111†Co-last authors

# Reaching the equilibrium state of frustrated triangular Ising magnet
Ca3Co2O6

Ivan Nekrashevich1,∗ Xiaxin Ding2,& Fedor Balakirev2 Hee Taek Yi3 Sang-Wook
Cheong3 Leonardo Civale1 Yoshitomo Kamiya4,† Vivien S. Zapf2,† 1Material
Physics and Applications-Quantum (MPA-Q), Los Alamos National Laboratory, Los
Alamos, NM, 87545, USA  2National High Magnetic Field Lab, Los Alamos National
Laboratory, Los Alamos, NM, 87545, USA  3Rutgers Center for Emergent Materials
and Department of Physics and Astronomy, Rutgers University, Piscataway, NJ
08854, USA  4School of Physics and Astronomy, Shanghai Jiao Tong University,
Shanghai, China  ∗Now at Fermi National Accelerator Laboratory  &Now at City
College New York

###### Abstract

Ca3Co2O6 is a frustrated magnet consisting of a triangular arrangement of
chains of Ising spins. It shows regular magnetization steps vs magnetic field
every 1.2 T that are metastable with very slow dynamics. This has puzzled the
community for many years and given rise to numerous potential theories. Here
we approach the problem by seeking the elusive magnetic equilibrium state at
$T=$ 2 K. To this end, we explore two approaches: (1) bypassing the slow
dynamics produced by changing fields by instead field-cooling directly to the
target temperature, and (2) quantum annealing in transverse magnetic fields.
While we observe no measurable effect of the quantum annealing in fields up to
7 T, which is likely due to the large Ising anisotropy of Co spins in this
material, we find that for the field cooling in longitudinal fields we achieve
the predicted equilibrium 1/3 magnetization. We perform Monte Carlo
simulations of the ground state phase diagram and we also simulate the quantum
annealing process and find good agreement between experiment and theory. Thus
we present an investigation of the elusive ground state properties of the
canonical frustrated triangular system Ca3Co2O6.

##  I Introduction

In Ising frustrated magnets, seemingly simple frustrated magnetic interactions
can give rise to complex magnetic behaviors including large unit cells,
incommensuration, fractal phase diagrams, spin liquid behavior, and very slow
dynamics Wannier (1950); Bak (1982); Selke (1988); Coppersmith (1985); Ramirez
(1994); Balents (2010). Ca3Co2O6 is an example of frustrated behavior in Ising
spins with slow dynamics that has remained a puzzle for 25 years Fjellvåg  _et
al._ (1996); Aasland  _et al._ (1997); Kageyama  _et al._ (1997). The
structure consists of ferromagnetically-coupled chains of Ising spins along
the $c$ axis with alternating prismatic Co3+ $S=2$ sites and octahedral $S=0$
Co3+ sites, and these chains are arranged in a triangular configuration in the
$ab$ plane with antiferromagnetic inter-chain coupling Fjellvåg  _et al._
(1996). The magnetization vs magnetic field, $M(H)$, shows regular steps every
1.2 T up to 3.6 T, followed by less distinct, irregular steps leading to the
saturation field of 7 T Har . However, these steps do not correspond to states
in equilibrium. At 4 K and above, magnetic fields applied at extraordinarily
slow sweep rates of 0.01 T/min cause the step phase to disappear in favor of a
single 1/3 plateau in $M(H)$ followed by a step to saturation Har . This 1/3
plateau state is naturally expected to be a ground state of a frustrated
triangular Ising system in a magnetic field Qin  _et al._ (2009), in which two
spins in a triangle point up and one points down. The sweep rate needed to
induce the metastable step phase varies with temperature, extending beyond
experimentally feasible time scale in temperatures below 4 K, such that the
equilibrium state below 4 K has never been measured.

In zero field, long-wavelength incommensuration along the $c$-axis chains and
short-range order has also been observed by neutron and X-ray diffraction
measurements below $T_{c}\sim$ 25K Agrestini  _et al._ (2008a, b); Mazzoli
_et al._ (2009); Fleck  _et al._ (2010); Agrestini  _et al._ (2011); Moyoshi
and Motoya (2011); Motoya  _et al._ (2018). The incommensurate wavevector
slides towards simple ferromagnetism along the chains as the temperature is
lowered, and also evolves with time over several days.

Many theories have been proposed to explain the peculiar magnetic behavior of
Ca3Co2O6. Some theories predicted a partially-disordered antiferromagnetic
state Kageyama  _et al._ (1997), where each ferromagnetic chain along the
$c$-axis is treated as a giant Ising spin that forms different patterns of up
and down in the ab plane Yao  _et al._ (2006); Kudasov (2006); Soto  _et al._
(2009). A different theory is inspired by single-molecule magnets where the
steps in the magnetization result from quantum tunneling Chapon (2009). These
theories all predate the observation of incommensuration in the $c$-axis
chains, which later on prompted Kamiya (co-author of this work) and Batista
Kamiya and Batista (2012) to propose a modified version of the axial next-
nearest-neighbor Ising (ANNNI) model. In the ANNNI model, frustration occurs
between ferromagnetic and antiferromagnetic nearest and next-nearest neighbor
interactions $J_{\mathrm{nn}}$ and $J_{\mathrm{nnn}}$ along the chains. This
gives rise to a fractal phase diagram with a theoretically infinite number of
phases as a function of temperature $T$ and exchange interactions with
different incommensurate wave vectors Bak (1982); Selke (1988) (in other
words, frictional sliding of the incommensurate wave vector). As the
temperature is lowered, the wave vector approaches commensuration, while at
the lowest temperatures a commensurate ground state with ’up up down’ or ’up
up down down’ spins appears depending on the ratio of
$J_{\mathrm{nn}}/J_{\mathrm{nnn}}$. In Ref. Kamiya and Batista, 2012, it was
argued that an effective mean-field description very similar to that of the
classic ANNNI model can be derived for Ca3Co2O6 by interleaving three chains
to form one effective chain with frustrated nearest, next-nearest and next-
next-nearest neighbor interactions. Quantum Monte Carlo (MC) simulations
performed in the original lattice for Ca3Co2O6 indeed reproduced both the
long-wavelength SDW order and the regular magnetization steps below 3.6 T
Kamiya and Batista (2012).

However, to bridge a missing link between theory and experiments, a new
experimental protocol has to be developed to reach the low-$T$ equilibrium
state (minimum of the free energy). In this work, we present the results of
our investigation of two methods to reach the ground state: the first method
is to cool the system in a longitudinal field at each magnetic field, thereby
avoiding the slow dynamics that occurs when the field is changed at low
temperatures. The second method is to quantum anneal the system with magnetic
fields transverse to the Ising axis Brooke  _et al._ ; King  _et al._ (2021),
applied either while cooling or after cooling to 2 K. We compare our
experimental data with our MC simulations for this system.

Figure 1: (color online) (a) Experimental Magnetization $M$. Black stars
(ground state): $M$ after cooling from 30 to 2 K in a longitudinal field
($H_{LFC}$) $H\parallel c$ and then measuring at the same field. Each star
represents a separate field cool. Blue squares (metastable): $M(H)$ measured
by cooling in zero field from 30 to 2 K and then sweeping the field to 7 T and
back to 0. (b) Ground state $M(H)$ obtained by classical MC simulations. (c)
Schematic picture of the magnetic lattice of Ca3Co2O6, with the intrachain
coupling $J_{1}$ and the interchain couplings $J_{2}$ and $J_{3}$,
superimposed on the crystal structure. For the sake of clarity, only $J_{2}$
and $J_{3}$ connected to the site in the center are shown.

##  II Experimental details

Magnetization was measured in a 7 T Magnetic Property Measurement System
(MPMS) Superconducting Quantum Interference Device (SQUID) by Quantum Design
with a horizontal-axis sample rotator. Magnetostriction measurements were
carried out using a piezoelectric strain gauge Ding  _et al._ (2018) in a 14 T
Physical Properties Measurement System by Quantum Design. The sample with a
crystalline face was mounted on the sample holder with less than half a degree
of misalignment. For magnetization measurements, additional alignment to
correct for the thermal contraction of the rotator was achieved for
longitudinal fields by measuring the sample’s remanent moment after field
cooling as a function of angle and seeking the maximum of the projection on
the vertical axis. For quantum annealing experiments, aligning the c-axis
precisely perpendicular to the field was more tricky and a greater degree of
alignment is needed to avoid accidental longitudinal fields. The procedure is
described in the quantum annealing section below.

##  III Equilibrium magnetization curve at low temperature

###  III.1 Experiments

In Fig. 1(a), we show magnetization data vs magnetic field. The star symbols
(black) show the magnetization measured after cooling from 30 K to 2 K in a
longitudinal field ($H_{LFC}$) to the final measurement temperature. In other
words, each data point is measured after separate field cooling at a different
magnetic field to avoid sweeping the field at low temperatures. The data shows
a plateau at 1/3 of the saturation magnetization, 1.66 $\mu_{B}$, before
saturating above 3.6 T. This data is in stark contrast with the previously-
published $M(H)$ curves measured by sweeping the field after zero-field
cooling (ZFC), as also reproduced here with square symbols (blue). As is well-
known, the field-sweep data shows a series of steps every 1.2 T due to the
very slow dynamics that traps the system into metastable states.Har  At 4 K,
Hardy et al. showed that the slow dynamics can be overcome and the 1/3 plateau
state can be reached by sweeping the field very slowly at less than 0.01 T/min
Har . However, the 1/3 plateau thus observed is not as flat as the one at 10
K, implying that slow dynamics could still be preventing the system from
showing true thermal equilibrium. Below 4 K, the dynamics diverge such that a
week-long field sweep still shows the steps. Our direct field-cooling
protocol, on the other hand, can bypass the slow dynamics and bring the system
to its equilibrium even at temperatures as low as 2 K.

###  III.2 MC simulations

In Fig. 1(b), we show magnetization obtained by classical MC simulations for
this system at low temperatures. Here we consider the same Hamiltonian that
was previously studied Kamiya and Batista (2012), defined in the same three-
dimensional lattice as Ca3Co2O6, though we include no transverse field term in
this study. Trigonally prismatic Co3+ ($3d^{6}$) sites with $S=2$ has large
easy-axis anisotropy, which permits a description with an effective Ising
model,

$\displaystyle\hat{H}=\sum_{\gamma=1,2,3}\sum_{\langle{ij}\rangle_{\gamma}}J_{\gamma}\sigma^{z}_{i}\sigma^{z}_{j}-h\sum_{i}\sigma^{z}_{i},$
(1)

where $\sigma^{z}_{i}=\pm 1$ represents the ground state doublet at site $i$,
$\langle{ij}\rangle_{\gamma}$, $\gamma\in\\{1,2,3\\}$ stands for neighboring
sites connected by $J_{\gamma}$, and $h=g\mu_{B}SH$ is the magnetic field.
$J_{1}<0$ is the ferromagnetic intrachain interaction, whereas $J_{2}$
($J_{3}$) is the antiferromagnetic interchain interactions with the vertical
shift of 1/3 (2/3) lattice units along the $c$ axis, as shown in Fig. 1(c). To
account for the highly anisotropic nature of the spin correlation in this
material, we consider a lattice of size $L\times L\times L_{c}$ with $L=4$ and
$L_{c}=40L$ ($N_{\text{spin}}=3L^{2}L_{c}=7680$ spins) with periodic boundary
conditions. We set $J_{2}=J_{3}=0.1\lvert{J_{1}}\rvert$ to obtain the data in
Fig. 1(b), which is motivated by the following considerations. Firstly,
$J_{2}\simeq J_{3}$ is suggested by an ab initio study Frésard  _et al._
(2004). Secondly, an NMR experiment reported $J_{1}=-23.9(2)$ K and
$J_{2}+J_{3}=2.3(2)$ K, further suggesting $J_{2}=1.1$ K and $J_{3}=1.2$ K to
explain the ordering vector Allodi  _et al._ (2014). Here, although the ratio
$J_{2}/\lvert{J_{1}}\approx J_{3}/\lvert{J_{1}}\approx 0.046$ suggested in
this way is even smaller than the one considered in this work, it has been
confirmed that the model can reproduce the same kind of long wavelength three-
sublattice SDW order (wavelength $\sim$ 80 sites) below
$T_{c}/\lvert{J_{1}}\rvert\approx 1.4$, as will be discussed elsewhere. The
data shown in Fig. 1(b) was obtained by combining single-spin and intra-chain
cluster updates Kamiya and Batista (2012) as well as replica exchanges
Hukushima and Nemoto (1996) performed every 10 MC steps. A few hundred
replicas and careful tuning of their individual temperatures are needed for
this system size to address equilibrium properties at low temperatures. The
temperature dependence of the raw data thereby obtained is smooth enough at
each field to allow us to perform interpolation to evaluate $M(H,T)$ at
several (equally separated) temperatures shown in Fig. 1(b). In this way, we
confirm that the system at low enough temperatures realizes the 1/3
magnetization over a wide field range. Here, the lowest $T$ is
$T/\lvert{J_{1}}\rvert=0.183792$, which amounts to $T/T_{c}\approx 0.13$. By
taking into account the transition temperature $T_{c}\approx$ 25 K in this
material, the lowest $T$ roughly corresponds to $\approx$ 3.3 K.

Figure 2: (color online) $M(H)$ measured while sweeping the longitudinal
field, after cooling from 30 to 2 K in different longitudinal fields as
labelled in the plot. Before measuring each M(H) loop, the FC field is
switched off. The magnetic field sweep rate was 10 Oe/s.

##  IV History necessary to reach the metastable step state or the 1/3 plateau
state

In this section we explore which history of field and temperature evolution is
needed to achieve the equilibrium 1/3 plateau vs the metastable step state. In
particular, we find that certain protocols yield a stable 1/3 plateau that
persists even after subsequent field sweeps. In Fig. 2, we first cool the
system in different longitudinal fields ($H||c$) from 30 K to 2 K, turn off
the field, and subsequently sweep the longitudinal $H||c$ magnetic field from
0 to 7 T at 10 Oe/s. For the initial magnetic field between 0 and 0.125 T
applied during the cooling stage, the step phases are observed on subsequent
field sweeps. However, when cooling in fields between 0.15 and 3.6 T (i.e.,
inside the region for the 1/3 plateau of Fig. 1), we observe the 1/3 plateau
state. It should be noted that here the step to saturation occurs at higher
fields than the equilibrium data in Fig. 1. This implies that the step to
saturation is also delayed by slow dynamics, as discussed recently Hegde  _et
al._ (2020).

In Fig. 3 we show the same data over a more extended region of fields and
temperatures in a 3D color plot. E.g. after field cooling to the target
temperature, the magnetic field was switched off and then the longitudinal
field was swept from 0 to 7 T. The target temperatures ranged from 1.8 K to
6.0 K and field-cooling fields of $H=$ 0, 1 T, 3.2 T and 5 T were used. In
Fig. 3 the color is the magnetization, the y-axis shows the swept field, the
temperature is the target temperature, and the inset shows the field-cooling
field.

Figure 3: (color online) $M(H)$ after field cooling in $H\parallel c$, showing
the conditions necessary to observe the 1/3 plateau. After field-cooling to
the target temperature, the magnetic field was switched off and subsequently,
longitudinal $H||c$ magnetization vs field was measured starting from 0 to 7
T. Upsweep date is shown. The color is the value of $M$. The magnetic fields
applied during the field-cooling step are shown in the top right corners of
the diagrams. Vertical axes show the swept longitudinal magnetic field of the
M(H) curves. The horizontal axis shows the target temperature at which M(H)
upsweep was performed after field cooling. The value of the magnetization at
1/3 saturation is light blue and labelled ”1/3 plateau”.

Our experiments show that a 1/3 plateau is achievable for fields swept at low
temperatures, if initially a high enough magnetic field $H\parallel c$ is
applied during the field cooling. Also, it can be noted that the onset of the
transition to saturation does not depend on either the magnitude of the field
during the cool down nor on the final temperature, and appears at 3.6 T during
the upsweep $M(H)$ measurement. Applying field higher than 3.6 T during the
initial field cooling brings the system in the saturated state and the
subsequent M-H upsweeps at different temperatures show re-emergence of the
step phase. For 5 T FC, we observe the 1/3 plateau at high temperatures, as is
consistent with previous works Har . However we were also surprised to observe
a narrow 1/3 plateau below 2 K and the steps phase is the most pronounced in
the region of temperatures 3 - 4 K.

In addition to magnetization we have also measured the magnetostriction
$\Delta L/L$ of Ca3Co2O6 with $H||c$ after zero field cooling at different
temperatures, as plotted in Fig. 4. Magnetostriction in insulating magnets
reflects the change in length of the sample as a result of changes in bond
lengths, which occur to minimize the magnetic energy via changes in the
exchange interaction and anisotropies. At 10 K, the magnetostriction shows all
the metastable steps seen in the $M(H)$ curve. However in the inset of Fig. 4,
the first jump ($\approx$0.15 T) in $\Delta L/L$ at 10 K is very small while
there is no jump at 2 K. The discrepancy between the $M(H)$ data and $\Delta
L(H)$ data is not due to the change in field sweep rate (10 Oe/s vs 100 Oe/s)
because the initial step is also seen at 100 Oe/s in previously published
magnetization data Har ; Kim  _et al._ (2018). One possible reason is that the
first step is due to domain alignment rather than due to a change in the
microscopic magnetic order. Magnetostriction is insensitive to domain
alignment since only the domain boundaries are affected and they occupy a very
small percentage of the sample’s volume. For example, the predicted
equilibrium 1/3 state in the magnetization has a net moment and thus could
form domains. T

Figure 4: (color online) Field dependence of magnetostriction at 2 and 10 K
with longitudinal magnetic field $H_{L}$ along $c$-axis and swept at 100 Oe/s.
Inset is the amplified view of low-field data showing that the initial step
that was seen in the $M(H)$ data is mostly absent in the $\Delta L/L$ data.

In order to further investigate whether the 1/3 plateau is the equilibrium
ground state, we performed magnetic relaxation experiments in the 1/3 state
and in the metastable step state. Some of these have also been reported
previously Moyoshi and Motoya (2011). After reaching each of the 3 steps, the
system’s magnetization was observed for 1 hour while temperature and magnetic
field were kept constant (black, red and blue curves on Fig. 5). At all three
steps, slow evolution of the magnetic moment is evident. In the case of the
1/3 plateau, however, the magnetization remained stable during the 1 hour
measurement. Such a high stability of the magnetization is indicative of
equilibrium achieved by the spin system and suggests that 1/3 plateau is the
global energy minimum state. A more detailed time-evolution of the CCO spin
system will be provided by us elsewhere.

Figure 5: (color online) Time-relaxation curves for four magnetic states of
the system. Magnetization steps 1 - 3 (see Fig. 1) are reached by cooling the
sample in zero field from 30K to 2K and then ramping $H||c$ to 0.4 T, 1.8 T
and 3.0 T respectively at the rate 10 Oe/s. Subsequently, $M(H)$ collected as
a function of time for one hour. The 1/3 plateau is reached by cooling the
sample from 30K to 2K in 3.4 T parallel to the c-axis. The vertical axis has 3
breaks to ease comparison of the relaxation curves, and the vertical scale is
constant throughout.

##  V Quantum Annealing

Besides exploring longitudinal field cooling to reach the equilibrium state,
we have also explored transverse field cooling to see if we can observe
quantum annealing. We are motivated in part by simulations of this compound
using a D-WAVE quantum annealing computer, which found the 1/3 equilibrium
state shown in Fig. 1 King  _et al._ (2021).

Here we explore quantum annealing on actual crystals of Ca3Co2O6. However we
do not see definitive evidence of quantum annealing for fields up to 7 T. We
attempted quantum annealing by two methods: in the first we zero-field-cooled
the sample to 2 K and then applied a transverse magnetic field
$H_{\mathrm{TFC}}$, $H\perp c$. In the second method, we cooled the sample in
$H_{\mathrm{TFC}}$ from 30 to 2 K. For this second method, the alignment of
the sample in the field is critical as any misalignment will create
longitudinal field cooling. To achieve optimal alignment we systematically
attempted transverse field cooling different alignments close to $H\perp c$.
We found that there was one orientation that produced no remanent
magnetization and produced no steps in the subsequent $H||c$ $M(H)$ data. We
took that field to be the transverse field. In Fig. 7a, we show the remanent
moment $M_{r}(H_{\mathrm{TFC}})$ data taken after field cooling in different
transverse fields $H_{\mathrm{TFC}}$ and for different nominal alignments. The
optimal angle is is purple triangles labeled 0.0∘ angle.

After field cooling in this optimal transverse orientation we measured $M(H)$
in longitudinal fields to see whether quantum annealing was effective at
producing the 1/3 plateau state, or if the sample would show the metastable
step state. In Fig. 7 (b) we show that the metastable step state resulted,
i.e. showing no evidence of quantum annealing towards the equilibrium state.
Fig. 7 (b) shows $M(H)$ upsweeps with longitudinal field ($H||c$) after CCO
crystal was cooled from 30K to 2K in the optimal transverse fields of either
0, 3, 4, 5, 6, or 7 T. Cooling in transverse fields up to 7 T produced no
effect on the magnetization curves - the metastable step state is observed for
all transverse field coolings.

Figure 6: (color online) Out-of-equilibrium magnetization computed by quantum
MC simulations for different values of $\Gamma$ and
$J_{2}=J_{3}=0.1\lvert{J_{1}}\rvert$. The system size is
12$\times$12$\times$120 (3 sites per unit cell). The system is initially
thermalized at $h=0$ and $T/\lvert{J_{1}}\rvert=0.3$ before performing an up-
sweep of $h$.

We have performed quantum MC simulations by including a transverse field term
$\hat{H}_{\Gamma}=-\Gamma\sum_{i}\sigma^{x}_{i}$ in Eq. (1). The system size
in this simulation is 12$\times$12$\times$120 and the coupling constants are
the same as above ($J_{2}=J_{3}=0.1\lvert{J_{1}}\rvert$). The protocol for
this numerical experiment follows one in Ref. Kamiya and Batista, 2012: we
first thermalize the system at $h=0$ and $T/\lvert{J_{1}}\rvert=0.3$ and then
perform a up-sweep of $h$. This indeed results in out-of-equilibrium
magnetization steps that becomes smaller for larger values of $\Gamma$ (Fig.
6). However, we find that the typical energy scale to achieve a sizable
quantum annealing effect is $\Gamma/\lvert{J_{1}}\rvert\gtrsim 0.5$ or more.
In experiments however, the size of $\Gamma=g\mu_{B}\mathbf{S}\cdot\mathbf{B}$
for a 7 T transverse field is 300 times smaller than $J_{1}$. This is because
the transverse spin $\mathbf{S}$ is less than 1% of the longitudinal spin as
measured in our samples. Thus the transverse field needed to achieve quantum
annealing in experiments would be at least 2,100 T.

Quantum annealing in a model version of Ca3Co2O6 was also explored by King et
al King  _et al._ (2021) using both quantum MC simulations and simulations on
a 2D D-wave quantum annealing computer. However there also the size of the
transverse field is much larger than in our experiments. Their value of
$\Gamma$ is comparable to the energy scale of the longitudinal field needed to
induce saturation, or about 100 times larger than in our experiments.

Figure 7: (color online) (a) Remanent magnetization measured after cooling
from 30K to 2K in various fields applied at orientations close to ab-plane
(from nominally -9.0° to +1.0°). Each data point represents the remanent
magnetization after cooling the sample (magnitude of the magnetic field during
the cool-down step is shown on horizontal axis) in a mostly transverse field
at various orientations (groups of points of different color) relative to the
ab-plane of the CCO crystal. The change of the sign of remanent magnetization
is due to the change of the projection of magnetic field on the c-axis during
the cool-down. The orientation of the field that didn’t yield transition to
1/3 plateau even at 7 T was assumed $H\perp$c. This orientation marked by the
curve obtained at 0.0° (nominal angle between H and ab-plane). Figure (b)
shows $M(H)$ loops obtained by longitudinal field sweep after cooling from 30K
to 2K in the transverse field identified in (a), showing no evidence of
quantum annealing.

##  VI Conclusion

In conclusion, we have measured and simulated the magnetic properties of the
triangular frustrated Ising spin chain system Ca3Co2O6. We have addressed the
long-standing puzzle of finding the experimental ground state at low
temperatures by field cooling at each temperature. Thereby we avoid the
extremely slow dynamics that so far have prevented reaching equilibrium when
the field is swept at low temperatures. We have performed MC simulations that
demonstrate good agreement with our experimental data showing 1/3 plateau as
the equilibrium state of the system. We have explored and identified the range
of the field cooling parameters such as longitudinal field amplitude and final
temperature that bring the system into the 1/3 equilibrium magnetic state.
Finally we note that after an extensive and careful effort to achieve the
necessary alignment, quantum annealing with a transverse magnetic field up to
7 T was not effective in reaching the equilibrium ground state, likely due to
the very small magnetic susceptibility transverse to the Ising axis. This is
in agreement with our quantum MC simulations of the quantum annealing process,
as well as simulations in the literature King  _et al._ (2021) that require
thousands of Tesla of transverse field to achieve quantum annealing. Thus we
have advanced our understanding of the equilibrium ground state of this
triangular frustrated Ising system, which is an implementation of the classic
ANNNI model for frustrated spins.

###### Acknowledgements.

We are grateful to Cristian Batista for valuable discussions. V.S.Z. and X. D.
were funded by the LDRD program at LANL. The facilities of the NHMFL are
funded by the U.S. NSF, the DOE and the State of Florida through Cooperative
Grant No. DMR-1157490. I.N. and L.C. were supported by the US D.O.E, Office of
Basic Energy Sciences, Division of Materials Sciences and Engineering. Single
crystal growth effort at Rutgers was supported by the DOE under Grant No. DOE:
DE-FG02-07ER46382. Y.K. acknowledges the support by the NSFC (No. 11950410507,
No. 12074246, and No. U2032213), NSFC U2032213 and MOST (No. 2016YFA0300500
and No. 2016YFA0300501) research programs.

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