# Three-dimensional Sandglass Magnet with Non-Kramers ions

Yan-Xing Yang  Yao Wang  Zhao-Feng Ding  State Key Laboratory of Surface
Physics, Department of Physics, Fudan University, Shanghai 200438, China  A.
D. Hillier  ISIS Facility, STFC Rutherford Appleton Laboratory, Chilton,
Didcot, Oxfordshire, OX110QX, United Kingdom  Lei Shu<EMAIL_ADDRESS>State Key Laboratory of Surface Physics, Department of Physics, Fudan
University, Shanghai 200438, China  Collaborative Innovation Center of
Advanced Microstructures, Nanjing 210093, China  Shanghai Research Center for
Quantum Sciences, Shanghai 201315, China

###### Abstract

Magnetic susceptibility, specific heat, and muon spin relaxation ($\mu$SR)
measurements have been performed on a newly synthesized three-dimensional
sandglass-type lattice Tm3SbO7, where two inequivalent sets of non-Kramers
Tm3+ ions (Tm${}^{3+}_{1}$ and Tm${}^{3+}_{2})$ show crystal electrical field
effect at different temperature ranges. The existence of an ordered or a
glassy state down to 0.1 K in zero field is excluded. The low-energy
properties of Tm3SbO7 are dominated by the lowest non-Kramers quasi-doublet of
$\rm Tm^{3+}_{1}$, and the energy splitting is regarded as an intrinsic
transverse field. Therefore, the low-temperature paramagnetic phenomenon in
Tm3SbO7 is explained by a transverse field Ising model, which is supported by
the quantitative simulation of specific heat data. In addition, the
perturbation from Tm${}^{3+}_{2}$ may play an important role in accounting for
the low temperature spin dynamics behavior observed by $\mu$SR.

##  I INTRODUCTION

In quantum materials, the concept of emergent phenomena due to strong-
correlations between electrons or magnetic moments, quantum entanglement,
topology, or frustration has attracted a lot of attention Keimer and Moore
(2017); Tokura  _et al._ (2017). Such a concept has been widely applied in
many interesting systems including high-temperature superconductor,
topological insulator, and quantum spin liquid (QSL). QSL is a novel quantum
system where the magnetic order is suppressed by quantum fluctuations even at
zero temperature. Considerable efforts in searching for QSL have been seen
since the concept was proposed by Anderson in 1973 Anderson (1973). Currently,
QSL has become a hot topic in condensed matter physics due to its potential
applications in quantum communication and computingKitaev (2006).

However, experimental identification of a QSL remains a great challenge since
one can not reach absolute zero temperature to identify a specific material’s
ground state. A compromise is to measure the enough low-temperature properties
by using a variety of methods, including magnetic susceptibility, specific
heat, muon spin relaxation ($\mu$SR), and nuclear magnetic resonance
measurements to exclude the magnetic ordering or freezing Broholm  _et al._
(2020); Wen  _et al._ (2019). However, the observation that a material does
not order magnetically at low temperatures may be caused by structure or
chemical component disorder Kimchi  _et al._ (2018a, b), or is only simply due
to a cooperative paramagnetic state Keren  _et al._ (2004); Ueland  _et al._
(2006). In addition to the absence of magnetic order, a recognized QSL
material needs to satisfy several conditions including fractional excitations
and long-range correlated dynamical spins. To demonstrate the existence of
fractional excitations, the residual linear term in low-temperature specific
heat Ding  _et al._ (2018); Ni  _et al._ (2019); Yamashita  _et al._ (2008a,
2011) and thermal conductivity Yamashita  _et al._ (2010, 2008b); Ni  _et al._
(2019), as well as the continuum magnetic excitation spectra in inelastic
neutron scattering measurements Coldea  _et al._ (2001); Banerjee  _et al._
(2017) are all expected. Besides, the low-temperature plateau of the muon spin
relaxation rates is an evidence of persistent spin dynamics in an entangled
spin system Ding  _et al._ (2018); Dunsiger  _et al._ (2011); Chang  _et al._
(2013).

So far numerous two-dimensional QSL candidates have been reported, among which
either geometry frustration Balents (2010); Zhou  _et al._ (2017) or Kitaev
interactions Kitaev (2006); Kitagawa  _et al._ (2018); Pei  _et al._ (2020)
introduces quantum fluctuations. However, promising three-dimensional (3D) QSL
candidates are still rare since higher dimensionality suppresses the quantum
fluctuations. Hyperkagome Na4Ir3O8Okamoto  _et al._ (2007); Chen and Balents
(2008); Zhou  _et al._ (2008) and pyrochlore Pr2Ir2O7 Nakatsuji  _et al._
(2006); Chen (2016); Yao and Chen (2018); Ni  _et al._ (2021) are the only two
representative 3D QSL candidates and have been the subject of extensive
studies. Even if the absence of magnetic order and spin dynamics are found, it
is not sufficient to claim a QSL material, since those observations can be
explained by other mechanism.

We report magnetic susceptibility, specific heat, and $\mu$SR studies of a
newly synthesized fluorite oxide Tm3SbO7, in which two sets of non-Kramers
Tm3+ ions (Tm${}^{3+}_{1}$ and Tm${}^{3+}_{2})$ form a three-dimensional
sandglass-type lattice. The absence of magnetic order or glassy state is
confirmed down to 0.1 K in zero magnetic field. The calculated magnetic
entropy shows a two-step release, indicates the inequivalent Tm${}^{3+}_{1}$
and Tm${}^{3+}_{2}$ play a part in different energy scales. The crystal
electric field (CEF) calculation suggests that the low-energy properties of
Tm3SbO7 are dominated by the lowest non-Kramers quasi-doublet of $\rm
Tm^{3+}_{1}$, and the energy splitting, which can be regarded as an intrinsic
transverse field, is about $h\sim 0.64$ meV. Therefore, the low-temperature
paramagnetic phenomenon in Tm3SbO7 can be described by the transverse field
Ising model (TFIM) Coldea  _et al._ (2010); Shen  _et al._ (2019); Cui  _et
al._ (2019); Liu  _et al._ (2020); Li  _et al._ (2020); Chen (2019); Bitko
_et al._ (1996), since the exchange interactions between effective $S=1/2$
spins is very small according to the small Curie-Weiss temperature at low
temperature. TFIM is further supported by the quantitatively simulation of
specific heat data. In addition, the perturbation from Tm${}^{3+}_{2}$ may
play an important role in accounting for the low temperature spin dynamics
behavior observed by $\mu$SR.

##  II EXPERIMENTS

Polycrystalline Tm3SbO7 and its non-magnetic analog Lu3SbO7 were synthesized
by the solid state reaction. Stoichiometric amounts of Sb2O3 and $Ln_{2}$O3
($Ln$ = Tm or Lu) were mixed, thoroughly grounded, and heated at 1500 ∘C for 7
days. Then polycrystalline samples were obtained after two additional re-
grindings and heating. The single phase of the two samples was checked by
powdered X-ray diffraction (XRD) measurements using a Bruker D8 advanced X-ray
diffraction spectrometer ($\lambda$ = 1.5418 Å). The Rietveld refinement of
XRD data was conducted using FullProf software. DC magnetic susceptibility
measurements were carried out in the temperature range from 2 K to 300 K by
using a Magnetic Property Measurement System (MPMS, Quantum Design). The
measurements of AC magnetic susceptibility from 0.1 K to 4 K and specific heat
from 0.1 K to 300 K were carried out in a Physical Property Measurement System
(PPMS, Quantum Design) equipped with dilution refrigerator. $\mu$SR
measurements with temperatures from 0.07 K to 43 K and longitudinal external
magnetic fields up to 0.3 T were performed on the MuSR spectrometer at ISIS
Neutron and Muon Facility, STFC, Rutherford Appleton Laboratory, UK.

##  III RESULTS

###  III.1 Crystal structure and CEF calculation

The fluorite-related structure has been reported in the rare earth rhenium
oxides $Ln_{3}$ReO7 ($Ln$ = Y, Er-Lu)Inabayashi  _et al._ (2018) and the
authors found that $Ln_{3}$ReO7 ($Ln$ = Y, Er, Tm) have the orthorhombic
structure with space group $C222_{1}$ while $Ln_{3}$ReO7 ($Ln$ = Yb, Lu) have
the cubic structure with space group $Fm\bar{3}m$. The XRD pattern of Tm3SbO7
is shown in Fig. 1(a). In the process of resolving the structure of this new
material, we have firstly tried the space group $Fm\bar{3}m$ with Tm and Sb
atoms randomly occupying the 4a Wyckoff position, and oxygen atoms occupying
the 4b position. Only a few obvious strong reflections are eligible and these
remaining weak reflections point out that the symmetry is lower than cubic
structure. As a result, the orthorhombic structure with space group $C222_{1}$
was used, and it matches the XRD data exactly, indicating the site-mixing is
not likely. The Rietveld refinement results are shown in TABLE I.

Table 1: Rietveld refinement results for Lu3SbO7. $R_{wp}=2.39\%,R_{p}=3.73\%,\chi^{2}=11.9$; $a=7.389$ Å, $b=10.398$ Å, $c=7.361$ Å; $\alpha=\beta=\gamma=90^{\circ}$; Space Group: $C222_{1}$. | Wyckoff |  |  |  |  |
---|---|---|---|---|---|---
Atom | positions | $x$ | $y$ | $z$ | B/Å2 | Occ.
Tm1 | 8c | 0.2424(8) | 0.2352(9) | 0.7431(6) | 0.1587(6) | 1
Tm2 | 4a | 0.0094(1) | 0.5 | 0.5 | 0.1598(7) | 1
Sb | 4a | 0.0129(8) | 0 | 0 | 0.0551(5) | 1
O1 | 8c | 0.2165(0) | 0.1351(4) | 0.4982(9) | 0 | 1
O2 | 8c | 0.2157(8) | 0.1166(8) | 0.0697(4) | 4.7098(3) | 1
O3 | 4b | 0 | 0.0840(5) | 0.75 | 1.8571(2) | 1
O4 | 4b | 0 | 0.3557(9) | 0.75 | 0.0822(5) | 1
O5 | 4b | 0 | 0.3640(6) | 0.25 | 0.5970 | 1

The unit cell of Tm3SbO7 is shown in Fig. 1(b). There are two different Tm
sites in Tm3SbO7 with the ratio of Tm1:Tm2 = 2:1. As shown in Figs. 1(b-d),
Tm1 atoms (dark blue dots) construct a twisted tetragonal lattice, while Tm2
(light blue dots) and Sb (brown dots) atoms reside in the center of Tm1
cuboids. Both Tm2 and Sb atoms form the one-dimensional chains parallel to the
[001] direction. As shown in Fig. 1(b), a Tm1 cuboid with a central Tm2 atom
form a sandglass-type unit. Ignoring the nonmagnetic Sb atoms, all the Tm
atoms construct the edge-shared sandglass-type structure. The bond lengths of
these neighboring Tm atoms vary in a small range from 3.53 Å to 3.87 Å,
indicating that Tm3SbO7 is a 3D magnet.

Figure 1: (a) Rietveld refinement of powder XRD pattern of Tm3SbO7 at room
temperature using the orthorhombic structure with space group $C222_{1}$. The
red dots, black line, and blue line are the experimental data, the calculated
patterns, and the differences, respectively. The green bars indicates the
Bragg reflections. (b) The unit cell of Tm3SbO7 as well as the Schematic
diagram of the sandglass-type configuration of Tm3+ ions. Dark blue: Tm1;
light blue: Tm2; brown: Sb; red: Oxygen. (c) Top view of two selected layers
of the unit cell. (d) Front view of the unit cell.

The coordinate oxygen atoms of both Tm1 and Tm2 are presented in Fig. 1(b).
Tm1 resides in an octahedral oxygen cavity, while Tm2 resides in an
8-coordinated oxygen polyhedron. Based on the structure obtained from XRD, we
did CEF calculation using the software PyCrystalFieldScheie (2021). The
13-fold degenerate 4f orbit of each Tm3+ will spilt due to the Coulomb
potential from their surrounding ions. The intuitive schematic of CEF
splitting of Tm1 and Tm2 is shown in Fig. 2. For Tm1, the ground state and the
first excited state form a quasi-doublet and the energy splitting gap $h$
between the two states is about 0.25 meV, which is much smaller than $\Delta$
($\approx$ 29 meV), the energy gap between $E_{1}$ and $E_{2}$. So the low-
temperature properties for Tm1 is qualitatively governed by this quasi-
doublet. For Tm2, The lowest 6 energy levels from $E_{0}$ to $E_{5}$ are 0
meV, 0.91 meV, 3.79 meV, 6.53 meV, 7.68 meV and 10.89 meV. They are relatively
evenly distributed and gapped by a large gap ($\Delta\approx$ 17 meV) from
$E_{6}$ and higher levels. Besides, according to the specific heat results
discussed in Sec. III.3, the ground state level $E_{0}$ of Tm2 is higher than
that of Tm1.

Figure 2: Schematic of the CEF splitting of (a) Tm1 and (b) Tm2 in Tm3SbO7.

###  III.2 Magnetic susceptibility

DC-magnetic susceptibility $\chi_{dc}$ of Tm3SbO7 measured under a magnetic
field of 0.5 T from 2 K to 300 K are shown in Fig. 3. No peak reflecting phase
transition or separation between zero-field cooling and field cooling (not
shown) is found down to 2 K. At high temperatures, $\chi_{dc}$ increases as
the temperature is reduced. The inset of Fig. 3 shows the inverse of
$\chi_{dc}$ as function of temperature. A fit of Curie-Weiss law is shown for
temperature between 100 K and 300 K (blue line) . The Curie-Weiss temperature
$\Theta_{\rm{CW}}$ is -23.3 K, and the effective magnetic moment
$\mu_{\rm{eff}}$ is 7.53 $\mu_{\rm{B}}$, which is close to the theoretical
value $\mu_{\rm{calc}}$ = 7.57 $\mu_{\rm{B}}$ for Tm3+ ions with the spin-
orbital coupling ground state ${}^{3}\rm{H}_{6}$. When the temperature is
cooled down below 100 K, $\chi_{dc}$ slowly deviates from the high temperature
Curie-Weiss law and forms to another Curie-Weiss behavior at low temperatures
(red line), which gives $\mu_{\rm eff}=6.25~{}\mu_{\rm B}$ and
$\Theta_{\rm{CW}}=-4.28$ K.

AC susceptibility was measured from 4 K to 0.1 K. $\chi^{\prime}_{ac}$ data
with different driving frequencies show a similar behavior, i.e., gradually
increases with lowering the temperature and finally saturates below 1 K
without showing any anomalies. Therefore, magnetic ordered state as well as
the spin glass state in Tm3SbO7 can be ruled out.

Figure 3: Temperature dependence of dc magnetic susceptibility $\chi_{dc}$
(circles) and the real part of ac susceptibility $\chi^{\prime}_{ac}$
(squares). $\chi_{ac}$ was measured in zero static field with a driven field
of 1 Oe from 0.1 K to 4 K. $\chi_{dc}$ was measured under $\mu_{0}H=0.5$ T
from 2 K to 300 K. $\chi_{dc}$ between 100-300 K was fitted using Curie-Weiss
law as shown in the picture. The temperature-independent $\chi_{0}$ is induced
by Van-Vleck susceptibility.

###  III.3 Specific heat

To further investigate the thermodynamics of Tm3SbO7, we measured the specific
heat down to about 0.1 K by applying various magnetic fields, as shown in Fig.
4(a). We subtract the phonon contribution, which is obtained from nonmagnetic
oxide Lu3SbO7 and depicted (black points) in the inset of Fig. 4(a), from the
total specific heat $C_{\rm{total}}/T$ of Tm3SbO7. Due to the uncertainty of
subtraction at high temperatures, the magnetic specific heat are only
exhibited below 120 K. The whole curve of $C_{\rm{M}}/T$ shows two overlapped
broad bumps. However, there are no sharp peaks throughout the full temperature
range . The characteristic of no phase transition is consistent with the
magnetic susceptibility above and the following $\mu$SR results.

Figure 4: (a) Temperature dependence of magnetic specific heat coefficient
$C_{\rm{M}}/T$ of Tm3SbO7 under several fields. Inset: Temperature dependence
of measured total specific heat coefficient $C_{\rm{total}}/T$ of Tm3SbO7 and
non-magnetic analog Lu3SbO7 (black points) under zero field. (b) Temperature
dependence of magnetic entropy $S_{\rm{M}}$ obtained by integrating
$C_{\rm{M}}/T$ from about $T=$ 0.1 K to $T$. (c) Temperature dependence of
$C_{\rm{M}}/T$ of Tm3SbO7 under zero field. Curves are guided by eye. (d) Two
step entropy increasing under zero magnetic field.

As displayed in Fig. 4(a), by increasing the external magnetic fields, the
lower-temperature bump is lowered and broadened, and the peak position moves
to higher temperatures until $\mu_{0}H$ = 5 T. The position of the higher-
temperature bump does not move but gets broader with increasing applied
magnetic fields, and there is no obvious bump at $\mu_{0}H$ = 9 T.

We calculated magnetic entropy $S_{\rm{M}}$ by integrating the $C_{\rm{M}}/T$
curve, as revealed in Fig. 4(b). With the temperature increasing, $S_{\rm{M}}$
rises up steeply in the beginning, and then gently climbs from 10 K to 120 K.
Since the entropy under various magnetic fields have a roughly 4$R$ln2 in
total for per mole sample except for a small deficiency at the highest field
we measured, we believe that the energy levels below 120 K keep unchanged with
magnetic field up to 9 T, which supports the CEF calculation that there is
indeed a large energy gap between the lower energy levels and the much higher
energy levels. As a result, in the following we only discuss the lower energy
levels.

To clarify the relationship between specific heat and the CEF energy levels,
we present zero-field $C_{\rm M}/T$ and $S_{\rm{M}}$ in Figs. 4(c, d). As
shown in Fig. 4(c), $C_{\rm M}/T$ curve is roughly divided into two parts,
labeled as $C_{\rm M,1}/T$ and $C_{\rm M,2}/T$, respectively. The
corresponding $S_{\rm{M,1}}$ and $S_{\rm{M,2}}$ are drawn in Fig. 4(d). The
CEF calculation in Sec. III.1 has revealed that, for per formula unit of
Tm3SbO7, there are 4 energy levels with small splitting for Tm1 and 6 energy
levels for Tm2. The 4 levels of Tm1 are 2-fold degenerate (two quasi-doublets)
due to two equivalent Tm1 atoms, while the 6 levels of Tm2 are non-degenerate.
Comparing the small gap $h$ between the quasi-doublet of Tm1 with the gaps
among Tm2’s 6 levels (for instance, the 3rd excited state of Tm2 has a
6.53-meV gap from $E_{\rm 0}$, which is one-order-of-magnitude larger than
$h$), we infer that $C_{\rm M,1}/T$ and $C_{\rm M,2}/T$ correspond to the
contribution from Tm1 and Tm2, respectively.

Now we can quantitatively describe the evolution of $S_{\rm{M}}$ in Fig. 4(d).
For Tm1, two quasi-doublets per formula unit can offer 2$R$ln2 entropy
increasing, which is exactly the saturation value of $S_{\rm{M,1}}$. For Tm2,
as shown in Fig. 4(d), $S_{\rm{M,2}}$ rises up from 10 K and reaches $R$ln4
around 120 K, without a sign of saturation. It is reasonable that only the
lowest 4 energy levels of Tm2 are covered below 120 K.

###  III.4 $\mu$SR

$\mu$SR is a low-frequency probe of spin dynamics and is particularly
sensitive to slow spin fluctuations Hayano  _et al._ (1979). It is therefore
ideally suited to studies of long-lived spin correlations in novel spin
systems. We continue to study the intrinsic magnetic properties of Tm3SbO7 by
performing $\mu$SR experiments. Both the zero-field (ZF)- and longitudinal-
field (LF)-$\mu$SR spectra are shown in Fig. 5(a). The ZF-$\mu$SR asymmetries
are well fitted by a sum of two damped Kubo-Toyabe functions originating from
two inequivalent muon sites:

$A(t)=A_{0}f_{1}e^{-\lambda_{1}t}G_{z}^{\rm{KT}}(\delta_{1},t)+A_{0}(1-f_{1})e^{-\lambda_{2}t}G_{z}^{\rm{KT}}(\delta_{2},t)$
(1)

where $A_{0}$ is the initial asymmetry and $f_{1}$ represents the fraction of
the first muon sites in sample. During the data processing, $A_{0}$ and
$f_{1}$ were found to be temperature independent and therefore are fixed at
the average values of 0.22 and 0.5, respectively. The exponential rates
$\lambda_{1,2}$ are the muon spin relaxation rates usually related to the
dynamic internal magnetic fields. $G_{z}^{\rm{KT}}(\delta,t)$ is the well-
known Kubo-Toyabe function in which the relaxation rates $\delta_{1,2}$
originate from static internal magnetic fields such as nuclear dipolar fields
Hayano  _et al._ (1979). As shown in Fig. 5(b) and (c), muons at two different
stopping sites sense the same internal magnetic fields and the only difference
is the strength. This indicates that two muon sites are reasonable and the
phase separation can be excluded.

Cooling from high temperatures, the dynamical relaxation rate $\lambda_{1,2}$
gradually goes up and shows broad peaks around 3 K and finally saturates below
1 K. The low-temperature plateau of muon relaxation rates is a sign of
persistent spin dynamics Dunsiger  _et al._ (2011); Ding  _et al._ (2018);
Chang  _et al._ (2013). The dynamical property of internal fields is further
confirmed by the LF-$\mu$SR results (see Fig. 5(a)), since the muon
depolarization would be completely decoupled under such an external
longitudinal field if the internal fields are static or quasi-static. Note
that although $\lambda_{2}$ is ten times smaller than $\lambda_{1}$, both
$\lambda_{1}$ and $\lambda_{2}$ show two peaks around the same temperatures.
The two peaks of $\lambda_{1,2}$ are also consistent with the two bumps
discovered in the specific heat measurements.

Figure 5: (a) Representative $\mu$SR asymmetry spectra (a constant background
is subtracted) measured in ZF and LF. Solid lines are fits to the data. (b)
Temperature dependence of ZF dynamic relaxation rate $\lambda$. The black line
is to guide the eyes. (c) Temperature dependence of ZF static relaxation rate
$\delta_{1}$ (red dots) and $\delta_{2}$ (green dots) at two different muon
stopping sites. Purple line: dc-magnetic susceptibility $\chi_{\rm dc}$.
Purple circle: ac-magnetic susceptibility $\chi^{\prime}_{ac}$, data from Fig.
2 and scaled with $\chi_{\rm dc}$. (d) Dependence of static relaxation rate
$\delta_{1,2}$ of $\chi_{dc}$ or $\chi^{\prime}_{ac}$ with temperature as an
implicit parameter.

As shown in Fig. 5(c), the static muon spin relaxation rates $\delta_{1,2}$,
similar to $\chi_{dc}$ and $\chi^{\prime}_{ac}$, increases with decreasing
temperature and saturates below 1 K. We argue that the temperature-dependent
$\delta_{1,2}$ is related to hyperfine-enhanced Tm nuclear moments MacLaughlin
_et al._ (2009). The enhanced value of the static Tm nuclear contribution,
$\delta$, has the formula:

$\delta=(1+k)\delta_{0}$ (2)

where $\delta_{0}$ is the un-enhanced value of the static Tm nuclear
contribution. $k=\rm{a_{4f}}\chi_{\rm{mol}}$ is the enhancement factor, where
$\rm{a_{4f}}$ is the atomic hyperfine coupling constant and $\chi_{\rm{mol}}$
is the DC magnetic susceptibility. By replacing $k$ in Eq. (2) with
$\rm{a_{4f}}\chi_{\rm{mol}}$, we obtain
$d\delta/d{\chi_{\rm{mol}}}=\rm{a_{4f}}\delta_{0}$, indicating that $\delta$
should be proportional to $\chi_{\rm{mol}}$ with temperature as an implicit
parameter. This is consistent with our experimental results as shown in Fig.
5(d). $\delta_{1,2}$ is proportional to $\chi_{dc}$ in a wide temperature
range.

##  IV DISCUSSION

###  IV.1 Dynamical muon spin relaxation rate, specific heat, and magnetic
susceptibility

If we compare the temperature dependence of $C_{\rm{M}}/T$ with the dynamical
muon relaxation rate $\lambda_{1,2}$, we find that the specific-heat bump is
consistent with the relaxation rate peak. When $T<<h$, the Tm electrons tend
to stay in the lowest level and the transition probability between different
levels is low. When $T$ is close to $h$, the probability gets sufficient and
hence leads to a maximum in $C_{\rm{M}}/T$. When $T>>h$, the probability drops
again because different levels are almost equally occupied. The temperature
dependence of electron transitions explains the change in specific heat.
Meanwhile, since Tm3+ ions are magnetic, the electron transitions sensed by
muons are considered as magnetic fluctuations. This is why the two different
methods possess some common features in the temperature dependence.

The low-temperature plateau of $\lambda$ below 1 K is a sign of persistent
spin dynamics. The spin dynamics is confirmed by LF-$\mu$SR experiment. We
speculate that the perturbation from Tm2 plays an important role and brings
about the dynamics. When cooling the temperature across $T_{s}$ = 1 K, Tm3SbO7
may evolves into quantum paramagnetic phase. In addition, we notice that both
$\chi^{\prime}_{ac}$ and $\chi_{dc}$ ($\propto\delta$) also saturate under 1
K. The fine uniformity indicates what we observed are all intrinsic.

###  IV.2 Transverse field Ising model

The low-temperature paramagnetic natrue of $\rm Tm_{3}SbO_{7}$ drives us to
focus on the low-energy physics, especially below 10 K. At low temperatures,
we can construct an effective model to describe the quasi-doublet of Tm1. In
general, considering the non-Kramers nature and interactions between Tm1 ions,
we can model the low-energy physics by using the transverse field Ising model
(TFIM), which is well-studied and successfully applied in many real
materialsColdea  _et al._ (2010); Chen (2019); Cui  _et al._ (2019); Li  _et
al._ (2020); Bitko  _et al._ (1996). As metioned in III.1, Tm1 forms a 3D
tetragonal lattice. The Hamiltonian can be generally written as

$H=\frac{1}{2}\sum_{ij}J_{ij}S_{i}^{z}S_{j}^{z}-\sum_{i}hS_{i}^{x}$ (3)

where the two energy levels in the quasi-doublet of Tm1 are regarded as the up
and down degrees of an Ising spin $S_{i}$. $J_{ij}$ is the interaction energy
between two spins, and $h$ is the energy splitting between the quasi-doublet
of Tm1 ions, acting as the intrinsic transverse field Coldea  _et al._ (2010);
Shen  _et al._ (2019); Cui  _et al._ (2019); Liu  _et al._ (2020); Li  _et
al._ (2020); Chen (2019); Bitko  _et al._ (1996).

Since a 3D tetragonal lattice hardly has geometrical frustration, if $J_{ij}$
is dominant, the system should become ordered at a finite temperature, while a
leading $h$ will prevent the system from ordering even at zero temperature.
The energy splitting $h$ between the quasi-doublet is about 0.64 meV, whose
order of magnitude is agreeable with the result estimated from point charge
model of CEF. All the experimental observations indicate that $h>J_{ij}$ in
$\rm Tm_{3}SbO_{7}$, in other words, the interaction between neighboring spins
is very weak. By now, we can not give the detailed pathway or strength of
$J_{ij}$ from the current data.

Figure 6: (a) Comparison between experimental results and numerical results of
$C_{\rm{M}}/T$ for Tm1 under zero magnetic field. (d) Numerical results of
$C_{\rm{M}}/T$ for Tm1 under different magnetic fields.

###  IV.3 Simulation of specific heat data

Based on the analysis in Sec IV.2, a single ion model is enough to capture the
principal low-temperature physics of $\rm Tm_{3}SbO_{3}$. To prove this view,
we model the quasi-doublet system quantitatively and compare with the
experimental data of specific heat.

Assuming the interaction between the neighboring spins is negligible, the
lower-temperature bump in $C_{\rm{M}}/T$ shown in Fig. 6(a) should be a
Schottky anomaly. For a two-level Schottky anomaly, the maximum occurs at
$T_{\rm M}\approx 0.42h$. Therefore, $h$ = 7.4 K ($\sim$0.64 meV) can be
derived, with the same order of magnitude of the one from CEF calculation. The
single-ion Hamiltonian can be expressed explicitly as

$H=\frac{1}{2}h\left(\begin{array}[]{ll}0&1\\\ 1&0\end{array}\right)+\mu
B_{0}\cos\theta\left(\begin{array}[]{ll}1&0\\\ 0&-1\end{array}\right)$ (4)

The first term in Eq. 4 is the Hamiltonian in zero field. The second term
represents the effect of external magnetic field, whose magnitude is $B_{0}$.
In a polycrystalline sample, the spins point to different directions. As a
result, an external magnetic field has different effects on these spins. In
our calculation, we asssum $\theta$, the angle between the easy axis of spins
and the external magnetic field, is uniformly distributed in $[0,\pi]$. $\mu$
= 7.53 $\mu_{\rm{B}}$ $\approx$ 5.058 K/T is obtained from magnetic
susceptibility experiments in Sec. III.2.

We simulate the specific heat by averaging over 105 random $\theta$ after
obtaining each thermodynamic $C_{V}$ with a fixed $\theta$. A comparison
between the experimental data and theoretical result in zero field is shown in
Fig. 6(a), which indicates that a single-ion Hamiltonian is appropriate for
the system, except for a slight inconsistency at low temperatures. The
theoretical results (Fig. 6(b)) are also consistent with the lower-temperature
bump of $C_{\rm{M}}/T$ under different external magnetic fields: the bump
becomes broader and lower with field increasing. One reason is that external
field expands the energy gap between the two levels, which will drive the bump
to higher temperatures and thus suppress the maximum of $C_{\rm{M}}/T$.
Another reason is that, in a polycrystalline sample, the distribution of
$\theta$ leads to different energy splitting under a same external field. As a
result, averaging all bumps with different maximum position give rise to a
broader and lower bump as the external field is increased.

##  V CONCLUSIONS

A new 3D Sandglass magnet Tm3SbO7 with non-Kramers ions has been discovered.
We have carried out magnetic susceptibility, specific heat, and $\mu$SR
experiments on polycrystalline samples. No long-range magnetic order and no
signature of spin freezing were observed down to 0.1 K. The low-energy
properties of Tm3SbO7 are dominated by the two lowest energy levels of
Tm${}^{3+}_{1}$ with a finite energy gap, and the CEF splitting can be
considered as an intrinsic transverse field. Due to the small exchange
interactions between effective $S=1/2$ spins signified by the small Curie-
Weiss temperature at low temperature, the TFIM with a quantum paramagnetic
state can be applied. Persistent spin dynamics are observed below 1 K and at
least 3 kOe. The perturbation fr0m Tm${}^{3+}_{2}$ may play an important role
to account for this dynamics behavior.

On the other hand, although geometry frustration is not found based on the
lattice structure of Tm3SbO7, the possibility that the absence of magnetic
order and the existence of spin dynamics are due to the competition of
exchange interactions between the nearest-neighbor and the next-nearest-
neighbor Tm3+ ions, can not be completely excluded.

Tm3SbO7 provides a new platform for studying quantum magnetism and dynamic
properties. To further identify the ground state of Tm3SbO7, high-quality
single crystals, careful measurements of CEF splitting using inelastic neutron
scattering are needed. Using pressure or element doping to regulate the energy
splitting $h$ can also be revealing. Besides, from the point-charge-model’s
calculation of CEF, we also find the size of unit cell along $c$ axis have a
significant influence on the several lowest energy levels of Tm2. Even 0.1 Å
can lead to a nearly degenerate doublet which offers a promising way to
regulate the energy level and investigate how exchange interaction come into
play in this system.

###### Acknowledgements.

We thank Gang Chen, Yang Qi, Yuan Wan, and Jianda Wu for fruitful discussion.
We are grateful to the ISIS cryogenics Group for their valuable help during
the $\mu$SR experiments (10.5286/ISIS.E.RB1820271). This research was funded
by the National Natural Science Foundations of China, No. 12034004 and
12174065, and the Shanghai Municipal Science and Technology (Major Project
Grant No. 2019SHZDZX01 and No. 20ZR1405300).

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