# Spin liquid state in a rare-earth hyperkagome lattice

J. Khatua  Department of Physics, Indian Institute of Technology Madras,
Chennai 600036, India  S. Bhattacharya  Université Paris-Saclay, CNRS,
Laboratoire de Physique des Solides, 91405, Orsay, France  Q. P. Ding  Ames
Laboratory, U.S. DOE, and Department of Physics and Astronomy, Iowa State
University, Ames, Iowa 50011, USA  S. Vrtnik  Jožef Stefan Institute, Jamova
c. 39, 1000 Ljubljana, Slovenia  A. M. Strydom  Highly Correlated Matter
Research Group, Department of Physics, University of Johannesburg, PO Box 524,
Auckland Park 2006, South Africa  N. P. Butch  NIST Centre for Neutron
Research, Gaithersburg, Maryland, USA  H. Luetkens  Laboratory for Muon-Spin
Spectroscopy, Paul Scherrer Institute, CH-5232 Villigen-PSI, Switzerland  E.
Kermarrec  Université Paris-Saclay, CNRS, Laboratoire de Physique des Solides,
91405, Orsay, France  M. S. Ramachandra Rao  Department of Physics, Nano
Functional Materials Technology Centre and Materials Science Research Centre,
Indian Institute of Technology Madras, Chennai-600036, India  Quantum Centre
for Diamond and Emergent Materials, Indian Institute of Technology Madras,
Chennai 600036, India.  A. Zorko  Jožef Stefan Institute, Jamova c. 39, 1000
Ljubljana, Slovenia  Faculty of Mathematics and Physics, University of
Ljubljana, Jadranska u. 19, 1000 Ljubljana, Slovenia  Y. Furukawa  Ames
Laboratory, U.S. DOE, and Department of Physics and Astronomy, Iowa State
University, Ames, Iowa 50011, USA  P.Khuntia<EMAIL_ADDRESS>Department
of Physics, Indian Institute of Technology Madras, Chennai 600036, India
Quantum Centre for Diamond and Emergent Materials, Indian Institute of
Technology Madras, Chennai 600036, India.  Functional Oxide Research Group,
Indian Institute of Technology Madras, Chennai 600036, India.

###### Abstract

Quantum fluctuations enhanced by frustration and subtle interplay between
competing degrees of freedom offer an ideal ground to realize novel states
with fractional quantum numbers in quantum materials that defy standard
theoretical paradigms. Quantum spin liquid (QSL) is a highly entangled state
wherein frustration induced strong quantum fluctuations preclude symmetry
breaking phase transitions down to zero temperature without any order
parameter. Experimental realizations of QSL in quantum materials with spin
dimensionality greater than one is very rare. Here, we present our
thermodynamic, nuclear magnetic resonance, muon spin relaxation and inelastic
neutron scattering studies of a new rare-earth hyperkagome compound
Li3Yb3Te2O12 in which Yb3+ ions constitute a three dimensional spin-lattice
without any detectable disorder. Our comprehensive experiments evince neither
signature of magnetic ordering nor spin freezing down to 38 mK that suggest
the realization of dynamic liquid-like ground state in this antiferromagnet.
The ground state of this material is interpreted by a low energy $J_{\rm eff}$
= 1/2 degrees of freedom with short range spin correlations. The present
results demonstrate a viable basis to explore spin-orbit driven enigmatic
correlated quantum states in a new class of rare-earth based three dimensional
frustrated magnets that may open new avenues in theoretical and experimental
search for spin liquids.

††preprint: APS

##  I Introduction

Quantum materials wherein superposition and entanglement are at play may
exhibit exotic physical phenomena, such as spin liquids with fractional
quantum numbers coupled to emergent gauge field that offer a novel paradigm in
advancing quantum science and technology Editorial. (2016); Tokura  _et al._
(2017); Basov  _et al._ (2017); Keimer and Moore (2017). The experimental
realization of quantum spin liquid with exotic collective excitations in
quantum materials set an outstanding track in modern condensed matter since
the seminal proposal of P. W. Anderson in 1973 Anderson (1973); Balents
(2010). Quantum spin liquids (QSLs) are characterized by absence of magnetic
order down to $T=0$, long-range entanglement, coherent fluctuation of spins
and the preservation of local symmetries despite strong exchange interaction
between spins Balents (2010); Gingras and McClarty (2014); Takagi  _et al._
(2019); Norman (2016). Fractionalization of quantum numbers is a fingerprint
of QSLs which are different from spin waves in conventional magnets. This is a
well established scenario in one-dimensional (1D) magnets. It has been
suggested that also 2D and 3D QSLs can host fractionalized deconfined
quasiparticles such as spinons and Majorana fermions the understanding of
which is essential in advancing fundamental physics and is highly relevant for
fault-tolerant quantum computing Balents (2010); Nayak  _et al._ (2008);
Savary and Balents (2016); Broholm  _et al._ (2020); Zhou  _et al._ (2017);
Klanjšek  _et al._ (2017); Li  _et al._ (2016); Khuntia  _et al._ (2020); Han
_et al._ (2012). However, the presence of perturbations, such as extra
exchange couplings, unavoidable disorder, and defects in real materials put a
strong constraint for the ideal realization of quantum spin liquids with D
$\geq$ 2 Bernu  _et al._ (1992); Zhu  _et al._ (2017); Khuntia  _et al._
(2016a).
Moreover, 3D spin lattices that are composed of small triangular motifs in
edge- or corner-sharing fashion, can host strong frustration similar to 2D
spin lattices. Namely, quantum fluctuations in such 3D frustrated lattices
with low value of spin could lead to unconventional ground states including
spin liquids Savary and Balents (2016); Broholm  _et al._ (2020). This has
motivated us to explore new quantum materials with rich potential to host such
states with fractional quantum numbers Knolle and Moessner (2019); Khuntia
_et al._ (2016b); Chillal  _et al._ (2020). The intertwining of spin-orbit
coupling, anisotropy, spin correlations and frustration in rare-earth magnets
offer another route to realize novel quantum states Arh  _et al._ (2022);
Gingras and McClarty (2014); Keimer and Moore (2017); Witczak-Krempa  _et al._
(2014); Maksimov  _et al._ (2019). In recent years, spin-orbit driven
frustrated magnets have provided a new light to realize novel quantum
phenomena ranging from spin ice state Ramirez  _et al._ (1999); Gingras and
McClarty (2014) to spin liquid Arh  _et al._ (2022); Li  _et al._ (2016);
Bordelon  _et al._ (2019) and anomalous and spontaneous Hall effect Taguchi
_et al._ (2001); Machida  _et al._ (2010) to Bose-Einstein condensate phase
Hester  _et al._ (2019); Witczak-Krempa  _et al._ (2014). In lanthanide
magnetic materials, the interplay between spin-orbit coupling and low-symmetry
crystal electric field leads to Kramers doublets where effective
$J_{\textnormal{eff}}$ = 1/2 moments of rare-earth ions appear, contrary to
pure $S$ = 1/2 moments in transition metal ions. The 3D spin lattice based
quantum magnets host a plethora of unconventional properties of 3D magnets
including non-trivial short-range spin correlation Dalmas de Réotier  _et al._
(2003), unconventional spin glass Silverstein  _et al._ (2014), and magnetic
Coulomb phase Fennell  _et al._ (2009); Petit  _et al._ (2016). For example,
the lanthanide family A2B2O7 where rare-earth ions form a pyrochlore lattice
is a rich reservoir that hosts a wide range of quantum phenomena owing to
complex interplay between competing degrees of freedom Gingras and McClarty
(2014); Bramwell and Gingras (2001). Quite recently, Ce2Zr2O7 was proposed to
be a 3D QSL candidate, where Ce3+ ions with $J_{\textnormal{eff}}$ = 1/2
moments decorate a network of corner-sharing tetrahedra Gao  _et al._ (2019).
The observation of $U$(1) QSL and emergent phases driven by multipolar
interactions in its analogue Ce2Sn2O7 also set a unique example of a 3D spin-
lattice for exploring quantum many-body phenomena Sibille  _et al._ (2020).
Furthermore, the garnet series R3B5O12 (R = rare-earth ions and B = Ga and Al)
where the rare-earth ions decorate two interpenetrating hyperkagome lattices
are known to harbor myriads of interesting quantum phenomena such as long-
range multipolar state Paddison  _et al._ (2015); Sandberg  _et al._ (2021)
and Ising antiferromagnets Cai  _et al._ (2019); Zhao  _et al._ (2022).
However, most of the garnets undergo a magnetic phase transition at lower
temperature Onn  _et al._ (1967); Deen  _et al._ (2015); Wawrzyńczak  _et al._
(2019); Wen  _et al._ (1989); Shindou (2016); Hermele  _et al._ (2004); Gong
_et al._ (2019). Therefore, the search for an ideal 3D spin lattice to realize
a QSL state poses a particularly challenging problem. For example no report
has been published so-far on a structurally perfect 4$f$-based hyperkagome
lattice in which high degree of spin frustration and anisotropic interactions
mediated by spin-orbit coupling could stabilize an enigmatic 3D spin liquid.
Here, we report a new rare-earth based quantum magnet Li3Yb3Te2O12 and
investigate its inherent physics of 3D frustrated spin-lattice with
$J_{\textnormal{eff}}$ = 1/2 moments of Yb3+ ions. Li3Yb3Te2O12 crystallizes
in the cubic space group $Ia\bar{3}d$ where Yb3+ ions form an interpenetrating
hyperkagome spin-lattice. Our thermodynamic results suggests the presence of a
Kramers doublet state and antiferromagnetic interaction between $J_{\rm eff}$
= 1/2 spins at low temperature. The absence of long-range magnetic ordering
and spin freezing down to at least 38 mK in our ac susceptibility, specific
heat and NMR experiments is the first fingerprint of a spin liquid state in
this frustrated quantum material. Furthermore, specific heat measurements show
the presence of short-range spin correlations in this antiferromagnet. Our
complementary experimental probes thus demonstrate a dynamic ground state and
a field-induced gapped behavior in the spin excitation spectrum in this spin-
orbit driven frustrated hyperkagome quantum material.

Figure 1: Crystal structure and bulk magnetic properties of LYTO. (a) The
position of the Yb3+ magnetic ions within the unit cell. These Yb3+ ions
decorate two interpenetrating chains of 3D corner-shared triangles known as
the hyperkagome lattice. (b) Rietveld refinement of powder X-ray diffraction
data recorded at room temperature. The experimentally observed points
(circles), the calculated Rietveld refinement profile (solid line), the Bragg
reflection positions (olive vertical bars) and the difference between the
observed and calculated intensity (blue line) are shown. (c) The temperature
dependence of magnetic susceptibility in two magnetic fields. The inset shows
the temperature dependent inverse magnetic susceptibility data with the orange
and red lines showing the Curie-Weiss fits for the low- and high-temperature
data, respectively. (d) The temperature dependence of the real part of ac
susceptibility at different frequencies down to 45 mK.

##  II METHODS

Sample synthesis: Polycrystalline samples of Li3Yb3Te2O12 (henceforth LYTO)
were synthesized by a standard solid-state reaction of Li2CO3 (99.0 %, Alfa
Aesar), Yb2O3 (99.998 %, Alfa Aesar), and TeO2 (99.9995 %, Alfa Aesar)
Cevallos  _et al._ (2018). Since the lanthanide-oxide and Li2CO3 are generally
hygroscopic, before being weighted, Yb2O3 and Li2CO3 were preheated at 900∘C
and 100∘C, respectively. The stoichiometric mixtures of starting materials
were pelletised and heated at 800∘C for 30 hours in air with several
intermittent grindings. We also used Y2O3 (99.999 %, Alfa Aesar) and followed
the same procedure to prepare the isostructural non-magnetic analogue
Li3Y3Te2O12. The phase purity was checked by room temperature powder X-ray
diffraction (XRD) using smartLAB Rigaku X-ray diffractometer with Cu K$\alpha$
radiation ($\lambda$ =1.54 Å). The crystal structure of LYTO was confirmed by
the Rietveld refinement of X-ray diffraction data using GSAS software Toby
(2001). All the XRD peaks could be indexed with the cubic space group
$Ia\bar{3}d$ and lattice parameter $a$ = 12.173 Å Cevallos  _et al._ (2018).
No secondary phase is detected in XRD and the Rietveld refinement of XRD data
rules out the presence of any inter-site disorder between the constituent
atoms. The observed sharp XRD peaks indicate high quality polycrystalline
samples investigated in this work.
Magnetization measurements: Magnetization data were acquired using a Quantum
Design, SQUID magnetometer in the temperature range 2 K $\leq$ T $\leq$ 350 K.
Magnetic susceptibility ($\chi(\textit{T})$) data were recorded in magnetic
fields 0.1 and 1 T and the zero-field cooled (ZFC) and field cooled (FC) data
were collected in a magnetic field of 0.01 T. In order to determine the
effective magnetic moment ($\mu_{\textnormal{eff}}$) and Curie-Weiss
temperature ($\theta_{\textnormal{CW}}$), the high-temperature ($>$160 K)
inverse susceptibility (1/$\chi(\textit{T})$) data were fitted with Curie-
Weiss (CW) law $\chi$ = $\chi_{\textnormal{0}}$\+ C/(T $-$
$\theta_{\textnormal{CW}}$). Here C is the Curie constant which is used to
calculate $\mu_{\textnormal{eff}}=\sqrt{8C}$ $\mu_{\textnormal{B}}$,
$\chi_{\textnormal{0}}$ is the temperature independent paramagnetic
susceptibility, and $\theta_{\textnormal{CW}}$ provides an estimate of
magnetic exchange interactions. The isotherm magnetization data are reproduced
well (see SI Fig. 3) by a function of the form $M$/$M_{\textnormal{s}}$ =
$B_{1/2}$ ($y$), where
$B_{J}(y)=[\frac{2J+1}{2J}coth[\frac{y(2J+1)}{2J}]-\frac{1}{2J}coth\frac{y}{2J}]$
is the Brillouin function, $M_{s}$ (= g$J\mu_{\textnormal{B}}$) is the
saturation magnetization and the parameter $y=g\mu_{\textnormal{B}}J\mu_{\rm
0}H/k_{\rm B}T$, where $\mu_{\textnormal{B}}$ is the Bohr magneton, $g$ is the
Landé g-factor. The Brillouin function fit with fixed $J$ = 1/2 to
magnetization data yields Landé $g$ factor, $g$ = 3.54 $\pm$ 0.02.
Specific heat measurements: Specific heat measurements were performed in the
temperature range 0.35 K $\leq$ T $\leq$ 270 K under magnetic fields 0 T
$\leq$ $\mu_{\textnormal{0}}H$ $\leq$ 7 T, using thermal-relaxation technique
provided by Quantum Design, PPMS. Furthermore, specific heat measurements were
carried out separately in the temperature range 0.054 K $\leq$ T $\leq$ 4 K in
zero-field using a dilution refrigerator which was also used to measure ac
susceptibility in the temperature range 0.045 K $\leq$ $T$ $\leq$ 4 K at six
different frequencies using a Dynacool PPMS instrument from Quantum Design.
The total specific heat of LYTO can be expressed as a sum of contributions
from the electronic spins in the ground state Kramers doublet ($C_{\rm mag}$),
lattice contribution ($C_{\rm lat}$) and nuclear contribution ($C_{n}$) i.e.,
$C_{\textnormal{tot}}(T)=C_{\textnormal{mag}}(T)+C_{\textnormal{lat}}(T)$ \+
$C_{n}$. We also measured specific heat of the Li3Y3Te2O12 which we used as a
non-magnetic analogue to subtract the lattice contribution. After subtraction
lattice contribution, we subtracted nuclear Schottky contribution $C_{n}$
$\propto$ $T^{-2}$ to obtain magnetic specific heat.
NMR measurements: Field-swept 7Li ($I$ = 3/2, and gyromagnetic ratio 16.54
MHz/T) NMR measurements down to 38 mK at several frequencies were carried out
on a homemade phase-coherent spin-echo pulse spectrometer equipped with a 9 T
Oxford magnet. The low temperature NMR measurements were performed with a
Oxford Kelvinox dilution refrigerator. NMR spectra measurements were carried
out using a standard Han-echo sequence while the spin lattice relaxation time
measurements were performed following saturation-recovery method. In LYTO, 7Li
is an NMR active nucleus which couples with Yb3+ ions via hyperfine
interactions. Therefore, 7Li NMR can probe the intrinsic magnetic
susceptibility as well as low-energy spin excitations via spectrum position,
and spin-lattice relaxation, respectively. The temperature dependence of the
NMR shift ($K$) at different frequencies was extracted from the fit of field-
swept NMR spectra. For an anisotropic magnetic material three different
components of NMR shift K(T) such as $K_{\textnormal{iso}}$ (isotropic shift),
$K_{\textnormal{ax}}$ (axial shift) and $K_{\textnormal{aniso}}$ (anisotropic
shift), arising from a general hyperfine coupling tensor, can be determined
from the observed experimental line shape. In this respect, the principal
components $K_{\textnormal{X}}$, $K_{\textnormal{Y}}$ and $K_{\textnormal{Z}}$
of tensor K(T) can be used to define $K_{\textnormal{iso}}$ =
1/3($K_{\textnormal{X}}$ $+$ $K_{\textnormal{Y}}$+K${}_{\textnormal{Z}}$),
$K_{\textnormal{aniso}}$ = 1/2($K_{\textnormal{Y}}$ $-$ $K_{\textnormal{X}}$)
and Kax = 1/6(2$K_{\textnormal{Z}}$ $-$ $K_{\textnormal{X}}$ $-$
$K_{\textnormal{Y}}$) Shimizu  _et al._ (2012); Dioguardi  _et al._ (2020).
The temperature dependence of the NMR shift scales with the magnetic
susceptibility as
$\textit{K}_{\textnormal{ax/iso}}(\textit{T})=\textit{K}_{0}+(A_{\textnormal{hf}}^{\rm
ax/iso}/{N_{\textnormal{A}}\mu_{\textnormal{B}}})\chi_{\textnormal{spin}}(\textit{T})$,
where $K_{0}$ is the temperature independent chemical and orbital shift,
$A_{\textnormal{hf}}$ is the hyperfine coupling constant between 7Li nucleus
and Yb3+ spins, $\mu_{\textnormal{B}}$ is the Bohr magneton and
$N_{\textnormal{A}}$ is the Avogadro number, respectively. To extract the
hyperfine couplings, the standard Clogston-Jaccarino plot (see SI Fig. 7) was
used to fit in two temperature regimes and the obtained fitting parameters are
given in SI note 5. The fit of NMR spectra below 1 K is not very good (see SI
Fig. 6) which could be due to an additional spectral weight in the low-field
side at low temperature,however, the origin is not clear at this moment. To
uncover the dynamics of Yb3+ spins in LYTO at very low-energy, we conducted
spin-lattice relaxation time measurement. In order to estimate $T_{1}^{-1}$,
the recovery of longitudinal nuclear magnetization data were fitted by the
single exponential function $M_{z}(t)=(M_{0}-M(t))/M_{0}=A\ \
\textnormal{exp}(-t/{\textit{T}_{1}})$ in the entire temperature range of the
investigation, where $M_{\textnormal{0}}$ is the equilibrium magnetization,
$M_{z}(t)$ is the magnetization at time t after the saturation pulse and $A$
is a constant. The validity of a single exponential function in a wide
temperature range suggests that the electronic moments are uniformly
distributed in the host lattice without disorder. The fits of the recovery
curves with a formula for $I$ = 3/2 nuclei were not as good as the single
exponential function.
$\mu$SR: Muon spin relaxation measurements were performed at the Paul Scherrer
Institute, on the GPS instrument. A weak transverse field of 50 Oe was applied
to determine the initial asymmetry parameter. Zero-field (ZF) and longitudinal
applied fields (LF) datasets were obtained using VETO mode resulting in
minimal background signal ($<0.02$).
Inelastic neutron scattering: Time-of-flight inelastic neutron scattering
measurements were carried out at the NIST Center for Neutron Research (NCNR,
Gaithersburg, MD) on the DCS spectrometer. In order to isolate the field
evolution of the inelastic spectrum, a field-independent background, made of
the average of the 0 and 2T datasets for $E>0.85$ meV and of the average of
the 7, 8, 9 and 10 T datasets for $E<0.85$ meV, was subtracted from the
$Q$-integrated energy cuts. Simulations of powder-averaged dynamical structure
factor for inelastic neutron scattering were done with the SpinW software Toth
and Lake (2015).

##  III Results

###  III.1 Magnetization

The magnetic susceptibility $\chi$(T) data do not exhibit any signs of a
magnetic phase transition down to 2 K (Fig. 1 (c)). The Curie-Weiss (CW) fit
to the high-temperature ($>$160 K) $\chi(T)$ data yields
$\theta_{\textnormal{CW}}$ $\approx$ $-$78 (1) K, $\mu_{\textnormal{eff}}$ =
4.4 $\mu_{\textnormal{B}}$, which is close to $\mu_{\textnormal{eff}}$ = 4.54
$\mu_{\textnormal{B}}$ of a free Yb3+ (J = 7/2) spin and $\chi_{\rm 0}$ = 4.97
$\times$ 10-4 cm3/mol. The estimated large negative $\theta_{\textnormal{CW}}$
is ascribed to the energy scale of excited crystal electric field levels Arh
_et al._ (2022). The interaction energy scale of 4f systems is much lower and
is in our case revealed at low-temperature where the CW fit of 1/$\chi$(T)
data in the temperature range 2 K $\leq$ T $\leq$ 10 K yields
$\mu_{\textnormal{eff}}$ = 3.0 $\mu_{\textnormal{B}}$ and
$\theta_{\textnormal{CW}}$ $\approx$ $-$0.3 K $\pm$ 0.03 K. The obtained
effective moment is smaller than the Yb3+ free-ion value, which suggests a
Kramers doublet state at low temperature, as also confirmed by specific heat
measurements (see below). The negative CW temperature indicates
antiferromagnetic interactions between the pseudo 1/2 moment of Yb3+ ions with
approximate interaction energy of the order of $\theta_{\textnormal{CW}}$. The
Curie-Weiss temperature of most 4$f$ systems is typically small and thus the
onset of spin correlation develops at very low temperature in these quantum
materials Sibille  _et al._ (2020, 2015); Gao  _et al._ (2019); Zhao  _et al._
(2022); Ross  _et al._ (2011). Furthermore, the absence of any bifurcation in
zero-field cooled (ZFC) and field cooled (FC) susceptibility data (see SI Fig.
2) in the field of 100 Oe suggests that Yb3+ spins are not frozen down to 2 K.
In order to confirm the absence of spin freezing, we performed ac
susceptibility measurement down to millikelvin temperatures at different
frequencies. Fig. 1 (d) depicts the real part of ac susceptibility as a
function of temperature down to 45 mK and it exhibits a broad maximum around
80 mK which suggests the short range interaction between Yb${}^{3}+$ moments
in LYTO Xing  _et al._ (2020); Sarte  _et al._ (2021); Zhong  _et al._ (2019).
Importantly, the absence of any frequency dependency of ac susceptibility
strongly rules out glassy behavior of Yb3+ spins in LYTO. The magnetization
isotherms at a few selected temperatures are shown in SI Fig. 3 and the
absence of any visible hysteresis rules out the presence of any ferromagnetic
signal. The calculated Landé $g$ factor, $g$ = 3.46 $\pm$ 0.02, from the low-
temperature Curie-Weiss fit of magnetic susceptibility, is close to that
obtained Landé $g$ factor from Brillouin function fit of the magnetization
isotherm. A similar value of powder average of Landé $g$ factor is also
observed in Yb based 3D pyrochlore system suggesting the single-ion properties
of LYTO are expected to be similar to the pyrochlore system Sarte  _et al._
(2021).

Figure 2: Specific heat of LYTO in various magnetic fields. (a) The
temperature dependence of specific heat ($C_{\rm p}$) of LYTO and its non-
magnetic analog Li3Y3Te2O12 in zero-magnetic field. (b) Low-temperature
magnetic specific heat ($C_{\rm mag}(T)$) after subtraction of lattice and
nuclear contributions in zero-field as well as non-zero field. (c) The
temperature dependence of entropy change ($\Delta$S = $\int C_{\rm mag}(T)/T$
d$T$, the temperatures of the color plots correspond to those in (b).) for
magnetic fields up to 7 T with the expected entropy (R ln 2 for spin-1/2)
indicated by a black dashed line.

###  III.2 Specific heat

Figure 3: $\mu$SR data confirming a dynamic ground state in LYTO. Muon
asymmetry decay obtained in ZF (a) and various LFs at 1.57 K (b) (symbols).
The datasets have been shifted vertically by 0.05 for clarity. Lines are fits
to a stretched exponential function (see text). The temperature (c) and
longitudinal field (d) dependence of the muon spin relaxation rate $\lambda$
extracted respectively from (a) and (b). The red line shows the
$1/T$-expansion of $\lambda(T)$ and indicates the presence of
antiferromagnetic correlation (see text). Error bars depict an uncertainty of
one standard deviation.

Fig. 2 (a) shows the total specific heat measured in zero-magnetic field. The
absence of any $\lambda$-type peak suggests that there is no phase transition
down to 54 mK. After subtracting the lattice contribution using the non-
magnetic analog Li3Y3Te2O12 and the nuclear contribution associated with the
hyperfine splitting of the Yb3+ nuclear spin multiplet Steppke  _et al._
(2010), the resulting temperature dependence of magnetic specific heat,
$C_{\rm mag}(T)$ is shown in Fig. 2 (b) in the temperature range 54 mK $\leq$
T $\leq$ 10 K. Below 2 K, the zero-field magnetic specific heat, $C_{\rm
mag}(T)$ starts increasing and shows a broad maximum around 0.18 K, which
indicates the presence of short-range spin correlations between Yb3+ moments
Bordelon  _et al._ (2019). The presence of broad maximum around 0.18 K agrees
well with the estimated CW temperature at low temperatures and it is
consistent with general expectation for a quantum spin liquid candidate Sarte
_et al._ (2021).
In finite applied magnetic fields, it is observed that the broad maximum
shifts to higher temperatures with increased field which opens gap owing to
Zeeman splitting of the lowest Kramers doublet state. It is worth noting that
we have not observed any suppression of $C_{\rm mag}(T)$ in presence of
magnetic field thus ruling out the presence of disorder Ding  _et al._ (2018).
Therefore, there is no uncertainties in Schottky contribution due to the
effect of disorder in the extracted $C_{\rm mag}(T)$ data. It is observed that
the gap size increases with the increasing magnetic fields linearly (see SI
Figs. 4 and 9) and the fit yields $g$ = 3.34 $\pm$ 0.01 which is consistent
with that obtained Landé $g$-factor from bulk magnetization measurements. The
magnetic field induced Zeeman gap in magnetic specific heat data is observed
at very low-temperature in several rare-earth based spin-liquid candidates
with relatively small exchange interactions compared to the Zeeman splitting
Gao  _et al._ (2019); Ding  _et al._ (2018).
In Fig. 2 (c), we present the estimated entropy change, $\Delta$S, by
integrating $C_{\rm mag}(T)/T$ data in the temperature range 0.054 K $\leq$ T
$\leq$ 16 K and 0.35 K $\leq$ T $\leq$ 16 K for zero-field and finite magnetic
fields, respectively. The total entropy ($R$ln2) expected for
$J_{\textnormal{eff}}$ = 1/2 state is fully recovered for zero-field as well
as for non-zero magnetic fields. This provides the concrete evidence of a
Kramers doublet ground state of Yb3+ spins in LYTO at low temperature.

Figure 4: 7Li NMR spectra and spin-lattice relaxation rate in various magnetic
fields reflecting the spin dynamics in the QSL state of LYTO. (a)7Li NMR
spectra measured at constant frequency $\nu$ = 20.5 MHz at a few
representative temperatures. The orange vertical line corresponds to zero-
shift reference line at 1.238 T. (b) The temperature dependence of isotropic
($K_{\textnormal{iso}}$) and axial ($K_{\rm ax}$) 7Li NMR shifts of LYTO where
the solid line is the scaled magnetic susceptibility measured in 1 T. (c) The
temperature dependence of the 7Li NMR spin-lattice relaxation rate ($T_{\rm
1}^{-\rm 1}$) for three different fields in log-log scale. (d) $T_{\rm
1}^{-\rm 1}$ as a function of inverse temperature (T-1) for different fields
in semi-log scale. The black line represents a fit to a phenomenological model
valid for thermally activated behavior of $T_{\rm 1}^{-\rm 1}$ as discussed in
the text. Error bars depict an uncertainty of one standard deviation. Figure
5: Inelastic neutron scattering. (a-d) Intensity maps of the dynamical
structure factor $S(Q,E)$ for polycrystalline LYTO measured at $T=1.5$ K under
zero applied field (a), $\mu_{0}H=2$ T (b), $\mu_{0}H=4$ T (c) and
$\mu_{0}H=5$ T (d). (e-f) Intensity versus energy for the $Q$-integrated
region $[0,1.5]$ Å-1 for the zero-field dataset (e) and some selected datasets
under applied fields (from which a field-independent background has been
subtracted) (f). Experimental data points are symbols, purple line the
estimated elastic contribution of the zero-field data and blue line is the
Lorentzian quasi-elastic contribution. Solid black lines in (f) are
simulations assuming an axially symmetric $g$-tensor (see text). (g) Median
position of the spectrum as a function of the applied field (black squares).
Red solid line is a linear fit.

###  III.3 Muon spin relaxation

We also performed muon spin relaxation ($\mu$SR), to confirm the absence of
static magnetism on a local scale. With its spin $I=1/2$ and gyromagnetic
ratio $\gamma_{\mu}/2\pi=135.5$ MHz/T, an implanted muon is an unbeatably
sensitive probe of magnetism. Fig. 3 shows the muon asymmetry obtained in zero
(ZF) and longitudinal (LF) applied field configurations down to 1.57 K. The ZF
relaxation shape is nearly exponential in the $T$-range 1.57-50 K. A fit of
the asymmetry using a stretched exponential function $A(t)=e^{-(\lambda
t)^{\beta}}$ leads to large $\beta\sim 0.8-1$ values. This is typical of
strongly fluctuating local fields and, indeed, relaxation under LF remains
almost unchanged, except for the highest field of 7500 Oe, corroborating the
dynamical character of Yb3+ moments. We further investigated this dynamics
through temperature dependence of the muon spin relaxation rate $\lambda$
(Fig. 3). Above 20 K, the strong relaxation ($\sim 1.85$ $\mu$s-1) is typical
of other related $4f$ rare-earth based magnetic materials, such as Yb3Ga5O12
or Yb2Ti2O7, and is consistent with fast fluctuations assuming a similar local
field amplitude Dalmas de Réotier  _et al._ (2003). The slight decrease of
$\lambda$ at low temperatures can be understood in the context of the presence
of antiferromagnetic spin correlations, and the first-order $1/T$-expansion
$\lambda(T)=\lambda^{(\infty)}(1+\theta_{\rm CW}/T)$ de Réotier and Yaouanc
(1997) gives a crude estimate of $\theta_{\rm CW}=-0.25(5)$ K (the red line in
Fig. 3 (c)), in agreement with magnetization measurements.

###  III.4 Nuclear magnetic resonance

We carried out NMR experiments in LYTO in order to confirm the absence of
magnetic ordering down to the lowest experimentally accessible temperatures
and to track static and dynamic susceptibilities. 7Li NMR spectra are
relatively narrow at room temperature (See SI Fig. 5), however, they exhibit
marked broadening below 10 K due to anisotropic interactions and the
development of Yb3+ electron spin correlations. We observed no detectable line
corresponding to the satellite transitions on both sides of the central
transition, which indicates the presence of a very weak quadrupole
interaction. Furthermore, the line shape was found asymmetric indicating the
presence of either asymmetry in the hyperfine coupling between the Li nucleus
and Yb3+ spins or asymmetry in spin susceptibility at low-temperature, both
leading to asymmetry in hyperfine fields. The smooth evolution of the NMR
spectra in the entire temperature range rules out the presence of a long-range
magnetic ordering at least down to 38 mK. The observation of clear shift of
spectra in Fig. 4 (a) uniquely determines the intrinsic uniform static spin
susceptibility on a microscopic scale. It is independent and saturates to a
finite value in the low-temperature limit due to the strong polarization of
the Yb3+ moments under the applied magnetic field.
Fig. 4 (b) presents the temperature dependence of the isotropic NMR shift,
$K_{\textnormal{iso}}$, and the axial shift $K_{\rm ax}$, that were extracted
from the fitting of NMR spectra. At high-temperatures both
$K_{\textnormal{iso}}$ and $K_{\rm ax}$ scale with bulk magnetic
susceptibility as expected due to significant hyperfine coupling between 7Li
nuclei and Yb atoms (see SI Fig. 7). The scaling of $K$ with $\chi$ indicates
that the macroscopic magnetization is intrinsic and originates from Yb3+
electronic moments while any impurity contributions are negligible. The NMR
shift reveals the presence of anisotropic hyperfine field as expected for 4$f$
quantum magnets possibly associated with spin-orbit interactions.
The nuclear spin-lattice relaxation rate ($T_{\rm 1}^{-\rm 1}$) probes the
wave-vector $q$-averaged low-energy spin excitations owing to dynamical
electron spin susceptibility in the ground state of correlated quantum
materials. Fig. 4 (c) shows the temperature dependence of $T_{\rm 1}^{-\rm 1}$
in three different magnetic fields, which exhibits the present of three
different relaxation regimes. Upon lowering the temperature from 300 K,
$T_{\rm 1}^{-\rm 1}$ increases in the intermediate temperature range and
exhibits a plateau around 70 K. Such increase of $T_{\rm 1}^{-\rm 1}$ is not
expected for paramagnetic Yb3+ spins but is rather due to crystal electric
field effects, where spin fluctuations $\nu$ slow down with decreasing
temperature and cause increased NMR relaxation in the fast fluctuation regime,
where $1/T_{1}\propto 1/\nu$ Zeng  _et al._ (2020). Below 70 K, $T_{\rm
1}^{-\rm 1}$ remains constant and field-independent down to 10 K, which
suggests that the relaxation is dominated by paramagnetic fluctuations of Yb3+
spins. At low T, i.e., below $\sim$ 10 K depending on the field, a dramatic
field dependence of $T^{-\rm 1}_{\rm 1}$ develops with a decrease of the
$T^{-\rm 1}_{\rm 1}$ value larger than two orders of magnitude in 4.8 T
compared to that in 1.24 T at 1.6 K. $T_{1}^{-1}$ decreases exponentially in
the temperature range 0.2 K $\leq$ T $\leq$ 10 K, suggesting a gapped
excitation spectrum. It is also observed that, $T_{1}^{-1}$ drops faster in
higher magnetic fields which again suggests a field-driven gap as the specific
heat data does. In Fig. 4 (d), we present $T_{\rm 1}^{-\rm 1}$ as a function
of inverse temperature ($T^{-\rm 1}$) in a semi-log plot where the
phenomenological model relevant for thermally activated behavior i.e., $T_{\rm
1}^{-\rm 1}$ $\propto$ exp ($-\Delta/k_{\rm B}T$) yield a straight line with a
slope proportional to the gap Zeng  _et al._ (2020); Quilliam  _et al._
(2012). We observe that the obtained gap indeed scales linearly with the
applied magnetic field (see SI Fig. 9), which is consistent with the specific
heat results. In addition, three distinct energy regimes are also observed in
the spin-spin relaxation rate ($T_{\rm 2}^{-\rm 1}$; see SI Fig. 8). The
absence of any peak feature in the temperature dependence of $T_{\rm 1}^{-\rm
1}$ rules out the presence of any phase transition also from the spin dynamics
perspective down to 200 mK, the lowest temperature that was reached in our
spin-lattice relaxation experiment. NMR results thus reveal the presence of a
dynamic ground state down to 38 mK with a field-induced gap owing to Zeeman
splittings of the Kramers doublet state.

###  III.5  Inelastic neutron scattering

In order to investigate in more detail the nature of excitations at low-
energy, we carried out inelastic neutron scattering measurements on the cold
neutron time-of-flight instrument (DCS) at the NCNR (Gaithersburg, MD). Fig. 5
shows the dynamical structure factor measured for an incident energy
$E_{i}=3.27$ meV with a resolution of $0.11$ meV (FWHM). The inelastic
spectrum of the zero-field dataset is essentially featureless (Fig. 5 (a)).
The $Q$-intregated energy cut (Fig. 5 (e)) can be well fitted to an elastic
Gaussian line plus a quasi-elastic Lorentzian profile,
$S(Q,E)\sim(1-e^{-E/k_{\rm B}T})^{-1}E\Gamma/(E^{2}+\Gamma^{2})$, accounting
for the magnetic contribution. In agreement with the NMR and $\mu$SR results,
the Yb3+ ions are found to be in a paramagnetic fluctuating regime
characterized by a quasi-elastic line with the linewidth $\Gamma=0.05(2)$ meV
setting an upper value of the energy scale of the interactions ($\sim 0.6$ K),
which is consistent with all other results. Upon increasing the field, a non-
dispersive mode emerges from the elastic line (Fig. 5 b-d) whose median
position depends linearly on the applied field (Fig. 5 g). It is ascribed to
the Zeeman splitting of the effective spin $J_{\rm eff}$ = 1/2 of the Kramers
doublet. Fig. 5 (f) shows $Q$-integrated energy cuts for different applied
fields, which further reveal the presence of two humps whose center to center
distance increases with fields. This spectrum is characteristic of an axially
symmetric $g$-tensor Abragam and Bleaney (2012). To further extract the values
of the $g$-tensor along the local anisotropy axis, $g_{z}=g_{\parallel}$, and
in the perpendicular plane, $g_{xx}=g_{yy}=g_{\perp}$, we simulated the
powder-averaged dynamical neutron structure factor (black lines in Fig. 5f)
and found $g_{\parallel}=4.0(1)$ and $g_{\perp}=2.85(5)$, that account for all
the datasets from 2 to 9 T. This leads to the powder-average value
$g=\sqrt{g_{\parallel}^{2}/3+2g_{\perp}^{2}/3}=3.27(7)$, which is in
reasonable agreement with that obtained from other experiments.

##  IV Summary

In the 3D rare-earth based quantum material Li3Yb3Te2O12 featuring a perfect
hyperkagome lattice constituted by Yb3+ ions, our results using several
complementary experimental techniques reveal that Yb3+ realizes a Kramers
doublet and hence an effective $J_{\rm eff}$ = 1/2 degrees of freedom captures
the essence of the low energy physics in the ground state. The Yb3+ spins
remain dynamic, which is characterized by quasi-elastic INS line in inelastic
neutron scattering experiments and $J_{\rm eff}$ = 1/2 moments interacting
with an energy of 0.6 K, which is rather low but typical for 4$f$-based
quantum materials. Furthermore, ac susceptibility data show no sign of spin
freezing down to 45 mK. The lack of phase transition down to 54 mK is
evidenced also by the magnetic specific heat data and a broad maximum around
0.18 K in zero-field is attributed to the presence of short-range spin-
correlations. The compound thus presents an interesting setting wherein
geometrical frustration conspires with quantum fluctuations to evade magnetic
long-range ordering as further revealed by our NMR results down to 38 mK that
is consistent with ac susceptibility and specific heat results, the primary
hallmark of a dynamic ground state. A field induced gap was evidenced in the
specific heat, NMR relaxation rate and neutron scattering results which is
reconciled with the Zeeman splitting of the effective spin
$J_{\textnormal{eff}}$ = 1/2 of the Kramers doublet. The 4$f$ moments on a
hyperkagome lattice- an emblematic frustrated three dimensional model offer a
rare realization of a spin liquid. Our comprehensive results demonstrate a
liquid-like dynamic state in a rare-earth hyperkagome, which might open the
possibility of stabilizing spin-orbit driven quantum disordered state in 4$f$
magnets. This new family of rare-earth hyperkagome quantum material
Li3RE3Te2O12 holds great potential to realize spin-orbit driven quantum
phenomena and opens new avenues for modern theoretical approaches in
establishing generic frameworks in the context of spin liquids.

##  V Acknowledgments

We acknowledge stimulating discussions with P. Mendels. PK acknowledges the
funding by the Science and Engineering Research Board, and Department of
Science and Technology, India through Research Grants. This research was
supported by the U.S. Department of Energy, Office of Basic Energy Sciences,
Division of Materials Sciences and Engineering. Ames Laboratory is operated
for the U.S. Department of Energy by Iowa State University under Contract No.
DE-AC02-07CH11358. EK and SB acknowledges the France Canada Research Fund for
financial support. AZ acknowledges the financial support of the Slovenian
Research agency under program No. P1-0125 and projects No. J1-2461 and No.
N1-0148. AMS thanks the URC/FRC of UJ and the SA-NRF for financial support.

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