# Interwoven atypical quantum states in CeLiBi2

Mitchell M. Bordelon  Materials Physics and Applications $-$ Quantum, Los
Alamos National Laboratory, Los Alamos, New Mexico 87545, USA  Clément Girod
Materials Physics and Applications $-$ Quantum, Los Alamos National
Laboratory, Los Alamos, New Mexico 87545, USA  Filip Ronning  Materials
Physics and Applications $-$ Quantum, Los Alamos National Laboratory, Los
Alamos, New Mexico 87545, USA  Km Rubi  National High Magnetic Field
Laboratory, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
Neil Harrison  National High Magnetic Field Laboratory, Los Alamos National
Laboratory, Los Alamos, New Mexico 87545, USA  Joe D. Thompson  Materials
Physics and Applications $-$ Quantum, Los Alamos National Laboratory, Los
Alamos, New Mexico 87545, USA  Clarina dela Cruz  Neutron Scattering Division,
Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA  Sean M. Thomas
Materials Physics and Applications $-$ Quantum, Los Alamos National
Laboratory, Los Alamos, New Mexico 87545, USA  Eric D. Bauer  Materials
Physics and Applications $-$ Quantum, Los Alamos National Laboratory, Los
Alamos, New Mexico 87545, USA  Priscila F.S. Rosa  Materials Physics and
Applications $-$ Quantum, Los Alamos National Laboratory, Los Alamos, New
Mexico 87545, USA

###### Abstract

We report the discovery of CeLiBi2, the first example of a material in the
tetragonal Ce$TX_{2}$ ($T$ = transition metal; $X$ = pnictogen) family wherein
an alkali cation replaces the typical transition metal. Magnetic
susceptibility and neutron powder diffraction measurements are consistent with
a crystal-field $\Gamma_{6}$ ground state Kramers doublet that orders
antiferromagnetically below $T_{N}=3.4$ K with an incommensurate propagation
wave vector ${\bf{k}}=(0,0.0724(4),0.5)$ that generates a nanometric
modulation of the magnetic structure. The best model of the ordered state is
an elliptical cycloid with Ce moments primarily residing in the $ab$ plane.
This is highly unusual, as all other $\Gamma_{6}$ Ce$TX_{2}$ members order
ferromagnetically. Further, we observe an atypical hard-axis metamagnetic
transition at $2$ T in magnetostriction, magnetization, and resistivity
measurements. CeLiBi2 is a rare example of a highly conductive material with
dominant skew scattering leading to a large anomalous Hall effect. Quantum
oscillations with five frequencies arise in magnetostriction and magnetic
susceptibility data to $T=30$ K and $\mu_{0}H=55$ T, which indicate small
Fermi pockets of light carriers with effective masses as low as 0.07$m_{e}$.
Density functional theory calculations indicate that square-net Dirac-like
Bi$-p$ bands are responsible for these ultralight carriers. Together, our
results show that CeLiBi2 enables multiple atypical magnetic and electronic
properties in a single clean material.

##  I Introduction

Rare-earth intermetallics show complex quantum behavior derived from the
competition of crystalline electric field (CEF) effects, Ruderman-Kittel-
Kasuya-Yosida (RKKY) exchange, and Kondo interactions [1, 2, 3, 4, 5, 6, 7, 8,
9, 10]. The CEF electric potential splits the strong spin-orbit coupled $f$
electron total angular momentum states, which defines the local moment size
and anisotropy across different local point group symmetries. In $f$ electron
metals, this CEF interaction is usually on the energy scale of 1$-$100 meV
[11, 12, 13, 14, 15], which implies that the thermal population of CEF states
is often relevant in determining magnetic properties. Importantly, the nature
of the local moment plays a key role in setting the basis for RKKY and Kondo
interactions [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. These competing interactions can
be further tuned with external parameters, and the fine balance between these
individually complex interactions leads to emergent phenomena including
superconductivity [16, 17, 18], quantum criticality [19, 20, 21, 22],
nematicity [23, 24, 25], non-Fermi liquid behavior [16, 26], and
incommensurate magnetic states [27, 28, 29].

The family of tetragonal intermetallic compounds Ce$TX_{2}$ ($T$ = transition
metal; $X$ = pnictogen) exemplifies this interconnection with magnetic and
electronic properties that are tunable in pressure and magnetic field [15, 24,
30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48,
49, 50]. In particular, this family shows an experimental link between the $f$
electron CEF ground state and the long-range magnetically ordered states [15,
46]. The CEF of the Ce3+ ions ($J=5/2$) are split within a $C_{4v}$ point
group into three Kramers doublets, wherein the ground state doublet can be
either $\Gamma_{6}$ (pure $m_{j}=\pm|1/2\rangle$) or $\Gamma_{7}$ (mixed
$m_{j}=\pm|5/2\rangle$ and $\pm|3/2\rangle$). Typically, $\Gamma_{6}$
materials are ferromagnets with Ce moments in the $ab$ plane, whereas
$\Gamma_{7}$ materials order antiferromagnetically with moments parallel to
the $c$ axis [15]. Notably, $\Gamma_{6}$ members have a larger $c/a$ ratio
compared to their $\Gamma_{7}$ counterparts.

Electrical and structural properties of the Ce$TX_{2}$ materials are
intertwined with the CEF ground state and its corresponding magnetically
ordered state [50]. For instance, in CeAgBi2 ($\Gamma_{7}$), antiferromagnetic
(AFM) order at $T_{N}=6.4$ K gives rise to successive metamagnetic transitions
that produce a large anomalous Hall component in an applied magnetic field
[49]. In CeAuSb2, an AFM stripe nematic state that breaks fourfold rotational
symmetry sets in at zero field, whereas a double-Q magnetic structure emerges
at high fields [24, 30, 31, 32, 33, 34].

Studying the multitude of electronic, structural, and magnetic states in
Ce$TX_{2}$ materials and their relation to the CEF-magnetism trend is
paramount to understanding and predicting their physical properties. For
instance, superconductivity has yet to be shown in the $P4/nmm$ Ce$TX_{2}$
series despite their apparent corollaries to $f$\- and $d$-based
superconductors that crystallize in the same space group [51, 52, 53, 54].
Tuning quantum materials toward superconductivity or other emergent states
will require a thorough description of the CEF relation not just to magnetic
order but also to electronic properties.

In particular, it is key to study Ce$TX_{2}$ materials wherein the
CEF$-$magnetic order trend breaks down, as this is where nearly degenerate
magnetic and electronic ground states may reside and generate unconventional
states. One notable example is CeAgSb2 ($\Gamma_{6}$ ground state) [38, 35,
36, 37], which contains a spontaneously ordered hard-axis FM moment along the
$c$ axis which contradicts the general $\Gamma_{6}$ $-$ FM trend. However, the
moment reorients to the $ab$ plane at $\mu_{0}H=$ 2.8 T. The unusual zero-
field FM structure has been recently suggested to derive from a dominant RKKY
exchange along the $c$ axis that competes with the CEF anisotropy to determine
the zero field magnetic ground state [36]. Recent reports on CeAgSb2 have
observed tubular shaped topological magnetic domains displaying hysteretic
features arising from patterning of the topological domains related to
competing $ab$ plane CEF easy axes [37]. This suggests that tuning the local
CEF in Ce$TX_{2}$ materials could produce competition and phase transitions
between magnetically ordered ground states and electronic states.

Here, we report on newly-synthesized CeLiBi2, a member of Ce$TX_{2}$ family
that contradicts the CEF$-$magnetic state trend in an unfamiliar manner.
CeLiBi2 replaces the typical transition metal with Li (Figure 1) and creates a
chemical variation in the Ce CEF environment. This material undergoes AFM
order at $T_{N}=3.4$ K as indicated by sharp signatures in magnetic
susceptibility, specific heat, thermal expansion, and electrical resistivity
measurements. Surprisingly, fits to anisotropic magnetic susceptibility show
that antiferromagnetic CeLiBi2 contains a $\Gamma_{6}$ CEF Kramers ground
state doublet, opposing the general Ce$TX_{2}$ trend for ferromagnetism tied
to the $\Gamma_{6}$ CEF ground states. We show that the AFM order is an
incommensurate cycloidal structure with propagation wave vector
${\bf{k}}=(0,0.0724(4),0.5)$ originating from competing exchange interactions.
We further reveal a field-induced transition along the hard $c$ axis, a large
anomalous Hall effect, and quantum oscillations from ultralight pockets
stemming from square-net Dirac-like Bi$-p$ bands. This material demonstrates
how nonmagnetic tuning of the local Ce environment results in a competing
balance of CEF anisotropy and magnetic order tied to uncommon electronic
properties.

##  II Methods

Table 1: Structural refinement parameters at room temperature collected on single crystals of CeLiBi2 with x-ray diffraction ($R_{1}=0.0359,wR_{2}=0.0882$). The structure was refined in the $P4/nmm$ space group setting 2, and the occupancy of Li was held constant. $T$ | 300 K
---|---
$a=b$ | 4.4843(3) Å
$c$ | 10.8791(12) Å
Atom | Wyckoff | x | y | z |  $U_{iso}$ (Å2) | Occupancy
Ce | 2c | 3/4 | 3/4 | 0.26455(7) | 0.0101(2) | 0.98(4)
Li | 2a | 3/4 | 1/4 | 0 | 0.027(9) | 1
Bi | 2b | 3/4 | 1/4 | 1/2 | 0.0104(2) | 0.98(4)
Bi | 2c | 1/4 | 1/4 | 0.16489(5) | 0.0110(2) | 0.99(4)

####  II.0.1 Synthesis

Single crystals of CeLiBi2 were obtained from a Li$-$Bi flux. Ce (99.95%,
AMES), Li (99.9%, Sigma Aldrich), and Bi (99.999%, Alfa Aesar) pieces in a
1:3.2:6.7 ratio were loaded into an alumina crucible and sealed under vacuum
in a quartz ampule. The reagents were slowly heated to 900 ∘C in 40 h to
mitigate Li loss. The melt was held at 900 ∘C for 12 h before slowly cooling
at 2 ∘C/h to 600 ∘C. Then, the ampule was inverted, and the flux was removed
via centrifuge. The plate-like CeLiBi2 crystals were 0.5-1 mm on a side and
showed surface oxidation if left in air. Single crystal x-ray diffraction data
were collected at room temperature on a Bruker D8 Venture diffractometer with
Mo K$\alpha$ radiation. CeLiBi2 produced 9360 reflections, and the data were
analyzed in the APEX 3 software suite with the Full matrix least squares
method [55]. The refined single crystal x-ray diffraction lattice and atomic
parameters at room temperature are displayed in Table 1 with
$R_{1}=0.0359,wR_{2}=0.0882$. Single crystals of LaLiBi2, which have
previously been reported [56], were prepared in a similar manner to CeLiBi2.

Figure 1: (a) Crystallographic structure of CeLiBi2 refined from single
crystal x-ray diffraction. Ce, Li, and Bi atoms are represented by green,
black, and orange spheres, respectively. (b) View along the $c$ axis with Ce
ions surrounded by eight Bi ions in a $C_{4v}$ point group.

Occasionally, three other phases were observed after removing the flux, and
single crystals of CeLiBi2 were judiciously selected from the products. One of
the phases is the known CeLi3Bi2 $P\bar{3}m1$ phase [57], which produced
hexagonal crystals that preferentially formed when the Li content was
increased and the maximum growth temperature was decreased. The two other
phases are unsolved structures and were always present in a small fraction
even in the optimized conditions for CeLiBi2. Though the full structure of
these crystals remains unknown, the lattice parameters were determined by
single crystal x-ray diffraction. Flat plates crystallized in the $C2/m$ space
group with lattice parameters $a=17.377(8)$ Å, $b=4.536(2)$ Å, $c=4.714(4)$ Å,
and $\beta=92.84(4)$∘. Needles crystallized in $C2/m$ with $a=23.0279(73)$ Å,
$b=10.2954(20)$ Å, $c=7.7661(20)$ Å, and $\beta=93.79(1)$∘. Both materials
appear to magnetically order between 3 and 4 K. In all of our single crystal
measurements, these unknown phases are not present.

Polycrystalline CeLiBi2 was made from a nearly stoichiometric mixture of
Ce:Li:Bi in a 1:1.2:1 ratio. The elements were loaded into an alumina
crucible, sealed under vacuum, and heated to 800 ∘C for 3 days. The furnace
was then shut off and cooled to room temperature. Phase analysis was conducted
on a Panalytical Empyrean powder diffractometer with Cu-K$\alpha$ radiation
and analyzed with the Rietveld method in the Fullprof software suite [58].

####  II.0.2 Magnetic susceptibility, heat capacity, resistivity, and
dilatometry

Magnetic properties of single crystals of CeLiBi2 were collected on a Quantum
Design Magnetic Properties Measurement System (MPMS3) equipped with a 7 T
magnet. Magnetic susceptibility and isothermal magnetization measurements were
collected with fields parallel and perpendicular to the crystallographic $c$
axis. Isothermal magnetization data were collected at $T=1.8,3$ and $10$ K,
and susceptibility data were collected under an applied field of
$\mu_{0}H=$0.02 T. In poor quality samples, a superconducting transition near
2.4 K was observed, likely caused by inclusions of BiLi inside of the
crystals. This transition could be suppressed under small magnetic fields less
than $\mu_{0}H=$ 0.02 T.

Specific heat measurements were collected in a Quantum Design Physical
Property Measurement System (PPMS) equipped with a 3He insert capable of
reaching 0.35 K and magnetic fields to $\mu_{0}H=9$ T. Data were collected
between 0.35 to 300 K. Data collected between $T=1.8$ to $2.4$ K above
$\mu_{0}H=5$ T were removed due to calibration issues. For magnetic entropy
analysis, the nonmagnetic analogue LaLiBi2 was measured between 1.8 to 300 K.
Magnetic entropy of CeLiBi2 was determined by integrating $C_{p}/T$ after
subtracting off the lattice contributions approximated by LaLiBi2.

Electrical resistivity measurements were obtained in a PPMS connected to a
low-frequency AC resistance bridge. Both four-point and Hall configurations
were measured with current in the $ab$ plane with external magnetic fields
applied perpendicular and parallel to the $c$ axis. The crystals showed
surface oxidation over time, so the surface was polished prior to attaching
leads. No solvent was used to prevent material degradation. Hall resistivity
was calculated by taking the difference of positive and negative applied
magnetic field data $(R_{H}^{+}-R_{H}^{-})/2$ containing the same field sweep
direction.

Linear thermal expansion and magnetostriction of CeLiBi2 samples were measured
in a PPMS above 1.8 K using a capacitive dilatometer as described in Ref.[59].
Length changes $\Delta L$ of samples with typical lengths $L=$ 300 $\mu$m and
800 $\mu$m were respectively measured along and perpendicular to the
crystallographic $c$ axis. Magnetic fields up to $\mu_{0}H=$ 16 T were applied
parallel to the measured length change. Linear thermal expansion coefficient
$\alpha=\frac{1}{L}\frac{\partial\Delta L(T)}{{\partial T}}$ and linear
magnetostriction coefficient $\beta=\frac{1}{L}\frac{\partial\Delta
L(H)}{{\partial H}}$ were respectively computed after smoothing the field and
temperature dependence of $\Delta L$.

Pulsed field de Haas-Van Alphen (dHvA) measurements were performed in a short-
pulsed magnet with the sample of dimensions $\sim$0.3 $\times$ 0.3 $\times$
0.15 mm3 immersed in 3He liquid or gas depending on the temperature, with the
temperature monitored before the pulse using a Cernox resistor. Additional
dHvA measurements made in pumped 4He liquid confirmed that sample self-heating
due to eddy currents was negligible. The detection coils of inner diameter
$\sim$0.5 mm used for susceptibility measurements were wound from ASW gauge 56
with 450 turns on the inner coil and $\sim$225 turns on the outer coil to
compensate for the signal originating from d$B/$d$t$. The coils were connected
to a differential amplifier with the gain set to 5000, the output of which was
connected to a digitizer.

####  II.0.3 Crystalline electric field analysis

In CeLiBi2, trivalent Ce ($L=3,S=1/2,J=5/2$) resides in a $C_{4v}$ point
group, and its CEF Hamiltonian is comprised of three parameters $B_{n}^{m}$
with Steven’s operators $\hat{O}_{m}^{n}$ [60]:

$H_{CEF}=B_{2}^{0}\hat{O}_{2}^{0}+B_{4}^{0}\hat{O}_{4}^{0}+B_{4}^{4}\hat{O}_{4}^{4}$
(1)

This CEF Hamiltonian generates three Kramers doublets. Determination of the
CEF scheme in CeLiBi2 from magnetic susceptibility was conducted in the
crystal field interface in Mantid Plot [61] combined with a numerical
minimization procedure. In Mantid Plot, the magnetic susceptibility is
calculated as:

$\begin{split}\chi(T)=\frac{N_{A}}{Z}\sum_{i}[\frac{|\langle\phi_{i}|g_{J}\mu_{B}\textbf{J}|\phi_{i}\rangle|^{2}}{k_{B}T}\\\
-2\sum_{i\neq
j}\frac{|\langle\phi_{i}|g_{J}\mu_{B}\textbf{J}|\phi_{j}\rangle|^{2}}{E_{i}-E_{j}}]e^{-\beta
E_{i}}\end{split}$ (2)

where $N_{A}$ is Avogadro’s constant, $k_{B}$ is Boltzmann’s constant, $Z$ is
the partition sum, $\phi_{i}$ is the $i^{th}$ wave function, $E_{i}$ is the
$i^{th}$ energy, and $\textbf{J}=J_{x}B_{x}+J_{y}B_{y}+J_{z}B_{z}$. Here, $z$
corresponds to the highest symmetry rotation axis along the $c$ axis in
CeLiBi2. The minimization procedure employed approaches a global minimum for
the CEF parameters according to the following process adapted from Bordelon et
al. [62]:

(1) Initialize the CEF Hamiltonian with a guess of CEF parameters, starting
from values of related Ce$TX_{2}$ materials.

(2) Diagonalize the CEF Hamiltonian to generate CEF energy eigenvalues, wave
functions, and magnetic susceptibility as a function of temperature.

(3) Calculate $X^{2}=(\chi_{calc}-\chi_{obs})^{2}/\chi_{calc}$ for $B\parallel
c$ and $B\perp c$. This reflects the deviations between observed and CEF
Hamiltonian calculated susceptibilities.

(4) Change the CEF parameters and recalculate $X^{2}$. If the new $X^{2}$ is
less than the old value, accept the new CEF parameters.

(5) Iterate from multiple initialization starting points to approach a global
minimum for CEF parameters.

####  II.0.4 Neutron powder diffraction

Neutron powder diffraction data were obtained on the HB2A diffractometer at
the High Flux Isotope Reactor at Oak Ridge National Laboratory, TN. The
$\sim$4 g CeLiBi2 powder sample was loaded into a vanadium canister that was
placed inside of a cryostat capable of reaching 1.5 K. Data were collected at
$T=300,10,$ and $1.6$ K with an incident neutron wavelength of $\lambda=2.41$
Å selected by a Ge(113) monochromator. Refinement of the diffraction patterns
was performed in the FullProf software suite [58]. Magnetic symmetry analysis
of the ordered phase was performed in the SARAh software suite [63]. The
structural Rietveld refinement parameters of CeLiBi2 at 10 K were fixed in
order to analyze the temperature-subtracted magnetic scattering at 1.64 K.

Magnetic structure refinement excluded the extra intensity near $|Q|=0.55$ Å-1
that only appears in the raw 1.64 K$-$10 K subtraction in Figure 9(a). It is
unlikely due to magnetic ordering of CeLiBi2. This intensity resides on the
shoulder of the lowest angle structural peak shown in the inset of Fig. 9(a).
If the intensity was due to CeLiBi2 magnetic ordering, an additional
incommensurate wave vector component would be necessary to include it.
However, adding intensity to the magnetic structure would cause the Ce moment
size to significantly exceed the maximally allowed value from the CEF
$\Gamma_{6}$ doublet. An estimation of the Ce moment with the included
intensity can be determined by comparing the integrated intensity of the extra
intensity versus the sum of the indexed CeLiBi2 magnetic peaks and the extra
intensity. The extra intensity accounts for roughly 40% of the total. The
expected Ce moment size would then proportionally increase by $\sim
1/0.6=1.67$ with a Ce moment maximal modulus of $1.67\times
1.24\mu_{B}=2.07\mu_{B}$. This far exceeds the $\Gamma_{6}$ CEF doublet
showing that the extra peak is not intrinsic to CeLiBi2 magnetic order. We
conclude the extra peak is not intrinsic to CeLiBi2, and two likely
explanations for the extra peak are magnetism from the impurity $C2/m$ phase
or structural broadening defects like stacking faults [64]. It was modeled at
$T=1.64$ K with increased peak broadening and asymmetry of the main structural
CeLiBi2 phase. The magnetic structure was fit to $1.64-10$ K data after
subtracting the broadened peak shape model as shown in Figure 9(c).

####  II.0.5 Density function theory

Density functional theory (DFT) calculations were performed using the PBE
exchange correlation functional[65] as implemented in the WIEN2K software
[66]. Spin orbit coupling without relativistic local orbitals was included
through a second variational method. To localize the $f$ electrons of Ce they
were treated as core states in the calculation.

##  III Experimental Results

###  III.1 Crystal structure

CeLiBi2 crystallizes in the $P4/nmm$ tetragonal crystal structure with lattice
parameters $a=4.48$ Å and $c=10.88$ Å shown in Figure 1. The refined single
crystal x-ray diffraction lattice and atomic parameters at room temperature
are displayed in Table 1 in the Appendix. Since Li is a light element with
weak x-ray scattering, the position of the Li atom was determined by
comparison with other Ce$TX_{2}$ materials and its occupancy was held constant
during refinement. Powder neutron diffraction results detailed below refined
the Li occupancy in powder samples to 0.96(6).

The CeLiBi2 structure contains trivalent Ce ions in a local $C_{4v}$ point
group. The nearest neighbor Ce-Ce ions reside 4.481 Å apart in a square
lattice, and these Ce lattices stack along the $c$ axis in a staggered
fashion. Between the Ce square nets are alternating layers of LiBi and two-
dimensional square lattices of Bi (Figure 1) that could form the basis for a
topological Dirac semimetallic ground state [67].

Figure 2: (a) Low-temperature magnetic susceptibility of CeLiBi2 collected in
a $\mu_{0}H=0.02$ T field parallel to the crystallographic $c$ axis. Filled
circles represent zero-field cooled (ZFC) data and open circles are field-
cooled (FC) data. (b) Inverse magnetic susceptibility with fields parallel
(orange) and perpendicular (purple) to the $c$ axis simultaneously fit (black)
to the crystalline electric field (CEF) Hamiltonian in Eq. (1). (c) Isothermal
magnetization with fields parallel (orange) and perpendicular (purple) to the
$c$ axis above and below $T_{N}$. (d) A closer look at isothermal
magnetization data for $T=1.8$ K reveals a subtle metamagnetic transition in
$B\parallel c$ data at $\mu_{0}H=2$ T. Above $\mu_{0}H_{FP}=3$ T ($6$ T) for
$B\perp c$ ($B\parallel c$), CeLiBi2 enters a field polarized state.

###  III.2 Magnetic properties and crystalline electric field analysis

The anisotropic magnetic susceptibility and isothermal magnetization data of a
CeLiBi2 single crystal are displayed in Figure 2, where $\chi_{\parallel}$
($\chi_{\perp}$) data are collected with an applied magnetic field parallel
(perpendicular) to the $c$ axis. A sharp decrease in susceptibility in Figure
2(a) occurs at $T_{N}=3.4$ K, which suggests that CeLiBi2 orders
antiferromagnetically. Below $T_{N}$, a small splitting between zero-field
cooled (ZFC) and field-cooled (FC) data arises. Figure 2(b) contains the
magnetic susceptibility of CeLiBi2 collected between 2 to 350 K.

The powder averaged magnetic susceptibility
$\chi_{pow.}~{}=~{}\frac{1}{3}\sqrt{\chi_{\parallel}^{2}+2~{}\chi_{\perp}^{2}}$
was fit to a Curie-Weiss law $\chi=C/(T-\theta_{CW})+\chi_{0}$ in the linear
region from $150-340$ K. The extracted parameters are $C=0.750(2)$ emu K
mol-1, $\mu_{eff}=\sqrt{8C}=2.45(1)\mu_{B}$, $\theta_{CW}=5.54(32)$ K, and
$\chi_{0}=0.00018(1)$ emu mol-1. The effective moment of 2.45 $\mu_{B}$ is
close to the expected value of $g_{J}\sqrt{J(J+1)}=2.54~{}\mu_{B}$ for Ce3+.
This value is reduced likely due to the highest CEF doublet near 45.3 meV,
determined from the full CEF analysis below, not being thermally occupied. The
value of $\theta_{CW}$ is nonphysical, as CEF induced curvature from thermal
depopulation of states offsets the Curie-Weiss law intercept.

To fully and more precisely analyze the anisotropic magnetic susceptibility of
CeLiBi2, a full CEF analysis was conducted. Trivalent Ce ($J=5/2$) resides in
a $C_{4v}$ point group, and its CEF Hamiltonian in Eq. (1) generates three
Kramers doublets. In the $J,\ m_{j}$ basis, two of the doublets contain mixed
$m_{j}=\pm|5/2\rangle$ and $m_{j}=\pm|3/2\rangle$ components ($\Gamma_{7}^{1}$
and $\Gamma_{7}^{2}$), and one doublet is pure $m_{j}=\pm|1/2\rangle$
($\Gamma_{6}$). The ground state doublet selection is predominantly determined
by the sign of the leading CEF Hamiltonian term, $B_{2}^{0}$. In the
Ce$TX_{2}$ materials family, typically when $B_{2}^{0}$ is positive, the
$\Gamma_{6}$ doublet is the ground state. If $B_{2}^{0}$ is negative, a
$\Gamma_{7}$ ground state doublet occurs. Correspondingly, the sign of the
$B_{2}^{0}$ term typically dictates the magnetic susceptibility anisotropy.
When $B_{2}^{0}<0$, usually $\chi_{\parallel}>\chi_{\perp}$, and when
$B_{2}^{0}>0$, usually $\chi_{\parallel}<\chi_{\perp}$. In CeLiBi2,
$\chi_{\parallel}<\chi_{\perp}$ with $B_{2}^{0}>0$, which suggests that its
ground state Kramers doublet is $\Gamma_{6}$.

To verify this, a fit of the CEF Hamiltonian in Eq. (1) to the anisotropic
magnetic susceptibility was performed with a procedure outlined in the Methods
section. The resultant fit in Figure 2(b) has a reduced
$X_{red.}^{2}=X^{2}/\nu=4.09$, where $\nu$ is the number of combined
observable data points in magnetic susceptibility $B\parallel c$ and $B\perp
c$. Extracted parameters are $B_{2}^{0}=1.301$ meV, $B_{4}^{0}=-0.056$ meV,
$B_{4}^{4}=0.728$ meV, and $\chi_{0}=0.000125$ emu mol-1. The ground state
doublet is $\Gamma_{6}=|\pm 1/2\rangle$, the first excited state doublet
$\Gamma_{7}^{1}=0.687|\pm 5/2\rangle-0.726|\pm 3/2\rangle$ is at 6.1 meV, and
the second excited state doublet $\Gamma_{7}^{2}=\pm 0.726|\pm 5/2\rangle\mp
0.687|\pm 3/2\rangle$ is at 45.3 meV.

Figure 3:  (a) Temperature dependence of specific heat $C_{p}(T)/T$ of CeLiBi2
(blue) and nonmagnetic analogue LaLiBi2 (orange). (b) Between 0.35 K to 10 K,
the corresponding integrated magnetic entropy ($S_{M}$) of CeLiBi2 (black
line) reaches 90% $Rln(2)$, suggestive of weak Kondo coupling. (c)-(d)
Tracking the specific heat anomaly as a function of external magnetic field
applied parallel (b) and perpendicular (c) to the $c$ axis. The sharp anomaly
peaked at $T_{N}=3.4$ K initially shifts to lower temperature with increasing
field. The sharp feature broadens out at $\mu_{0}H=5$ T and $\mu_{0}H=3$ T in
$B\parallel c$ and $B\perp c$ data, respectively, corresponding to field
polarized behavior consistent with isothermal magnetization data.

The projected anisotropic $g$-factor components may be calculated for the
$i^{th}$ Kramers doublet as:

$\begin{split}g_{\parallel}=2g_{J}|\langle\phi_{i}^{\pm}|J_{z}|\phi_{i}^{\pm}\rangle|,\\\
g_{\perp}=g_{J}|\langle\phi_{i}^{\pm}|J_{\pm}|\phi_{i}^{\mp}\rangle|.\end{split}$
(3)

For the $\Gamma_{6}$ doublet, this results with $g_{\perp}=2.571$ and
$g_{\parallel}=0.857$. When the doublet is thermally isolated, the expected
maximal ground state anisotropic magnetic moment is calculated as
$m=g\mu_{B}J_{eff}$ where $J_{eff}=1/2$. For the ground state $\Gamma_{6}$
doublet, this corresponds to $m=$1.29 $\mu_{B}$ perpendicular and $m=$0.43
$\mu_{B}$ parallel to the $c$ axis.

Isothermal magnetization M($H$) data collected above and below $T_{N}=3.4$ K
are displayed in Figure 2(c)-(d). No hysteresis is observed, which
corroborates an AFM ground state. The data collected at $T=10$ K are linear
and do not saturate up to $\mu_{0}H=7$ T. At $T=1.8$ K, the data show CeLiBi2
enters a field polarized (FP) regime for $B\perp c$ above $\mu_{0}H_{FP}=3$ T
and for $B\parallel c$ above $\mu_{0}H_{FP}=6$ T. The positive non-zero slopes
above these fields arise from Van Vleck susceptibility between neighboring CEF
levels. The moments reach $m_{FP}=$1.26 $\mu_{B}$ with $B\perp c$ and
$m_{FP}=$0.39 $\mu_{B}$ with $B\parallel c$, indicating that the anisotropic
$\Gamma_{6}$ Ce moments reside primarily within the $ab$ plane and is in good
agreement with the CEF scheme. A closer look at the $T=1.8$ K data in Figure
2(d) reveals a small hard-axis transition just above $\mu_{0}H=2$ T that
reappears in other thermodynamic measurements below.

Figure 4: (a) In-plane electrical resistivity $\rho(T)$ collected on a single
crystal of CeLiBi2 between 2 to 250 K. Nonlinearity at high temperature
($T>10$ K) is caused by depopulation of the two excited state CEF doublets. A
sharp decrease in resistivity appears at $T_{N}=3.4$ K. (b) $\rho(T)$ between
0.35 to 6 K plotted versus $T^{2}$ is fit to Fermi liquid behavior (red line)
with $\rho_{0}=1.40$ $\mu\Omega$ cm and $A=0.50$ $\mu\Omega$ cm K-2.
Deviations from Fermi liquid behavior due to magnetic scattering may reside
below the measured temperature range. (c)-(d) Resistivity as a function of
increasing magnetic field perpendicular (c) and parallel (d) to the $c$ axis
shows an initial downward shift of $T_{N}$ and an increase in in-plane
resistivity below $T_{N}$ with increasing field. (e) Field dependence of
$\rho_{xx}$ and $\rho_{xy}$ below $T_{N}$ in a magnetic field parallel to the
$c$ axis. $\rho_{xx}$ (purple) and $\rho_{xy}$ (orange) show two main features
near $\mu_{0}H=2$ and 6 T. While $\rho_{xx}$ remains positive in the full
field range measured, $\rho_{xy}$ changes sign above $\mu_{0}H=7$ T. (f)
Magnetoresistance of CeLiBi2 with varying $T$. At $T=0.4$ K, a large positive
magnetoresistance is observed that reaches nearly 650% by $\mu_{0}H=9$ T. (g)
Magnetoresistance (MR) obtained at $T=0.4,25,50,100,$ and $200$ K. Below
$T_{N}=3.4$ K, CeLiBi2 exhibits large MR that approaches 650%. h)
$\sigma_{xx}$ and $\sigma_{xy}$ obtained at $T=0.4$ K. CeLiBi2 is a highly
conductive material with $|\sigma_{xx}|\sim 0.55\times 10^{6}$ ($\Omega$
cm)-1.

###  III.3 Specific heat

Low-temperature specific heat data were obtained between 0.35 to 300 K in
varying magnetic fields of $\mu_{0}H=0,1,3,5,7,9$ T. Figure 3(a) shows the
specific heat of CeLiBi2 and nonmagnetic analogue LaLiBi2 to $T=200$ K. As
shown in Figure 3(b), a sharp transition occurs at zero field with its peak at
$T_{N}=3.4$ K (midpoint: $T=3.5$ K). The calculated magnetic entropy was
determined by subtracting off the lattice contribution to specific heat
approximated by the nonmagnetic analogue LaLiBi2. The magnetic entropy
approaches 60% of $Rln(2)$ at $T_{N}$ and 90% of $Rln(2)$ by 7 K, suggesting
that the trivalent Ce moments are well localized and short range correlations
develop well above $T_{N}$. The 10% entropy unaccounted for is caused by a
small Kondo screening, residual entropy below the lowest measured temperature
of $T=0.35$ K, or by errors in the estimation of the magnetic entropy. Between
7 to 15 K, the $C/T$ versus $T$ data were fit to $C/T=\gamma+\beta T^{2}$ to
reveal $\gamma=42.01$ mJ mol-1 K-2, further indicating only a small
enhancement of the effective mass $m^{*}$ in CeLiBi2 compared to other
Ce$TX_{2}$ materials such as CeAgSb2 where $\gamma$ approaches 250 mJ mol-1
K-2 [68].

In external magnetic fields shown in Figure 3(c)-(d), the sharp magnetic
transition shifts down in temperature, again suggestive of AFM order. The
feature broadens above $\mu_{0}H=3$ T and $\mu_{0}H=6$ T for $B\perp c$ and
$B\parallel c$, respectively, whereupon CeLiBi2 enters into a field polarized
state. The broadened feature shifts up with increasing magnetic field in this
state, as is typical of fluctuations above the ordered regime in a magnetic
field [69]. With increasing magnetic field, the magnetic specific heat
component diminishes at low temperatures, and the electronic specific heat
component may be more accurately determined. Analyzing the linear portion of
$C_{p}/T$ versus $T^{2}$ between 0.35 to 3 K at $\mu_{0}H=7$ and $9$ T with
$B\parallel c$ ($B\perp c$) shows decreased $\gamma$ of 18 (10) and 5 (2) mJ
mol-1 K-2, respectively. The decreased values of $\gamma$ compared with the
zero-field value in the paramagnetic region suggests that the carriers become
lighter in an external field as Kondo hybridization effects decline further.

###  III.4 Resistivity

####  III.4.1 Longitudinal

In-plane resistivity data $\rho(T)$ collected on a single crystal of CeLiBi2
is presented in Figure 4. At high temperatures, a broad feature centered
around 100 K could indicate thermal depopulation of the excited CEF level. At
low temperatures, $\rho(T)$ decreases sharply on cooling below $T_{N}=3.4$ K
as CeLiBi2 magnetically orders and magnetic scattering is reduced. Below
$T_{N}$, no magnon scattering component is observed down to $T=0.35$ K. Figure
4(b) shows $\rho(T)$ follows a $T^{2}$ dependence. A fit to
$\rho(T)=\rho_{0}+AT^{2}$ yields a residual resistivity of $\rho_{0}=1.40$
$\mu\Omega$ cm and a Fermi-liquid coeffcient of $A=0.50$ $\mu\Omega$ cm K-2.
No deviation from $\rho(T)\propto T^{2}$ was observed to 0.35 K, which,
combined with a very small residual resistivity, indicates Fermi-liquid
behavior and minimal magnetic scattering in a clean, highly conductive
material. The estimated conductance at $T=0$ K is
$\sigma_{0}=1/\rho_{0}=0.71\times 10^{6}$ ($\Omega$ cm)-1, which indicates
that CeLiBi2 is in the high-purity regime of $\sigma>0.5\times 10^{6}$
($\Omega$ cm)-1 dominated by skew scattering [70].

Additional properties of $\rho(T)$ were studied in external magnetic fields in
Figure 4(c)-(d) between $\mu_{0}H=[0,9]$ T and $T=[0.35,6]$ K. In general,
external fields shifted the ordering transition down in temperature, which is
again an indication of antiferromagnetism. Increasing magnetic field also
resulted in increased resistivity that eventually leads to an upturn in
resistivity below $T=2$ K above $\mu_{0}H=7$ T with $B\parallel c$ and
flattening off of resistivity with $B\perp c$. This reappears as a large
positive magnetoresistance (MR) in Figure 4(g) that approaches 650% at
$\mu_{0}H=9$ T. The origin of increased resistivity in a magnetic field will
be discussed below.

####  III.4.2 Hall resistivity

Hall resistivity $\rho_{xy}(T)$ measurements are displayed in Figure 4(e)-(f).
As $T$ decreases, $\rho_{xy}(T)$ of CeLiBi2 changes from hole-like to
electron-like character. Below $T_{N}=3.4$ K, comparison of $\rho_{xy}$ and
$\rho_{xx}$ reveals two similar features. First, the metamagnetic transition,
originally seen in Figure 2(d) at $\mu_{0}H=2$ T, is observed in both
$\rho_{xy}$ and $\rho_{xx}$ data. Second, another inflection arises near
$\mu_{0}H=5$ to 7 T, above which the sign of $\rho_{xy}$ changes. This second
feature appears just below the field polarized state in $B\parallel c$ and
$T=1.8$ K magnetization data in Figure 2(d). This result suggests that another
phase transition occurs at $T=0.4$ K prior to entering the field polarized
regime as was similarly seen in CeAgBi2 [49]. These phase transitions reappear
in $T=0.4$ K MR data in Figure 4(g). Calculated $\sigma_{xx}$ and
$\sigma_{xy}$ are additionally shown in Figure 4h) and further reveal that
CeLiBi2 is in the high-purity regime with $|\sigma_{xx}|\sim 0.55\times
10^{6}$ ($\Omega$ cm)-1 at $\mu_{0}H=0$ T and $T=0.4$ K. These results further
suggest Hall resistivity in CeLiBi2 primarily originates from skew scattering
[70].

Figure 5: (a) Magnetic field dependence of linear thermal expansion
coefficient $\alpha$ perpendicular to the $c$ axis tracking the evolution of
$T_{N}$. (b) Temperature dependence of linear magnetostriction coefficient
$\beta$ measured with external magnetic field and length change perpendicular
to the $c$ axis. A sharp second-order phase transition appears above
$\mu_{0}H=3$ T at $T=1.8$ K that shifts to lower fields and eventually
broadens with increasing temperature. The transition corresponds to entering a
field polarized regime as similarly observed in $B\perp c$ magnetization data
in Figure 2(d). (c) Magnetic field dependence of linear thermal expansion
coefficient $\alpha$ parallel to the $c$ axis with an anomaly at $T_{N}$.
Correlated noise pattern on the thermal expansion data come from the cell
effect subtraction [59], that is identical between different temperature runs.
(d) Temperature dependence of linear magnetostriction coefficient $\beta$ for
an external magnetic field and length change parallel to the $c$ axis. A
first-order-like anomaly appears at $\mu_{0}H=2$ T coincident with the
$B\parallel c$ hard-axis metamagnetic transition observed in Figure 2(d).
Quantum oscillations with a frequency $F\approx 23$ T are clearly visible
above $\mu_{0}H=3$ T and overlap with the second-order magnetic transition
near $\mu_{0}H\approx 6-7$ T at $T=1.8$ K that shifts to lower fields and
broadens with increasing temperature. Figure 6: (a) Linear magnetostriction
coefficient $\beta$ measured with field and length change along the $c$ axis
at the indicated temperatures. Light colors correspond to data of Figure 5 (c)
and dark colors to data measured on a different sample. Curves at different
temperatures are offset for clarity. (b) $\beta$ above $T=3.4$ K as a function
of $1/B$ between $\mu_{0}H=3$ T and $\mu_{0}H=16$ T, displaying QO at the
lowest frequency $F=24$ T. (c) $\beta$ as a function of $1/B$ between
$\mu_{0}H=11$ T and $\mu_{0}H=16$ T, displaying QO in the frequency range
$F=400-600$ T. (d) Lifshitz-Kosevich fits to the amplitudes of the $F=24$ T
(red) and $F=430$ T (blue) QO from panels (b) and (c), resulting in the
indicated effective masses $m^{*}$. (e) Frequency spectrum of $\beta(1/B)$
above $T=3.4$ K calculated between $\mu_{0}H=3$ T and $\mu_{0}H=16$ T showing
the lowest frequency $F=24$ T and its associated harmonics at 48 and 72 T. (f)
Frequency spectrum of $\beta(1/B)$ calculated between $\mu_{0}H=12$ T and
$\mu_{0}H=16$ T showing three nearby frequencies at $F=430,500$ and $580$ T.

###  III.5 Thermal expansion, magnetostriction, and de Haas-Van Alphen effect

####  III.5.1 Phase analysis

Dilatometry experiments are summarized in Figure 5. Figures 5(a) and (c) show
the temperature dependence of the linear thermal expansion coefficient
$\alpha$ at different applied magnetic fields, with $\Delta L$ and $H$
respectively perpendicular and parallel to the $c$ axis. As for the other
thermodynamic probes, we observe second-order-like anomalies at the AFM order
temperature indicated by a sharp decrease in $\alpha$ near 3.4 K. The zero-
field $T_{N}\approx 3.4$ K is in agreement with thermodynamic probes discussed
above. As magnetic field increases, the ordering temperature decreases similar
to the heat capacity results.

Figure 7:  (a) An example of the raw susceptibility signal measured for a
sample of CeLiBi2 in a pulsed magnetic field up to $\mu_{0}H=55$ T and at
$T=545$ mK. (b) The cluster of $\sim 5$ dHvA frequencies obtained between 400
and 600 T obtained by taking a Fourier transform of signals versus $1/B$ for
$\mu_{0}H=55$ T pulses made at different temperatures as indicated. The
magnetic field interval interval runs from $\mu_{0}H=5$ to 45 T and the signal
is modulated by a Hanning window in $1/B$. (c) The dHvA amplitude for the four
dominant frequencies versus $T$ together with fits to the $T$-dependent term
in the LK formula [71]. The average inverse magnetic field used for the fits
is $0.111$ $T^{-1}$. The fitted masses are shown together with errors.

Figure 5(b) and (d) show the field dependence of the linear magnetostriction
coefficient $\beta$ at different temperatures for applied field and measured
length change $\Delta L$ respectively perpendicular and parallel to the $c$
axis. Typical second-order anomalies appear along both applied field
directions as sharp decreases in $\beta$. This appears between 3$-$4 T for
$B\perp c$ and between 6$-$8 T for $B\parallel c$ and is associated with the
transition from AFM order to the field polarized regime. The value of the
field where CeLiBi2 enters a field polarized state obtained for $B\perp c$ at
our lowest temperature $T=1.8$ K is $\mu_{0}H\approx 3$ T, in agreement with
the results presented above. For $B\parallel c$, clear quantum oscillations
(QO) in $\beta$ are observed above $\mu_{0}H=3$ T. These oscillations, which
will be discussed in following subsection, render the estimation of the onset
of field polarization more difficult, especially at the base temperature
$T=1.8$ K, for which a maximum of the oscillations seats close to the magnetic
anomaly. Nevertheless, the onset of field polarization can be approximated
between $\mu_{0}H_{FP}\approx 6-7$ T that coincides with the behavior observed
in $B\parallel c$ magnetization data in Figure 2(d). For $B\parallel c$, an
additional first-order-like anomaly is found around $\mu_{0}H=2$ T, arising
from the hard-axis metamagnetic transition inferred from the $B\parallel c$
magnetization data in Figure 2(d). With increasing temperature, the onset of
field polarization decreases, as expected for the weakening of the magnetic
order by thermal fluctuations.

####  III.5.2 Quantum oscillations

Extended measurements to $\mu_{0}H=16$ T were carried out on a different
sample to investigate the evolution of QO. A comparison between the low-field
and high-field magnetostriction data of two samples is shown in Figure 6(a).
The almost identical amplitudes of the QO over the overlapping field range
suggests that intrinsic carriers of CeLiBi2, and not impurities or parasitic
phases, are responsible for these QO. The low frequency QO persist at least up
to $T=30$ K. Above $\mu_{0}H=10$ T, additional higher frequency oscillations
are observed, but not fully resolved, to $T=5$ K. This reveals the presence of
multiple orbits with different effective masses. Figure 6(b) shows the lowest
frequency oscillations at $F=24$ T as a function of $1/B$ between $\mu_{0}H=3$
T and $\mu_{0}H=16$ T. Figure 6(c) shows the highest oscillation frequencies
$430-500-580$ T between $\mu_{0}H=11$ T and $\mu_{0}H=16$ T. Frequency spectra
after a polynomial background subtraction contain peaks at the indicated
frequencies as shown in Figure 6(e) and Figure 6(f), respectively for the low-
and high-frequency ranges. The frequency of the slowest QO, $F=24$ T, suggests
a small Fermi surface on the order of $3\times 10^{-3}$ of the unreconstructed
first Brillouin zone area perpendicular to the $c$ axis. By fitting the
temperature evolution of the amplitude of the QO displayed in Figure 6(e)-(f)
to the Lifshitz-Kosevich (LK) formula [71], we obtain effective masses on the
order of $m^{\star}\approx 0.07m_{e}$ for the $F=24$ T pocket and effective
masses $m^{\star}\approx 0.26m_{e}$ for the larger $F=430-500-580$ T pockets.

Additional pulsed field to $\mu_{0}H=55$ T de Haas-Van Alphen (dHvA)
measurements are shown in Figure 7 on another CeLiBi2 sample. These
measurements confirm the existence of the $F=24$ T QO and further reveal
details of the highest QO frequencies with greater precision, indicating a
total of five distinct frequencies. Quantum oscillations of $F=23$ T begin at
fields around $\mu_{0}H=1$ T, but are interrupted by the magnetic phase
transitions occurring at around $\mu_{0}H=$ 2 and $7$ T. At high magnetic
fields, dHvA oscillations for a cluster of frequencies between 400 and 600 T
emerge. These can be seen clearly up to $\mu_{0}H\approx 30$ T whereupon they
become overwhelmed by much larger dHvA oscillations originating from the Cu
that comprises the detection coils. The presence of frequencies between 400
and 600 T continues to be visible in Fourier transforms that are made of the
data at fields extending above $\mu_{0}H=30$ T. In fact, the high frequency QO
is comprised of four QO frequencies split into two doublets at $F=$ 445 and
467 T and $F=$ 551 and 567 T. Their respective effective masses are shown in
Figure 7(c) and range from $m^{\star}\approx 0.50-0.71m_{e}$. These high
frequency QO $m^{\star}$ are larger than the $m^{\star}\approx 0.26m_{e}$ mass
from magnetostriction where the four frequencies are not fully resolved.

Pure bismuth is known to possess a small Fermi surface with light charge
carriers [72]. However, as stated above, QO with the same amplitudes were
measured on different samples, therefore it is unlikely that these QO are
coming from any type of impurity. In fact, light carriers in CeLiBi2 are
consistent with other thermodynamic measurements that suggest well localized
Ce moments with minimal Kondo screening.

###  III.6 Density functional theory analysis

Figure 8(a) shows our first-principles band structure analysis with itinerant
and localized $f$ electrons. The band structure calculations include spin
orbit coupling, and in general follow the previous results on structurally
similar materials in (Ba,Ca,Sr)Mn(Sb,Bi)2, LaAgBi2, and Ce$TX_{2}$ materials
[73, 74, 75, 76, 77, 78, 79, 80, 81]. In the absence of spin orbit coupling,
linear bands cross at Dirac points along the $\Gamma-M$, $Z-R$, and $A-Z$
directions and at the $X$ point in the Brillouin zone. The primary character
of the bands near the Fermi level ($E_{\textrm{F}}$) are from the Bi$-$5$p$
square nets. With spin orbit coupling, however, the crossings are gapped,
leaving linear dispersions near $E_{\textrm{F}}$ but no evidence of massless
Dirac fermions. The small natural electron pockets near $X$ and along $Z-R$ in
Figure 8 that, depending on the tuning of $E_{\textrm{F}}$, are the likely
source of QO data presented above.

Figure 8: (a) The band structure of CeLiBi2 with itinerant (left) and
localized (right) $4f$ electrons where the line at zero energy indicates the
Fermi level position ($E_{\textrm{F}}$). (b) The Fermi surface plot for the
localized scenario reveals small pockets that likely generate the QO.

###  III.7 Elastic neutron diffraction

####  III.7.1 Crystal structure refinement

Neutron powder diffraction results at $T=300,10,$ and $1.64$ K are displayed
in Figure 9(a)-(d). Structural refinement of the 10 K data in Figure 9(b)
required three phases, wherein CeLiBi2 is the main phase ($\sim$ 81% by mass).
We emphasize that, though impurities were present in polycrystalline samples,
our single crystal measurements above do not contain these phases.

The neutron powder diffraction impurity peaks were indexed to $Cmcm$ CeBi2 and
the lattice parameters of the needle-shaped $C2/m$ Ce$-$Li$-$Bi crystal
structure (see Methods: Synthesis). No other known binary or ternary within
the Ce$-$Li$-$Bi phase space could account for the observed impurity
reflections. In order to model the neutron diffraction data, CeLiBi2 and CeBi2
were fit with the Rietveld method while $C2/m$ Ce$-$Li$-$Bi was modeled with
the LeBail method [82]. An estimate of the impurity phase fraction was
determined by comparing the integrated scattering intensity corresponding to
CeLiBi2 and the left over intensity, resulting in roughly 81% of the scattered
intensity corresponding to CeLiBi2 and 19% to CeBi2 and $C2/m$ Ce$-$Li$-$Bi.
The refined lattice parameters of CeLiBi2 at 10 K are $a=4.46689(4)$ Å and
$c=10.88299(15)$ Å. Occupancies of Ce and Bi atoms refined to full within
resolution of the experiment, and the occupancy of Li refined to nearly
stoichiometric at 0.96(6). The overall fit had $\chi^{2}=9.63$.

Structurally-similar compounds $R$Li2$Pn_{2}$ ($R=$ lanthanide; $Pn=$
pnictide) have been previously reported to crystallize in the $P4/nmm$ space
group with the CaBe2Ge2 structure type [83, 84]. This structure is nearly
identical to that of CeLiBi2 but contains a second Li site residing directly
above and below the Ce ions. For CeLiBi2, x-ray diffraction was insufficient
to delineate these two structures due to the weak scattering intensity of
light Li relative to heavy Ce and Bi. In neutron powder diffraction, Li has a
significant scattering cross section and allowed for the comparison of the two
structures. The Rietveld refinement of the neutron powder diffraction data in
Figure 9 rules out the CaBe2Ge2 structure type as the second Li site occupancy
refines to zero.

####  III.7.2 Magnetic structure determination

Figure 9: (a) Elastic neutron powder diffraction data collected on CeLiBi2 at
$T=300,10,$ and $1.64$ K revealing new magnetic peaks at $T=1.64$ K. (b)
Neutron powder diffraction data at $T=10$ K. Impurity phases of CeBi2 and an
unsolved $C2/m$ Ce$-$Li$-$Bi compound are present, but the main phase is
CeLiBi2. (c) Temperature subtracted data reveal new magnetic peaks appearing
below CeLiBi2 $T_{N}=3.4$ K fit to an incommensurate propagation wave vector
${\bf{k}}=(0,0.0724,0.5)$ (black ticks). These data are modeled with an
elliptical cycloidal structure (orange) and a pure sinusoid structure (blue).
(d) Depiction of the elliptical cycloid model where the Ce moments rotate
within the $ac$ plane with the largest modulus in the $ab$ plane. The solid
black lines represent the chemical unit cell. The entire magnetic unit cell
doubles the chemical unit cell along the $c$ axis and is approximately 62 Å
long.

New magnetic peaks arise at $T=1.6$ K as shown in Figure 9(b). These peaks can
be indexed to the antiferromagnetic incommensurate propagation wave vector
${\bf{k}}=(0,0.0724(4),0.5)$ except for one peak near $|Q|=0.55$ Å-1, which is
not intrinsic to CeLiBi2 magnetic order as described in the Methods.

With a propagation wave vector of ${\bf{k}}=(0,0.0724(4),0.5)$, symmetry
analysis of the magnetic structure of CeLiBi2 with two unique magnetic Ce
atoms in SARAh [63] produced four nonzero irreducible representations
$IR_{1}$, $IR_{2}$,$IR_{3}$, and $IR_{4}$ (Kovalev scheme). The $IR_{1}$ and
$IR_{3}$ irreducible representations contain one basis vector each while
$IR_{2}$ and $IR_{4}$ contain two basis vectors each. The best fit to the data
can equivalently be represented by $IR_{2}$ or $IR_{4}$, where the main
difference is the stacking sequence of antiferromagnetism along the $c$ axis.
$IR_{2}$ forms $ABBA$ coupling (AFM coupling within the atomic unit cell)
while $IR_{4}$ forms $AABB$ coupling (AFM coupling between atomic unit cells).

Here, the fits presented in Figure 9(b) were produced from $IR_{4}$ with basis
vectors shown in Table 2. Coefficients of the basis vectors $C_{1}$ and
$C_{2}$ correspond to $bv^{1}$ and $bv^{2}$, respectively, and can be real or
imaginary. Previous reports have shown Ce$TX_{2}$ materials form an elliptical
cycloid [85] or sinusoidally modulated spin density wave [33, 34] structure. A
modulation of the Ce moment within the $IR_{4}$ basis can be formed by a
mixture of real $C_{1}$ and $C_{2}$ of varying magnitudes (fixed moment
orientation) or real $C_{1}$ and imaginary $C_{2}$ of varying magnitudes
(elliptical modulation of a cycloidal structure). For reference, an equal
moment cycloid forms when the magnitude of $|C_{1}|=|C_{2}|$ while $C_{1}$ is
real and $C_{2}$ is imaginary.

Figure 9(c)-(d) and Table 2 show the depiction of the best fit to the data
with an antiferromagnetic elliptical cycloid magnetic structure. The
wavelength of the modulated cycloid along the $b$ axis is $1/0.0724\approx
13.8$ atomic unit cells or $\lambda=b/0.0724\approx 62$ Å. The elliptical
cycloidal model refines $C_{1}=0.621(9)$ and $C_{2}=0.290(25)$ with a
$\chi^{2}=1.99$. This model produces a Ce moment with its maximum modulus
nearly parallel to the $ab$ plane ($1.24(6)$ $\mu_{B}$) and minimum modulus
nearly parallel to the $c$ axis ($0.58(3)$ $\mu_{B}$), which are close to the
expected anisotropic CEF values. For comparison, a pure sinusoidal model is
also fit to the data in Figure 9(c) with $C_{1}=0.539(10)$, $C_{2}=0.353(43)$,
producing a marginally lower fit quality relative to the elliptical cycloid
model with $\chi^{2}=2.06$.

Table 2: Refined coefficients of the magnetic basis vectors of $IR_{4}$
creating the ellipical cycloid magnetic structure in CeLiBi2 below $T_{N}=3.4$
K. $T=1.64$ K
---
${\bf{k}}=(0,0.0724(4),0.5)$
atom (x, y, z) |  $bv^{1}$ |  $bv^{2}$
|  $C_{1}=0.621(9)$ |  $C_{2}=0.290(25)$
Ce1 (3/4, 3/4, 0.26424) |  $(0,2,0)$ |  $(0,0,2)$
|  $+i(0,0,0)$ |  $+i(0,0,0)$
Ce2 (1/4, 1/4. 0.73576) |  $(0,1.552,0)$ |  $(0,0,-1.552)$
|  $+i(0,1.262,0)$ |  $+i(0,0,-1.262)$

##  IV Discussion

From our magnetic susceptibility and neutron diffraction results, CeLiBi2 must
contain a $\Gamma_{6}$ CEF ground state Kramers doublet. To our knowledge, all
other $\Gamma_{6}$ CEF ground state Ce$TX_{2}$ materials exhibit ferromagnetic
ground states, and most of them contain Ce moments within the $ab$ plane.

The notable exception to the latter pattern is CeAgSb2 [38, 35, 36, 37],
wherein ferromagnetism arises at zero field from a $\Gamma_{6}$ ground state
with Ce moments oriented along the $c$ axis [38, 35, 36, 37]. Importantly, the
low-energy spin dynamics of CeAgSb2 measured by inelastic neutron scattering
can be described within linear spin-wave theory by a combination of nine
anisotropic exchange interactions between local pseudo spin $S=1/2$ Ce moments
within the $\Gamma_{6}$ ground state doublet [36]. These results point to the
importance of long-range exchange interactions that go well beyond nearest
neighbors. More recently, a microscopic minimal Kondo-Heisenberg model
uncovered the central role of long-ranged RKKY contributions in understanding
magnetic order and magnon dispersion in AFM CeIn3 [86].

In the case of CeLiBi2, wherein Kondo or quantum critical effects are
negligible, our results unravel the delicate balance between the crystal
electric field ground state and multiple competing exchange interactions,
which likely contain substantial long-range exchange contributions. In fact,
long-range RKKY interactions in generic Fermi surfaces usually give rise to
incommensurate magnetic structures [86], and we find evidence of an
incommensurate AFM ordered state in CeLiBi2 contrasting with the $\Gamma_{6}$
CEF$-$FM order trend.

The best model of our neutron diffraction data is an elliptically modulated
cycloidal structure where the Ce moments primarily reside within the $ab$
plane (Figure 9). Ce moments are ferromagnetically coupled along the $a$ axis
and antiferromagnetically modulated along $\bf{k}$ in the $b$ and $c$ axes.
The size of the modulation within the magnetic unit cell varies from $1.24(6)$
$\mu_{B}$ Ce moments nearly parallel to the $ab$ plane to $0.58(3)$ $\mu_{B}$
Ce moments nearly parallel to the $c$ axis, producing a $\approx 0.66$
$\mu_{B}$ modulus magnitude. Because the Ce moments in CeLiBi2 are highly
localized with minimal Kondo coupling, the modulation of the Ce moment is not
accomplished via itinerant $4f$ electrons near $E_{\textrm{F}}$ with a Fermi
nesting vector in a spin density wave. Instead, the elliptical cycloid
naturally forms as a consequence of the $\Gamma_{6}$ $g$-factor anisotropy and
competing RKKY exchange interactions including short- and long-range
components as discussed above.

The magnetic peaks determined in Figure 9(c) for CeLiBi2 were described by a
single incommensurate propagation wave vector along $ab$ plane as
${\bf{k}}=(0,0.0724(4),0.5)$ that produces an elliptical cycloidal magnetic
structure. We note that neutron powder diffraction data are insensitive to the
difference of multi-$\bf{k}$ magnetism (e.g., ${\bf{k_{1}}}=(0,0.0724(4),0.5)$
and ${\bf{k_{2}}}=(0.0724(4),0,0.5)$) that could generate a modulated magnetic
structure as seen in elemental Nd [87]. However, the single$-{\bf{k}}$
propagation wave vector is similar to those of CeAgBi2 [85] and CeAuSb2 [33],
as AFM members of this family contain an ordering wave vector near
${\bf{k}}=(0,0,0.5)$ with incommensurability within the $ab$ plane. Notably,
the modulation of these magnetic phases occurs in a length scale that is an
order of magnitude larger than the lattice spacing, which again likely arises
due to competitive long-range RKKY interactions. However, the zero-field
propagation wave vector of CeLiBi2 curiously coincides with the $b$ axis (or
equivalently the $a$ axis), whereas in CeAgBi2 and CeAuSb2 the wave vector
points between $a$ and $b$ as ${\bf{k}}=(0.18,0.22,0.5)$ and
${\bf{k}}=(0.136,\pm 0.136,0.5)$, respectively. Similar wave vectors were
attempted to model CeLiBi2 but were unsuccessful. Interestingly, CeAuSb2 under
$b$ axis uniaxial pressure adopts a commensurate wave vector
${\bf{k}}=(0,0.25,0.5)$ where the in-plane component resides along the $b$
axis [34]. In this case, however, uniaxial stress explicitly breaks the
underlying tetragonal symmetry of the lattice, which allows additional terms
in the Landau free energy, but such symmetry breaking is not observed in
CeLiBi2. Future single crystal neutron scattering measurements could readily
improve our understanding of this material by further refining the ordering
wave vector with elastic measurements and investigate the balance between
local CEF anisotropy and itinerant magnetic exchange in dynamic inelastic
scattering experiments.

Another atypical characteristic of CeLiBi2 is the hard-axis metamagnetic
transition at $\mu_{0}H=2$ T observed in magnetization and electrical
resistivity data in Figure 4. In general, a hard-axis transition is not a
common property of $f$ electron materials, especially when observed at a lower
field compared to metamagnetic transitions along the easy axis [88, 89]. For
CeLiBi2, two likely scenarios of hard-axis metamagnetism are a change in the
moment size or magnetic structure. Magnetic moment size change could result
from population of neighboring CEF states in a magnetic field, but the change
in magnetization of CeLiBi2 in Figure 2(d) is not significant enough to
support this, which agrees with the well separated CEF ground state.

If not a change in the CEF contribution, a natural scenario is that the
multiple exchange interactions present in CeLiBi2 are field dependent, and the
out-of-plane magnetic field therefore alters the zero-field cycloidal magnetic
structure. Magnetic field tuning of the competition between the single-ion CEF
anisotropy and the anisotropy of exchange interactions has been recently
proposed for YbRh3Si7, wherein metamagnetic transitions also occur for fields
along the hard $c$ axis [88]. In addition, spin density wave materials such as
Sr3Ru2O7 [90] and URu2Si2 [91] undergo wave vector changes in an applied field
as the Fermi surface is modified. Magnetism in CeLiBi2 is not a consequence of
a spin density wave, but the magnetic structure is formed via extended RKKY
exchange interactions that would be susceptible to Fermi surface changes.
Alternatively, the hard-axis magnetic field may push CeLiBi2 towards a
commensurate ’lock-in’ phase transition as seen in CeAuSb2 under pressure
[34], in spin density wave materials like CaFe4As3 as function of temperature
[92], and CeRh3Si2 [93] or other incommensurate spiral magnetic materials like
LiYbO2 [94] in a magnetic field. Finally, a change in the spin texture is a
related viable explanation of this hard-axis transition in CeLiBi2 akin to
other Ce$TX_{2}$ materials as a function of temperature, pressure, or field.
For example, magnetic skyrmion formation has been recently observed in CeAgBi2
in an external magnetic field above $\mu_{0}H=4$ T in polarized neutron
scattering experiments [85]. Possible development of a multi$-\bf{k}$
structure in a magnetic field in CeLiBi2 prompts further single crystal
neutron scattering measurements in a magnetic field to fully determine the
origin of this metamagnetic transition.

Figure 10: Phase diagram of CeLiBi2 obtained by combining thermal expansion
($\alpha$), magnetostriction ($\beta$), specific heat ($C_{p}$), and
electrical resistivity ($\rho$) measurements. Blue (red) curves represent data
collected in magnetic fields parallel (perpendicular) to the $c$ axis. Solid
lines represent the phase boundaries between the antiferromagnetically ordered
elliptical cycloid phase to the field polarized regime. The dashed blue line
indicates the location of the hard-axis metamagnetic transition. The golden
star indicates where elastic neutron powder diffraction data determined the
elliptical cycloid magnetic structure in CeLiBi2.

The hard-axis metamagnetic transition also appears in resistivity and
magnetostriction measurements in Figures 4(e)-(f) and 5(c) and is summarized
in Figure 10. These ubiquitous signatures suggest that the electronic and
magnetic properties of CeLiBi2 are moderately, but undoubtedly, intertwined.
CeLiBi2 resides within the high-conductivity regime ($\sigma_{0}\sim
0.71\times 10^{6}$ ($\Omega$ cm)-1), which suggests skew scattering is the
dominant process in this clean material. Skew scattering is in stark contrast
to magnetic scattering, which would cause a reduction in the resistivity of
CeLiBi2 as a function of magnetic field due to increased spin polarization.
Instead, we observe a large positive MR $\sim 650\%$ at $\mu_{0}H=9$ T (Figure
4(c)-(d)). Nonsaturating MR can arise, for instance, in a nearly compensated
semimetal with quadratic (linear) band dispersion producing quadratic (linear)
MR field dependence [95, 96]. This appears in semimetals such as
(Ta,Nb)(Sb,As)2 and (Ta,Nb)P [96, 97]. Alternatively, nonsaturing linear MR
appears in clean metals displaying quantum oscillations in the limit
$\omega_{c}\tau>>1$ [70, 98, 99] as is found in the high-field regime of
CeCoIn5 [100, 101]. CeLiBi2 is a clean metal with light carriers in the high-
conductivity limit and therefore the condition $\omega_{c}\tau>>1$ is likely
to be satisfied.

We determined the mass of the carriers in CeLiBi2 via magnetostriction and
dHvA measurements displaying QO up to $T=30$ K and $\mu_{0}H=55$ T. Further,
the frequency of the slowest oscillations and associated effective mass of
$m^{*}\approx 0.07\ m_{e}$ agree with results from other materials of the same
structure type such as (Ba,Ca,Sr)Mn(Sb,Bi)2, LaAgBi2, and other Ce$TX_{2}$
materials [73, 74, 75, 76, 77, 78, 79, 80, 81]. In the absence of spin-orbit
coupling, the square nets of Bi ions form Dirac cones with $p$ orbital
character near $E_{\textrm{F}}$. As has been noted previously in SrMnBi2, the
presence of strong spin-orbit coupling from the heavy Bi atoms can create gaps
at the Dirac points of around 40 meV [73]. These band crossings become avoided
band crossings, generating small electron pockets near $E_{\textrm{F}}$ near
($\pi$,0,0) and along (0,0,$\pi$) to ($\pi$,0,$\pi$) in Figure 8.
Consequently, these electron pockets are not expected to contain Dirac
fermions, but their highly dispersive bands can contain small effective mass
carriers that, depending on the exact tuning of $E_{\textrm{F}}$, are the most
likely origin of the light carriers in CeLiBi2. The coexistance of light,
mobile carriers based on Bi square nets and AFM order makes CeLiBi2 a possible
platform for developing spintronic devices[102] as has been proposed for GdTe3
containing an analogous Te square net and AFM order [103]. To explore this
further, angle-resolved photoemission spectroscopy and angle-resolved dHvA
measurements are desired.

##  V Conclusions

CeLiBi2 expands the Ce$TX_{2}$ family into the alkali metal phase space and
opens a new chemical pathway to tune the physical properties in this family.
Our combined single crystal physical properties measurements and elastic
neutron powder diffraction analyses show that CeLiBi2 hosts a breadth of
atypical magnetic and electronic properties. First, contradicting $\Gamma_{6}$
CEF and antiferromagnetic incommensurate cycloidal order below $T_{N}=3.4$ K
suggests a delicate balance between CEF anisotropy and competing long-ranged
magnetic exchange interactions. Second, a hard-axis metamagnetic transition
occurs near $\mu_{0}H=2$ T in magnetization, magnetostriction, and transport
measurements implying intertwined changes in magnetic structure or texture
with electronic properties. Third, CeLiBi2 is a rare example of a clean
material with high conductance, a large anomalous Hall effect, and positive
linear magnetoresistance, indicating that skew scattering dominates in the
material. Fourth, we find evidence of quantum oscillations at five different
frequencies arising from light carriers from square-net Bi bands. All of these
properties separately are uncommon, but not unique. However, their coexistence
places newly-synthesized CeLiBi2 as a prime candidate for investigating their
complex interplay in a single clean material.

###### Acknowledgements.

We would like to thank M. Janoschek, D. Yahne, W. Simeth, and C. Batista for
fruitful discussions. Work at Los Alamos was performed under the auspices of
the U.S. Department of Energy, Office of Basic Energy Sciences, Division of
Materials Science and Engineering. MB, CG, and SMT acknowledge support from
the Laboratory Directed Research and Development program. Scanning electron
microscope and energy dispersive X-ray measurements were performed at the
Center for Integrated Nanotechnologies, an Office of Science User Facility
operated for the U.S. Department of Energy Office of Science. Pulsed field
measurements (NH and RK) were supported by the Department of Energy (DoE) BES
project ‘Science of 100 tesla.’ The National High Magnetic Field Laboratory,
which hosts the high magnetic field magnets, is funded by NSF Cooperative
Agreements DMR-1157490 and 1164477, the State of Florida and DoE. A portion of
this research used resources at the High Flux Isotope Reactor a DOE Office of
Science User Facility operated by the Oak Ridge National Laboratory.

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