving a small fraction of static Cr3+ moments.
Simultaneously, the rest of the Cr3+ moments maintain a dynamic state most
likely due to the interplay of competing magnetic interactions. In
geometrically frustrated magnets, the coexistence of dynamic and static
electronic moments is a common scenario, as also observed in two coupled
trillium lattice K2Ni2(SO4)3 and pyrochlore lattice NaCaNi2F7 Cai  _et al._
(2018). $\Delta_{0}$ undergoes an order-parameter-like decrease as the
temperature increases from 1.9 K towards $T_{\rm L}$. It may be noted that the
relative distribution width $R$ steadily increases and reaches a value of one
at $T_{\rm L}$ (inset of Fig. 6 (f)). The critical-like behavior of the local
field distribution near $T_{\rm L}$ is described by phenomenological
expression Lee  _et al._ (2023); Baker  _et al._ (2011)
$\Delta_{0}(T)=\Delta_{0}(T=0\ \ \text{K})\left[1-\frac{T}{T_{\rm
L}}\right]^{\beta}$ that yields $\beta$ = 0.37 that is typical for 3D
Heisenberg magnets (see the dashed line in Fig. 6 (f)). Using $\Delta_{0}$ =
35 MHz at $T$ = 1.9 K, the calculated internal magnetic field is found to be
$\langle B_{\rm loc}\rangle$ $\approx$ 400 G. In order to confirm such static
magnetic field is of electronic origin, we also performed $\mu$SR measurements
at 3.3 K in longitudinal field (not shown here), which decouples the static
field owing to nuclear origin. It is observed that approximately 2 kG $>$
$\langle B_{\rm loc}\rangle$ longitudinal field is required to recover full
polarization, confirming the presence of both dynamic and static moments of
Cr3+ ions in KCTPO. Similar to KCTPO, the observed muon spin asymmetry of
triangular lattice antiferromagnet NiGa2S4 was characterized by a stretched
exponential function at high temperatures, associated to multichannel
relaxation, while the data at low temperatures were primarily influenced by
static local fields Yaouanc  _et al._ (2008). The contrasting time evolution
of the muon spin polarization function, observed both above and below the
characteristic temperature of $T_{\rm L}$, may be associated with distinct
magnetic phases in KCTPO, similar to those found in other 3D frustrated
magnets Forslund  _et al._ (2020); Hodges  _et al._ (2002); Lee  _et al._
(2021); Frandsen  _et al._ (2020).

##  IV Conclusion

In summary, we have thoroughly investigated the structural and magnetic
properties of nearly perfect $S$ = 3/2 trillium lattice K2CrTi(PO4)3 through
magnetization, specific heat, as well as ESR and $\mu$SR techniques. Magnetic
susceptibility results suggest that the presence of antiferromagnetic
interactions between $S$ = 3/2 moments of Cr3+ ions. Specific heat
measurements unveil two consecutive phase transitions, one at $T_{\rm L}$ =
4.3 K, corresponding to the canted antiferromagnetic phase, and another at
$T_{\rm H}$ = 8 K due to an antiferromagnetic phase. Above $T_{\rm H}$, the
existence of short-range magnetic order is supported by the observation of
maximum entropy release and the manifestation of critical-like behavior in the
ESR line width. Furthermore, $\mu$SR provides concrete evidence of the
presence of static and dynamic local magnetic fields, particularly below
$T_{\rm L}$, as well as a critical slowing down of spin dynamics above $T_{\rm
L}$ and persistent spin dynamics to the lowest temperatures. Further
investigations on single crystals is essential to gain a detailed
understanding of the low-energy excitations associated to the short range
order above the transition temperature and the multi-stage spin dynamics in
this three-dimensional frustrated antiferromagnet.

## Acknowledgments

P.K. acknowledges the funding by the Science and Engineering Research Board,
and Department of Science and Technology, India through Research Grants. The
work at SKKU was supported by the National Research Foundation (NRF) of Korea
(Grant no. RS-2023-00209121, 2020R1A5A1016518). We further acknowledge the
support of HLD at HZDR, member of the European Magnetic Field Laboratory
(EMFL).

## References

  * Balents (2010) L. Balents, Nature 464, 199 (2010).
  * Mermin and Wagner (1966) N. D. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133 (1966).
  * Khuntia (2019) P. Khuntia, J. Magn. Magn. Mater. 489, 165435 (2019).
  * Khatua  _et al._ (2023) J. Khatua, B. Sana, A. Zorko, M. Gomilšek, K. Sethupathi, M. R. Rao, M. Baenitz, B. Schmidt, and P. Khuntia, Phys. Rep. 1041, 1 (2023).
  * Zhou  _et al._ (2017) Y. Zhou, K. Kanoda, and T.-K. Ng, Rev. Mod. Phys. 89, 025003 (2017).
  * Wen  _et al._ (2019) J. Wen, S.-L. Yu, S. Li, W. Yu, and J.-X. Li, npj Quantum Mater. 4, 12 (2019).
  * Savary and Balents (2016) L. Savary and L. Balents, Rep. Prog. Phys. 80, 016502 (2016).
  * Broholm  _et al._ (2020a) C. Broholm, R. J. Cava, S. A. Kivelson, D. G. Nocera, M. R. Norman, and T. Senthil, Science 367, eaay0668 (2020a).
  * Anderson (1973) P. W. Anderson, Mater. Res. Bull. 8, 153 (1973).
  * Wen (1991) X. G. Wen, Phys. Rev. B 44, 2664 (1991).
  * Baskaran  _et al._ (1987) G. Baskaran, Z. Zou, and P. Anderson, Solid State Commun. 63, 973 (1987).
  * Anderson (1987) P. W. Anderson, science 235, 1196 (1987).
  * Jiang and Kivelson (2021) H.-C. Jiang and S. A. Kivelson, Phys. Rev. Lett. 127, 097002 (2021).
  * Fradkin  _et al._ (2015) E. Fradkin, S. A. Kivelson, and J. M. Tranquada, Rev. Mod. Phys. 87, 457 (2015).
  * Castelnovo  _et al._ (2012) C. Castelnovo, R. Moessner, and S. Sondhi, Annual Review of Condensed Matter Physics 3, 35 (2012).
  * Nayak  _et al._ (2008) C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Rev. Mod. Phys. 80, 1083 (2008).
  * Phillips  _et al._ (2020) P. W. Phillips, L. Yeo, and E. W. Huang, Nature Physics 16, 1175 (2020).
  * Göbel  _et al._ (2021) B. Göbel, I. Mertig, and O. A. Tretiakov, Phys. Rep. 895, 1 (2021), beyond skyrmions: Review and perspectives of alternative magnetic quasiparticles.
  * Jeon  _et al._ (2023) S. Jeon, D. Wulferding, Y. Choi, S. Lee, K. Nam, K. H. Kim, M. Lee, T.-H. Jang, J.-H. Park, S. Lee, S. Choi, C. Lee, H. Nojiri, and K.-Y. Choi, (2023), arXiv:2311.11619 [cond-mat.str-el] .
  * Arh  _et al._ (2022) T. Arh, B. Sana, M. Pregelj, P. Khuntia, Z. Jagličić, M. D. Le, P. K. Biswas, P. Manuel, L. Mangin-Thro, A. Ozarowski, and A. Zorko, Nat. Mater. 21, 416 (2022).
  * Khuntia  _et al._ (2016) P. Khuntia, F. Bert, P. Mendels, B. Koteswararao, A. V. Mahajan, M. Baenitz, F. C. Chou, C. Baines, A. Amato, and Y. Furukawa, Phys. Rev. Lett. 116, 107203 (2016).
  * Chillal  _et al._ (2020) S. Chillal, Y. Iqbal, H. O. Jeschke, J. A. Rodriguez-Rivera, R. Bewley, P. Manuel, D. Khalyavin, P. Steffens, R. Thomale, A. T. M. N. Islam, J. Reuther, and B. Lake, Nat. Commun. 11, 2348 (2020).
  * Shockley  _et al._ (2015) A. C. Shockley, F. Bert, J.-C. Orain, Y. Okamoto, and P. Mendels, Phys. Rev. Lett. 115, 047201 (2015).
  * Dally  _et al._ (2014) R. Dally, T. Hogan, A. Amato, H. Luetkens, C. Baines, J. Rodriguez-Rivera, M. J. Graf, and S. D. Wilson, Phys. Rev. Lett. 113, 247601 (2014).
  * Plumb  _et al._ (2019) K. W. Plumb, H. J. Changlani, A. Scheie, S. Zhang, J. W. Krizan, J. A. Rodriguez-Rivera, Y. Qiu, B. Winn, R. J. Cava, and C. L. Broholm, Nat. Phys. 15, 54 (2019).
  * Krizan and Cava (2015) J. W. Krizan and R. J. Cava, Phys. Rev. B 92, 014406 (2015).
  * Cai  _et al._ (2018) Y. Cai, M. N. Wilson, A. M. Hallas, L. Liu, B. A. Frandsen, S. R. Dunsiger, J. W. Krizan, R. J. Cava, O. Rubel, Y. J. Uemura, and G. M. Luke, J. Condens. Matter Phys. 30, 385802 (2018).
  * Gao  _et al._ (2019) B. Gao, T. Chen, D. W. Tam, C.-L. Huang, K. Sasmal, D. T. Adroja, F. Ye, H. Cao, G. Sala, M. B. Stone, C. Baines, J. A. T. Verezhak, H. Hu, J.-H. Chung, X. Xu, S.-W. Cheong, M. Nallaiyan, S. Spagna, M. B. Maple, A. H. Nevidomskyy, E. Morosan, G. Chen, and P. Dai, Nat. Phys. 15, 1052 (2019).
  * Khatua  _et al._ (2022a) J. Khatua, S. Bhattacharya, Q. P. Ding, S. Vrtnik, A. M. Strydom, N. P. Butch, H. Luetkens, E. Kermarrec, M. S. R. Rao, A. Zorko, Y. Furukawa, and P. Khuntia, Phys. Rev. B 106, 104404 (2022a).
  * Broholm  _et al._ (2020b) C. Broholm, R. J. Cava, S. A. Kivelson, D. G. Nocera, M. R. Norman, and T. Senthil, Science 367 (2020b).
  * Kivelson and Sondhi (2023) S. Kivelson and S. Sondhi, Nat. Rev. Phys. 5, 368 (2023).
  * Lancaster (2023) T. Lancaster, Contemp. Phys. 0, 1 (2023).
  * Khatua  _et al._ (2022b) J. Khatua, M. Gomilŝek, J. C. Orain, A. M. Strydom, Z. Jagliĉiĉ, C. V. Colin, S. Petit, A. Ozarowski, L. Mangin-Thro, K. Sethupathi, M. S. R. Rao, A. Zorko, and P. Khuntia, Commun. Phys. 5, 99 (2022b).
  * Nguyen and Cava (2021) L. T. Nguyen and R. J. Cava, Chemical Reviews 121, 2935 (2021).
  * Bradley and Cracknell (2010) C. Bradley and A. Cracknell,  _The mathematical theory of symmetry in solids: representation theory for point groups and space groups_ (Oxford University Press, 2010).
  * Kakihana  _et al._ (2017) M. Kakihana, K. Nishimura, Y. Ashitomi, T. Yara, D. Aoki, A. Nakamura, F. Honda, M. Nakashima, Y. Amako, Y. Uwatoko, T. Sakakibara, S. Nakamura, T. Takeuchi, Y. Haga, E. Yamamoto, H. Harima, M. Hedo, T. Nakama, and Y. Ōnuki, Journal of Electronic Materials 46, 3572 (2017).
  * Redpath and Hopkinson (2010) T. E. Redpath and J. M. Hopkinson, Phys. Rev. B 82, 014410 (2010).
  * Chizhikov and Dmitrienko (2013) V. A. Chizhikov and V. E. Dmitrienko, Phys. Rev. B 88, 214402 (2013).
  * Dzyaloshinskii (1964) I. Dzyaloshinskii, Sov. Phys. JETP 19, 960 (1964).
  * Ding  _et al._ (2021) L. Ding, X. Xu, H. O. Jeschke, X. Bai, E. Feng, A. S. Alemayehu, J. Kim, F.-T. Huang, Q. Zhang, X. Ding, N. Harrison, V. Zapf, D. Khomskii, I. I. Mazin, S.-W. Cheong, and H. Cao, Nat. Commun. 12, 5339 (2021).
  * Ramakrishnan  _et al._ (2019) M. Ramakrishnan, E. Constable, A. Cano, M. Mostovoy, J. S. White, N. Gurung, E. Schierle, S. d. Brion, C. V. Colin, F. Gay, P. Lejay, E. Ressouche, E. Weschke, V. Scagnoli, R. Ballou, V. Simonet, and U. Staub, npj Quantum Materials 4, 60 (2019).
  * Izyumov (1984) Y. A. Izyumov, Soviet Physics Uspekhi 27, 845 (1984).
  * Qian  _et al._ (2018) F. Qian, L. J. Bannenberg, H. Wilhelm, G. Chaboussant, L. M. Debeer-Schmitt, M. P. Schmidt, A. Aqeel, T. T. M. Palstra, E. Brück, A. J. E. Lefering, C. Pappas, M. Mostovoy, and A. O. Leonov, Science Advances 4, eaat7323 (2018).
  * Mühlbauer  _et al._ (2009) S. Mühlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch, A. Neubauer, R. Georgii, and P. Böni, Science 323, 915 (2009).
  * Yu  _et al._ (2011) X. Z. Yu, N. Kanazawa, Y. Onose, K. Kimoto, W. Z. Zhang, S. Ishiwata, Y. Matsui, and Y. Tokura, Nat. Mater. 10, 106 (2011).
  * Kanazawa  _et al._ (2011) N. Kanazawa, Y. Onose, T. Arima, D. Okuyama, K. Ohoyama, S. Wakimoto, K. Kakurai, S. Ishiwata, and Y. Tokura, Phys. Rev. Lett. 106, 156603 (2011).
  * Higa  _et al._ (2021) N. Higa, T. U. Ito, M. Yogi, T. Hattori, H. Sakai, S. Kambe, Z. Guguchia, W. Higemoto, M. Nakashima, Y. Homma, A. Nakamura, F. Honda, Y. Shimizu, D. Aoki, M. Kakihana, M. Hedo, T. Nakama, Y. Ōnuki, and Y. Tokunaga, Phys. Rev. B 104, 045145 (2021).
  * Kneidinger  _et al._ (2019) F. Kneidinger, I. Zeiringer, A. Siderenko, E. Bauer, H. Michor, P. Rogl, and J. G. Sereni, Phys. Rev. B 100, 134442 (2019).
  * Pfleiderer  _et al._ (1997) C. Pfleiderer, G. J. McMullan, S. R. Julian, and G. G. Lonzarich, Phys. Rev. B 55, 8330 (1997).
  * Hopkinson and Kee (2006) J. M. Hopkinson and H.-Y. Kee, Phys. Rev. B 74, 224441 (2006).
  * Živković  _et al._ (2021) I. Živković, V. Favre, C. Salazar Mejia, H. O. Jeschke, A. Magrez, B. Dabholkar, V. Noculak, R. S. Freitas, M. Jeong, N. G. Hegde, L. Testa, P. Babkevich, Y. Su, P. Manuel, H. Luetkens, C. Baines, P. J. Baker, J. Wosnitza, O. Zaharko, Y. Iqbal, J. Reuther, and H. M. Rønnow, Phys. Rev. Lett. 127, 157204 (2021).
  * Yao  _et al._ (2023) W. Yao, Q. Huang, T. Xie, A. Podlesnyak, A. Brassington, C. Xing, R. S. D. Mudiyanselage, H. Wang, W. Xie, S. Zhang, M. Lee, V. S. Zapf, X. Bai, D. A. Tennant, J. Liu, and H. Zhou, Phys. Rev. Lett. 131, 146701 (2023).
  * Boya  _et al._ (2022) K. Boya, K. Nam, K. Kargeti, A. Jain, R. Kumar, S. K. Panda, S. M. Yusuf, P. L. Paulose, U. K. Voma, E. Kermarrec, K. H. Kim, and B. Koteswararao, APL Materials 10, 101103 (2022).
  * Bulled  _et al._ (2022) J. M. Bulled, J. A. M. Paddison, A. Wildes, E. Lhotel, S. J. Cassidy, B. Pato-Doldán, L. C. Gómez-Aguirre, P. J. Saines, and A. L. Goodwin, Phys. Rev. Lett. 128, 177201 (2022).
  * Huai and Tran (2023) X. Huai and T. T. Tran, Annu. Rev. Mater. Res. 53, 253 (2023).
  * Hall  _et al._ (2023) A. E. Hall, P. Manuel, D. D. Khalyavin, F. Orlandi, D. A. Mayoh, L.-J. Chang, Y.-S. Chen, D. G. C. Jonas, M. R. Lees, and G. Balakrishnan, Phys. Rev. Mater. 7, 114402 (2023).
  * Suter and Wojek (2012) A. Suter and B. Wojek, Phys. Procedia 30, 69 (2012), 12th International Conference on Muon Spin Rotation, Relaxation and Resonance ($\mu$SR2011).
  * Toby (2001) B. H. Toby, J. Appl. Crystallogr. 34, 210 (2001).
  * Isasi and Daidouh (2000) J. Isasi and A. Daidouh, Solid State Ion. 133, 303 (2000).
  * Boudjada and Perret (1977) A. Boudjada and R. Perret, J. Appl. Crystallogr. 10, 129 (1977).
  * Norberg (2002) S. T. Norberg, Acta Crystallographica Section B 58, 743 (2002).
  * Isakov  _et al._ (2008) S. V. Isakov, J. M. Hopkinson, and H.-Y. Kee, Phys. Rev. B 78, 014404 (2008).
  * Khan  _et al._ (2019) M. S. Khan, A. Bandyopadhyay, A. Nag, V. Kumar, A. V. Mahajan, and S. Ray, Phys. Rev. B 100, 064423 (2019).
  * Lee  _et al._ (2022) S. Lee, T. Zhu, Y. Oshima, T. Shiroka, C. Wang, H. Luetkens, H. Yang, M. Lü, and K.-Y. Choi, Phys. Rev. B 105, 094439 (2022).
  * Zhu  _et al._ (2021) T. Zhu, S. Lee, X. Zhang, H. Yang, Y. Jin, Y. Jin, K.-Y. Choi, and M. Lü, Inorg. Chem. 60, 13707 (2021).
  * Bukowski  _et al._ (2022) Z. Bukowski, D. Rybicki, M. Babij, J. Przewoźnik, Ł. Gondek, J. Żukrowski, and C. Kapusta, Sci. Rep. 12, 14718 (2022).
  * Lai  _et al._ (2021) Y. Lai, L. Ke, J. Yan, R. D. McDonald, and R. J. McQueeney, Phys. Rev. B 103, 184429 (2021).
  * Cao  _et al._ (2013) G. Cao, A. Subedi, S. Calder, J.-Q. Yan, J. Yi, Z. Gai, L. Poudel, D. J. Singh, M. D. Lumsden, A. D. Christianson, B. C. Sales, and D. Mandrus, Phys. Rev. B 87, 155136 (2013).
  * Verrier  _et al._ (2020) A. Verrier, F. Bert, J. M. Parent, M. El-Amine, J. C. Orain, D. Boldrin, A. S. Wills, P. Mendels, and J. A. Quilliam, Phys. Rev. B 101, 054425 (2020).
  * Schmidt and Thalmeier (2017) B. Schmidt and P. Thalmeier, Phys. Rep. 703, 1 (2017), frustrated two dimensional quantum magnets.
  * Kimura  _et al._ (2022) S. Kimura, S. Imajo, M. Gen, T. Momoi, M. Hagiwara, H. Ueda, and Y. Kohama, Phys. Rev. B 105, L180405 (2022).
  * Khatua  _et al._ (2021) J. Khatua, T. Arh, S. B. Mishra, H. Luetkens, A. Zorko, B. Sana, M. S. R. Rao, B. R. K. Nanda, and P. Khuntia, Sci. Rep. 11, 6959 (2021).
  * Murugan  _et al._ (2023) G. S. Murugan, C. Lee, R. Kalaivanan, I. P. Muthuselvam, Y. Oshima, K.-Y. Choi, and R. Sankar, Phys. Rev. B 107, 214411 (2023).
  * Abragam and Bleaney (1970) A. Abragam and B. Bleaney,  _Electron paramagnetic resonance of transition ions_ (Clarendon Press, Oxford, 1970).
  * Glamazda  _et al._ (2017) A. Glamazda, Y. S. Choi, S.-H. Do, S. Lee, P. Lemmens, A. N. Ponomaryov, S. A. Zvyagin, J. Wosnitza, D. P. Sari, I. Watanabe, and K.-Y. Choi, Phys. Rev. B 95, 184430 (2017).
  * Lee  _et al._ (2016) S. Lee, S.-H. Do, W.-J. Lee, Y. S. Choi, M. Lee, E. S. Choi, A. P. Reyes, P. L. Kuhns, A. Ozarowski, and K.-Y. Choi, Phys. Rev. B 93, 174402 (2016).
  * Yaouanc and De Reotier (2011) A. Yaouanc and P. D. De Reotier,  _Muon spin rotation, relaxation, and resonance: applications to condensed matter_ , 147 (OUP Oxford, 2011).
  * Nocerino  _et al._ (2023) E. Nocerino, S. Kobayashi, C. Witteveen, O. K. Forslund, N. Matsubara, C. Tang, T. Matsukawa, A. Hoshikawa, A. Koda, K. Yoshimura, I. Umegaki, Y. Sassa, F. O. von Rohr, V. Pomjakushin, J. H. Brewer, J. Sugiyama, and M. Månsson, Commun. Mater. 4, 81 (2023).
  * Papadopoulos  _et al._ (2022) K. Papadopoulos, O. K. Forslund, E. Nocerino, F. O. L. Johansson, G. Simutis, N. Matsubara, G. Morris, B. Hitti, D. Arseneau, P. Svedlindh, M. Medarde, D. Andreica, J.-C. Orain, V. Pomjakushin, L. Börjesson, J. Sugiyama, M. Månsson, and Y. Sassa, Phys. Rev. B 106, 214410 (2022).
  * Campbell  _et al._ (1994) I. A. Campbell, A. Amato, F. N. Gygax, D. Herlach, A. Schenck, R. Cywinski, and S. H. Kilcoyne, Phys. Rev. Lett. 72, 1291 (1994).
  * Yaouanc  _et al._ (2008) A. Yaouanc, P. Dalmas de Réotier, Y. Chapuis, C. Marin, G. Lapertot, A. Cervellino, and A. Amato, Phys. Rev. B 77, 092403 (2008).
  * Koo  _et al._ (2018) C. Koo, J. Werner, M. Tzschoppe, M. Abdel-Hafiez, P. K. Biswas, R. Sarkar, H.-H. Klauss, G. V. Raganyan, E. A. Ovchenkov, A. Y. Nikulin, A. N. Vasiliev, E. A. Zvereva, and R. Klingeler, Phys. Rev. B 97, 224416 (2018).
  * Weinhold  _et al._ (2023) T. Weinhold, C. Wang, F. Seewald, V. Grinenko, Y. Imai, F. Sato, K. Ohgushi, H. H. Klauss, and R. Sarkar, “Magnetism in kitaev quantum spin liquid candidate rubr3,” (2023), arXiv:2311.03800 [cond-mat.str-el] .
  * Li  _et al._ (2021) Y. Li, Q.-Y. Li, W. Li, T. Liu, D. J. Voneshen, P. K. Biswas, and D. Adroja, npj Quantum Materials 6, 34 (2021).
  * Khuntia  _et al._ (2017) P. Khuntia, S. Manni, F. R. Foronda, T. Lancaster, S. J. Blundell, P. Gegenwart, and M. Baenitz, Phys. Rev. B 96, 094432 (2017).
  * Zorko  _et al._ (2010) A. Zorko, F. Bert, P. Mendels, K. Marty, and P. Bordet, Phys. Rev. Lett. 104, 057202 (2010).
  * Noakes and Kalvius (1997) D. R. Noakes and G. M. Kalvius, Phys. Rev. B 56, 2352 (1997).
  * Do  _et al._ (2018) S.-H. Do, W.-J. Lee, S. Lee, Y. S. Choi, K.-J. Lee, D. I. Gorbunov, J. Wosnitza, B. J. Suh, and K.-Y. Choi, Phys. Rev. B 98, 014407 (2018).
  * Lee  _et al._ (2023) S. Lee, Y. S. Choi, S.-H. Do, W. Lee, C. H. Lee, M. Lee, M. Vojta, C. N. Wang, H. Luetkens, Z. Guguchia, and K.-Y. Choi, Nat. Commun. 14, 7405 (2023).
  * Baker  _et al._ (2011) P. J. Baker, I. Franke, F. L. Pratt, T. Lancaster, D. Prabhakaran, W. Hayes, and S. J. Blundell, Phys. Rev. B 84, 174403 (2011).
  * Forslund  _et al._ (2020) O. K. Forslund, K. Papadopoulos, E. Nocerino, G. Morris, B. Hitti, D. Arseneau, V. Pomjakushin, N. Matsubara, J.-C. Orain, P. Svedlindh, D. Andreica, S. Jana, J. Sugiyama, M. Månsson, and Y. Sassa, Phys. Rev. B 102, 144409 (2020).
  * Hodges  _et al._ (2002) J. A. Hodges, P. Bonville, A. Forget, A. Yaouanc, P. Dalmas de Réotier, G. André, M. Rams, K. Królas, C. Ritter, P. C. M. Gubbens, C. T. Kaiser, P. J. C. King, and C. Baines, Phys. Rev. Lett. 88, 077204 (2002).
  * Lee  _et al._ (2021) S. Lee, S.-H. Do, W. Lee, Y. S. Choi, J. van Tol, A. P. Reyes, D. Gorbunov, W.-T. Chen, and K.-Y. Choi, npj Quantum Materials 6, 47 (2021).
  * Frandsen  _et al._ (2020) B. A. Frandsen, K. A. Petersen, N. A. Ducharme, A. G. Shaw, E. J. Gibson, B. Winn, J. Yan, J. Zhang, M. E. Manley, and R. P. Hermann, Phys. Rev. Mater. 4, 074405 (2020).