# High-field phase diagram of the chiral-lattice antiferromagnet
Sr(TiO)Cu4(PO4)4

Toshihiro Nomura<EMAIL_ADDRESS>Tokyo Denki University, Adachi,
Tokyo 120-8551, Japan  Hochfeld-Magnetlabor Dresden (HLD-EMFL) and Würzburg-
Dresden Cluster of Excellence ct.qmat, Helmholtz-Zentrum Dresden-Rossendorf,
01328 Dresden, Germany  Institute for Solid State Physics, The University of
Tokyo, Kashiwa, Chiba 277-8581, Japan  Yasuyuki Kato
<EMAIL_ADDRESS>Department of Applied Physics, The University
of Tokyo, Tokyo 113-8656, Japan  Yukitoshi Motome  Department of Applied
Physics, The University of Tokyo, Tokyo 113-8656, Japan  Atsushi Miyake
Institute for Solid State Physics, The University of Tokyo, Kashiwa, Chiba
277-8581, Japan  Institute for Materials Research, Tohoku University, Oarai,
Ibaraki 311-1313, Japan  Masashi Tokunaga  Institute for Solid State Physics,
The University of Tokyo, Kashiwa, Chiba 277-8581, Japan  Yoshimitsu Kohama
Institute for Solid State Physics, The University of Tokyo, Kashiwa, Chiba
277-8581, Japan  Sergei Zherlitsyn  Hochfeld-Magnetlabor Dresden (HLD-EMFL)
and Würzburg-Dresden Cluster of Excellence ct.qmat, Helmholtz-Zentrum Dresden-
Rossendorf, 01328 Dresden, Germany  Joachim Wosnitza  Hochfeld-Magnetlabor
Dresden (HLD-EMFL) and Würzburg-Dresden Cluster of Excellence ct.qmat,
Helmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden, Germany  Institut für
Festkörper- und Materialphysik, TU-Dresden, 01062 Dresden, Germany  Shojiro
Kimura  Institute for Materials Research, Tohoku University, Katahira 2-1-1,
Sendai 980-8577, Japan  Tsukasa Katsuyoshi  Department of Advanced Materials
Science, The University of Tokyo, Kashiwa, Chiba 277-8561, Japan  Tsuyoshi
Kimura  Department of Applied Physics, The University of Tokyo, Tokyo
113-8656, Japan  Department of Advanced Materials Science, The University of
Tokyo, Kashiwa, Chiba 277-8561, Japan  Kenta Kimura<EMAIL_ADDRESS>Department of Advanced Materials Science, The University of Tokyo, Kashiwa,
Chiba 277-8561, Japan  Department of Materials Science, Osaka Metropolitan
University, Osaka 599-8531, Japan

###### Abstract

We studied the high-field phase diagram of a chiral-lattice antiferromagnet
Sr(TiO)Cu4(PO4)4 by means of ultrasound, dielectric, and magnetocaloric-effect
measurements. These experimental techniques reveal two new phase transitions
at high fields, which have not been resolved by previous magnetization
experiments. Specifically, the $c_{66}$ acoustic mode shows drastic changes
with hysteresis for magnetic fields applied along the $c$ axis, indicating a
strong magneto-elastic coupling. Combined with cluster mean-field theory, we
discuss the origin of these phase transitions. By considering the chiral-twist
effect of Cu4O12 cupola units, which is inherent to the chiral crystal
structure, the phase diagram is reasonably reproduced. The agreement between
experiment and theory suggests that this material is a unique quasi-two-
dimensional spin system with competing exchange interactions and chirality,
leading to a rich phase diagram.

##  I Introduction

Figure 1:  (a) Crystal structure of SrTCPO with magnetic interactions;
nearest-neighbor exchange ($J_{1}$), next-nearest-neighbor exchange ($J_{2}$),
inter-cupola exchanges ($J^{\prime}$ and $J^{\prime\prime}$), and DM vector
($D$). $\theta$ is the angle of the DM vector from the $c$ axis. The site
index $\ell$ (0–15) is denoted on each Cu2+. (b) In-plane interactions of
SrTCPO with the twist angle $\phi$. (c, d) Strained unit cells with (c)
$\epsilon_{xy}$ and (d) $\epsilon_{xx}$. The shorter bonds are shown by
thicker lines.

Chiral magnets are of great interest because of their exotic properties such
as helical magnetism, multiferroicity, and skyrmionic textures [1, 2, 3, 4].
Dzyaloshinskii-Moriya (DM) interaction, inherent to the chiral structure
without inversion symmetry, leads to canted spin textures, which are rarely
observed in usual magnets. Helical magnetism can also be realized in the
presence of geometrical frustration (or competing exchange interactions),
hindering trivial long-range order [5, 6]. In the presence of quantum
fluctuations, exotic ground states such as quantum spin liquids and magnon
Bose-Einstein condensates may emerge [7, 8]. A system with chirality and
geometrical frustration is an attractive platform for novel exotic states of
matter, although the appropriate material design is challenging.

A series of chiral-lattice crystals, $A$(TiO)Cu4(PO4)4 ($A$TCPO with $A$ = Ba,
Sr, Pb) with space group $P42_{1}2$ show fascinating properties originating
from the Cu4O12 square-cupola units [9, 10] [Fig. 1(a)]. The upward ($\alpha$)
and downward ($\beta$) square cupolas alternatively align in the $ab$ plane,
and the compounds can be regarded as quasi-two-dimensional systems. On each
Cu4O12 unit, the four $S=1/2$ spins of the Cu2+ ions form a sort of spin
tetramer and can host magnetic multipole moments (monopole, quadrupole, and
toroidal moments) [11, 12, 13]. Under external magnetic fields, these magnetic
multipoles order and/or disorder in a ferroic or antifferoic manner, leading
to successive magnetic phase transitions. Since these magnetic multipoles
simultaneously break space-inversion and time-reversal symmetries,
magnetoelectric (ME) coupling appears in $A$TCPO. Recent experimental and
theoretical studies revealed a relation between the magnetic structure and the
ME responses in $A$TCPO [11, 15, 12, 10, 14, 13, 16, 17, 18, 19]. Noteworthy,
the cluster mean-field (CMF) approach has clarified the evolution of the
magnetic multipoles in the Cu4O12 units as a function of magnetic field and
allowed to predict the ME responses. However, the experimental study on the
magnetic field–temperature ($B$–$T$) phase diagram of $A$TCPO up to the
saturation field has not yet been completed.

In this study, we present the experimental $B$–$T$ phase diagram of SrTCPO
using the combined results of ultrasound, dielectric, and magnetocaloric-
effect (MCE) measurements. SrTCPO orders antiferromagnetically at
$T_{\mathrm{N}}=6.3$ K, and an antiferroic ME response is simultaneously
observed in the dielectric susceptibility [11]. The high-field magnetization
results suggest that the magnetic saturation field $B_{\mathrm{s}}$ for the
crystallographic [001], [100], and [110] directions are
$B_{\mathrm{s}}^{[001]}=37.0$ T, $B_{\mathrm{s}}^{[100]}=41.0$ T, and
$B_{\mathrm{s}}^{[110]}=38.9$ T, respectively [12]. Before the saturation
fields, spin-flop-like transitions are observed at
$B_{\mathrm{c1}}^{[001]}=27.4$ T, $B_{\mathrm{c1}}^{[100]}=13.5$ T, and
$B_{\mathrm{c1}}^{[110]}=15.0$ T, respectively [12]. Our experimental
techniques allow us to detect two additional phase transitions which are not
observed in the magnetization. We discuss the origin of these new phase
transitions based on the CMF theory.

This paper is organized as follows. In Sec. II, we explain our experimental
and theoretical methods. In Sec. III, we present the experimental results of
the ultrasound, dielectric, and MCE measurements together with the
experimental $B$–$T$ phase diagram. In Sec. IV, we show the results of
theoretical calculations corresponding to the experiments. In Sec. V, we
discuss the phase diagram by comparing the experimental and theoretical
results. In Sec. VI, conclusive remarks are given.

##  II Methods

###  II.1 Experimental methods

We grew single crystals of SrTCPO by the flux method [9]. We performed powder
x-ray diffraction (XRD) on crushed single crystals and confirmed a single
phase. We determined the crystal orientation by Laue x-ray scattering. The
crystal used in this study has L-type chirality, determined by optical-
activity measurement at a wavelength of 450 nm. The size of the investigated
crystal was $4\times 4\times 2$ mm3.

We performed ultrasound-velocity measurements based on the transmission pulse-
echo technique with a phase-sensitive detection [21, 20, 22]. We attached two
LiNbO3 transducers (41∘-X cut for transverse and 36∘-Y cut for longitudinal
acoustic modes) to the polished surfaces of the single crystal. We analyzed
the phase of the echo signal and obtained the relative change in the sound
velocity. The ultrasound frequency was typically 20–180 MHz. For some
experimental geometries, the ultrasound frequency was increased up to 500 MHz
to investigate the nonreciprocal properties (magnetochiral effect) [23, 24,
25]. The elastic modulus $c_{ij}$ was calculated as $c_{ij}=\rho v_{ij}^{2}$,
using the density $\rho=4.109$ g/cm3 [9]. The density change below 200 K is
negligibly small [16]. The uncertainty of $c_{ij}$ is around $\pm$5 %. The
experimental geometries (propagation and polarization vectors, ${\bf k}$ and
${\bf u}$, respectively) for each acoustic mode are summarized in Table 1 with
the sound velocity and the calculated elastic modulus at 2 K. Here,
$c_{\mathrm{T}}=(c_{11}-c_{12})/2$.

Table 1:  Experimental geometries for each acoustic mode. ${\bf k}$ and ${\bf u}$ are the propagation and polarization vectors, respectively. The sound velocity $v_{ij}$ and elastic modulus $c_{ij}$ at 2 K are summarized. The irreducible representations (IR) for $D_{4}$ symmetry and the related strains are also shown. | ${\bf k}$ | ${\bf u}$ | $v_{ij}$ | $c_{ij}$ |  | IR | Strain
---|---|---|---|---|---|---|---
|  |  | (km/s) | (GPa) |  |  |
$c_{66}$ | [100] | [010] | 3.32 | 45 |  | $B_{2}$ | $\epsilon_{xy}$
$c_{11}$ | [100] | [100] | 6.47 | 172 |  | $A_{1}\oplus B_{1}$ |  $\epsilon_{xx}$, $\epsilon_{yy}$
$c_{\mathrm{T}}$ | [110] | [1$\overline{1}$0] | 4.20 | 78 |  | $B_{1}$ | $\epsilon_{xx}-\epsilon_{yy}$
$c_{44}$ | [100] | [001] | 2.41 | 24 |  | $E$ |  $\epsilon_{yz}$, $\epsilon_{zx}$
$c_{33}$ | [001] | [001] | 4.47 | 82 |  | $A_{1}$ | $\epsilon_{zz}$

We measured the dielectric constant $\varepsilon$ up to 25 T by using a
superconducting magnet and up to 50 T in a pulsed magnet. For these
measurements we utilized an LCR meter (Agilent E4980) and a capacitance bridge
for the static- and pulsed-field experiments, respectively [26]. The pulse
duration for the dielectric experiment was $\sim 35$ ms, which was shorter
than that of the ultrasound experiment $\sim 150$ ms. Because of the shorter
duration, the temperature of the crystal changed during the pulsed magnetic
fields. Therefore, we used only the static-field data for the phase diagram.

We obtained the adiabatic temperature change in the pulsed magnetic field
(magnetocaloric effect, MCE) using a RuO2 thermometer (900 $\Omega$,
$0.6\times 0.3\times 0.1\ \mathrm{mm}^{3}$) [27, 28, 29]. We glued the
thermometer on the polished surface of the single crystal ($4\times 4\times 2\
\mathrm{mm}^{3}$). The sample was placed in vacuum to ensure adiabatic
conditions. We measured the resistance of the thermometer using a standard ac
four-probe method and a numerical lock-in technique at a frequency of 50 kHz,
typical for pulsed-field experiments.

###  II.2 Theoretical methods

We consider an effective spin model for the $S=1/2$ spin degrees of freedom of
the Cu2+ cations, which was first introduced for BaTCPO [12] and later applied
to PbTCPO [10] and SrTCPO [13]. The Hamiltonian reads

$\displaystyle\mathcal{H}=\sum_{\langle i,j\rangle}\left[J_{1}{\bf
S}_{i}\cdot{\bf S}_{j}-{\bf D}_{ij}\cdot({\bf S}_{i}\times{\bf
S}_{j})\right]+J_{2}\sum_{\langle\langle i,j\rangle\rangle}{\bf
S}_{i}\cdot{\bf S}_{j}$ $\displaystyle+J^{\prime}\sum_{(i,j)}{\bf
S}_{i}\cdot{\bf S}_{j}+J^{\prime\prime}\sum_{((i,j))}{\bf S}_{i}\cdot{\bf
S}_{j}-g\mu_{\rm B}\sum_{i}{\bf B}\cdot{\bf S}_{i},~{}~{}$ (1)

where ${\bf S}_{i}=(S^{x}_{i},S^{y}_{i},S^{z}_{i})$ represents the $S=1/2$
spin at site $i$, $J_{1}$, $J_{2}$, $J^{\prime}$, and $J^{\prime\prime}$
represent four dominant antiferromagnetic exchange interactions [Fig. 1(a)],
${\bf D}_{ij}$ is the Dzyaloshinskii-Moriya vector, and the last term
represents the Zeeman coupling with an isotropic $g$-factor $g$ and the Bohr
magneton $\mu_{\rm B}$. The sums for $\langle i,j\rangle$, $\langle\langle
i,j\rangle\rangle$, $(i,j)$, and $((i,j))$ run over the $J_{1}$, $J_{2}$,
$J^{\prime}$, and $J^{\prime\prime}$ bonds, respectively. We use the model
parameters for SrTCPO found in Ref. [13] as $J_{1}=0.6$, $J_{2}=1/6$,
$J^{\prime}=1/2$, $J^{\prime\prime}=1/100$, $D=0.7$, and $\theta=90^{\circ}$.
While the effect of the chiral twist of the square cupolas has been neglected
in Ref. [13], we take into account a nonzero chiral twist to explain the
magnetic transitions found in this study in the high-field regime for ${\bf
B}\parallel[100]$. In the present calculations, we take $\phi=1^{\circ}$ [30],
which leads to a $\pm 1^{\circ}$ tilt of ${\bf D}_{ij}$ from the $\langle
110\rangle$ directions [Figs. 1(a) and 1(b)].

As in the previous studies [12, 10, 13], we performed calculations based on
the cluster mean-field (CMF) theory, which is suitable for cluster-based
magnetic materials. In the CMF method, the intracupola interactions and the
Zeeman coupling are dealt with by exact diagonalization, and, therefore,
quantum fluctuations in each cupola are fully taken into account; the other
weaker intercupola interactions ($J^{\prime}$ and $J^{\prime\prime}$ terms)
are taken into account by means of conventional mean-field approximation.
Following the previous studies [12, 13], we consider four square cupolas
allocated as shown in Fig. 1(a) in the CMF method, namely we consider 16
sublattices.

To characterize the magnetically ordered phases, we consider the same order
parameter as defined in the previous studies [12, 10, 13]:

$\displaystyle{\bf m}_{\rm AF}=\frac{1}{N_{\rm
spin}}\sum_{\ell}(-1)^{\ell}p_{\ell}\langle{\bf S}_{\ell}\rangle,$ (2)

where $\langle{\bf S}_{\ell}\rangle$ is the expectation value of the spin
operator ${\bf S}_{\ell}$, $p_{\ell}=+1(-1)$ for the upper (lower) layer in
Fig. 1(a) and $N_{\rm spin}$ is the number of spins. We also compute the
entropy $S$ that is obtained from the temperature integral of the specific
heat as

$\displaystyle S(T)=\int_{T_{0}}^{T}\frac{C(\tilde{T})}{\tilde{T}}d\tilde{T},$
(3)

with a sufficiently low $T_{0}=0.01$, where $C(T)$ is estimated by a
derivative of the cubic spline interpolation of the internal energy as
$C(T)=(1/N_{\rm spin})d\langle\mathcal{H}\rangle/dT$.

The elastic constants $c_{66}$ and $c_{11}$ are given by the second derivative
of the free energy in terms of the strain as
$c=\partial^{2}F/\partial\epsilon^{2}$. Since the spin interactions linearly
change with the strain in general, we consider the second derivatives of the
free energy with respect to the interactions instead. We take into account a
deformation of $J_{1\pm}$ bonds [Fig. 1(c)] as $J_{1}\to(1\pm r)J_{1}$, ${\bf
D}_{ij}\to(1\mp r){\bf D}_{ij}$ for the strain of $\epsilon_{xy}$ and a
deformation of $J^{\prime}_{-}$ bonds [Fig. 1(d)] as $J^{\prime}_{-}\to
J^{\prime}-j^{\prime}$ for the strain of $\epsilon_{xx}$. For simplicity, it
is assumed that $J_{1}$ and ${\bf D}_{ij}$ have the same rate of change for
$\epsilon_{xy}$, and the effects on the other interactions by the strain are
also neglected. Then, we compute the second derivatives of the free energy as

$\displaystyle\Delta_{r}^{(2)}=$ $\displaystyle\frac{1}{N_{\rm
spin}}\frac{\partial^{2}F}{\partial r^{2}},$ (4)
$\displaystyle\Delta_{j^{\prime}}^{(2)}=$ $\displaystyle\frac{1}{N_{\rm
spin}}\frac{\partial^{2}F}{\partial{j^{\prime}}^{2}}.$ (5)

One might think that the deformation of $J^{\prime}$ bonds for the strain of
$\epsilon_{xy}$ is also important as indicated in Fig. 1(c). However, the
effect is canceled out in the CMF calculation and, therefore, is expected to
be not significant. In practical calculations, the first derivatives,

$\displaystyle\Delta_{r}^{(1)}=$ $\displaystyle\frac{1}{N_{\rm
spin}}\frac{\partial F}{\partial r},$ (6)
$\displaystyle\Delta_{j^{\prime}}^{(1)}=$ $\displaystyle\frac{1}{N_{\rm
spin}}\frac{\partial F}{\partial j^{\prime}},$ (7)

are obtained by linear combinations of $\langle{\bf S}_{i}\cdot{\bf
S}_{j}\rangle$ or $\langle{\bf S}_{i}\rangle\cdot\langle{\bf S}_{j}\rangle$,
and do not require numerical derivatives; the second derivatives are computed
by numerical derivatives of $\Delta_{r}^{(1)}$ and $\Delta_{j^{\prime}}^{(1)}$
as

$\displaystyle\Delta_{r}^{(2)}=$
$\displaystyle\frac{1}{2\delta}\left[\left.\Delta_{r}^{(1)}\right|_{r=\delta}-\left.\Delta_{r}^{(1)}\right|_{r=-\delta}\right],$
(8) $\displaystyle\Delta_{j^{\prime}}^{(2)}=$
$\displaystyle\frac{1}{2\delta}\left[\left.\Delta_{j^{\prime}}^{(1)}\right|_{j^{\prime}=\delta}-\left.\Delta_{j^{\prime}}^{(1)}\right|_{j^{\prime}=-\delta}\right],$
(9)

with a sufficiently small $\delta=10^{-4}$.

##  III Experimental results

###  III.1 Temperature dependence of the acoustic properties

Figure 2:  (a–e) Temperature dependence of the elastic constants. Solid lines
show the results at zero field. The gray dashed line shows the fitted
background due to the anharmonic phonon contribution based on Ref. [31]. (f–j)
Relative change of the elastic constants $\Delta c/c$ near $T_{\mathrm{N}}$.
(f) Results for zero and applied fields
($\bf{B}\parallel\bf{k}\parallel[100]$) as labeled for each curve. The inset
shows the enlarged result at zero field.

Figures 2(a)–2(e) show the temperature dependence of each elastic constant at
zero field. All acoustic modes exhibit an anomaly at $T_{\mathrm{N}}=6.3$ K,
evidencing the long-range magnetic ordering. We comment that the acoustic
attenuation does not show any anomaly at $T_{\mathrm{N}}$ for all acoustic
modes within our experimental resolution. The elastic softening towards
$T_{\mathrm{N}}$ reflects developing short-range correlations via magneto-
elastic coupling. The $c_{66}$ mode shows the largest softening of 1.9 %,
indicating the strong spin-lattice coupling with the $ab$-plane shear strain
$\epsilon_{xy}$. The softening of the other acoustic modes are one or two
orders of magnitude smaller than that of the $c_{66}$ mode. The background
curve of the $c_{66}$ mode due to the phonon anharmonicity is estimated by the
dashed line in Fig. 2(a) [20, 31].

Generally, for two-dimensional spin systems, the in-plane acoustic modes
($c_{66}$, $c_{11}$, $c_{\mathrm{T}}$) show larger spin-lattice coupling than
the acoustic modes with out-of-plane strains ($c_{44}$, $c_{33}$) [20, 32,
33]. This is because the in-plane strains ($\epsilon_{xy}$, $\epsilon_{xx}$,
$\epsilon_{xx}-\epsilon_{yy}$) may linearly modulate the Cu-Cu distance, while
the out-of-plane strains ($\epsilon_{yz}$, $\epsilon_{zz}$) only modulate with
higher-order coupling. Thus, the weaker elastic anomalies for the $c_{44}$ and
$c_{33}$ modes are understandable. The differences between the in-plane modes
($c_{66}$, $c_{11}$, $c_{\mathrm{T}}$) reflect how the corresponding strains
are coupled to the spin degrees of freedom.

Next, we focus on the anomalies of the elastic constants near $T_{\mathrm{N}}$
in Figs. 2(f)–2(j). For the $c_{11}$, $c_{\mathrm{T}}$, $c_{44}$, and $c_{33}$
modes, the elastic constants show a single anomaly at the ordering
temperature. In contrast, the $c_{66}$ mode at zero field [enlarged in the
inset of Fig. 2(f)] shows a sharp minimum at $T_{\mathrm{N}}$ and a broad
minimum at around 5 K. The hardening of $c_{66}$ below $T_{\mathrm{N}}$ is
smaller than for the other modes. When the magnetic field is applied with
$\textbf{B}\parallel\textbf{k}\parallel[100]$, the broad minimum gradually
disappears towards 12 T. At higher magnetic fields, $c_{66}(T)$ shows
qualitatively different behavior due to a field-induced phase transition,
which is discussed later together with the phase diagram.

Figure 3:  Temperature dependence of the specific heat divided by temperature
$C/T$ at zero field. A broad hump is indicated by the arrow.

Here, we discuss the origin of the broad minimum observed for $c_{66}$ in zero
field [Fig. 2(f)]. Below $T_{\mathrm{N}}$ in zero field, no phase transition
was reported from magnetic-susceptibility, heat-capacity, and dielectric
measurements [16, 13]. However, the spin-lattice relaxation rate of 31P-NMR
exhibits a small discontinuity at around 5 K [16]. Figure 3 shows the $T$
dependence of the specific heat divided by $T$ at zero field, which is
obtained by combining the data taken from Refs. [14, 34] and new data taken
with a narrower $T$ interval around $T_{\mathrm{N}}$. There is a clear hump
around 5.5 K. As discussed later in connection with the MCE results,
irreversible heating is observed only below 5 K. Such irreversible heating
indicates slow dynamics of this system.

A similar temperature dependence of the elastic constants below
$T_{\mathrm{N}}$ is also observed for the frustrated magnet GeCo2O4 [35, 36]
and the charge-ordering system $\alpha^{\prime}$-NaV2O5 [37]. In these
materials, the characteristic hardening below $T_{\mathrm{N}}$ was attributed
to domain-wall stress and to spin-singlet to triplet excitations.

Here, we discuss the effect of possible magnetic domains in SrTCPO. The
magnetic order at zero field is of antiferro-quadrupolar type
($q_{x^{2}-y^{2}}$), where $q_{x^{2}-y^{2}}$ magnetic quadrupoles align in a
ferroic manner in the two-dimensional magnetic layer, but in an antiferroic
manner along the $c$ axis. For simplicity, we consider only the single
magnetic layer in the $ab$ plane. The upward ($\alpha$) and downward ($\beta$)
cupola alternatively take “udud” and “dudu” spin configurations, where “u” and
“d” denote up and down spin configurations, respectively. Note that the
opposite configuration (“udud” for $\beta$ and “dudu” for $\alpha$) has the
same energy. Thus, two domains with opposite magnetic quadrupoles can coexist
and form magnetic domains. At low temperatures, the domain motions are
restricted due to pinning and possibly affect the elastic properties. The
anomaly at around 5 K could be related to the freezing of the antiferro-
quadrupolar domains.

Figure 4:  (a-c) Relative changes of the elastic constants $\Delta c/c$ at 1.8
K as a function of magnetic field for the field directions (a) ${\bf
B}\parallel[001]$, (b) ${\bf B}\parallel[100]$, and (c) ${\bf
B}\parallel[110]$. Enlarged results of (d) $\Delta c/c$ and (e) the relative
change of the acoustic attenuation of the $c_{66}$ mode ($f=122$ MHz) with
${\bf B}\parallel[001]$. The dashed lines show results obtained using lower
maximum-field pulses. The results are shifted for clarity.

###  III.2 Magnetic-field dependence of the acoustic properties

Figure 4 summarizes the relative changes of the elastic constants $\Delta c/c$
at 1.8 K up to the saturation magnetic fields along the [001], [100], and
[110] axes. For each direction, one or two field-induced phase transitions are
observed as denoted by the gray dashed lines and the gray band
($B_{\mathrm{c2}}^{[100]}$). We also performed measurements at 4.2 K in liquid
4He, however, the results show extrinsic hysteresis due to a strong MCE (not
shown). Only below 2 K, where the sample was immersed in superfluid 4He, an
isothermal condition was realized. The hysteresis observed in Fig. 4 is
considered to be intrinsic.

First, the results for ${\bf B}\parallel[001]$ [Fig. 4(a)] are discussed. We
observed clear discontinuities of $\Delta c/c$ at
$B_{\mathrm{c}1}^{[001]}=27.6$ T, $B_{\mathrm{c}2}^{[001]}=29.5$ T, and
$B_{\mathrm{s}}^{[001]}=36.5$ T. In magnetization measurements, the field-
induced transition at $B_{\mathrm{c}1}^{[001]}$ and saturation of the
magnetization at $B_{\mathrm{s}}^{[001]}$ were observed [13]. In $c_{66}$,
$c_{11}$, and $c_{44}$, we clearly detect another transition at
$B_{\mathrm{c}2}^{[001]}$ with remarkable hysteresis. In particular, the
$c_{66}$ mode ($\epsilon_{xy}$) shows drastic changes with a complex
hysteresis loop, indicating strong in-plane spin-strain correlations. Figures
4(d) and 4(e) show the field dependence of $\Delta c_{66}/c_{66}$ and the
relative change of the acoustic attenuation $\Delta\alpha$ near
$B_{\mathrm{c}2}^{[001]}$. Here, results for different peak fields
$B_{\mathrm{max}}$ are presented for 4.2 and 1.8 K. The complex hysteresis
loops are reproduced for these conditions and observed in $\Delta\alpha$ as
well. Note, that the acoustic attenuation becomes minimal at the fields
between $B_{\mathrm{c}1}^{[001]}$ and $B_{\mathrm{c}2}^{[001]}$, indicating
that the state in this region is a thermodynamically stable phase, not just a
transient state.

Second, we discuss the results for ${\bf B}\parallel[100]$ [Fig. 4(b)]. A
field-induced transition at $B_{\mathrm{c}1}^{[100]}=13.5$ T and saturation of
the magnetization at $B_{\mathrm{s}}^{[100]}=40.0$ T are clearly observed. In
addition to these anomalies, a slight change of elasticity is observed at
$B_{\mathrm{c}2}^{[100]}\sim 24$ T with hysteresis as indicated by the gray
band. The magnetization shows only a tiny anomaly at $B_{\mathrm{c}2}^{[100]}$
[13]. At $B_{\mathrm{c}2}$, none of the acoustic modes shows a drastic change,
indicating that the symmetry of the magnetic state only slightly changes at
this critical field. Therefore, this phase transition is related to a slight
rearrangement of spins in the cupola.

Third, we discuss the results for ${\bf B}\parallel[110]$ [Fig. 4(c)]. Again,
a field-induced transition at $B_{\mathrm{c}1}^{[110]}=15.0$ T and saturation
of the magnetization at $B_{\mathrm{s}}^{[110]}=38.2$ T are observed. For this
field direction, no additional feature is detected compared to the
magnetization results [13].

Last, we note on the possible nonreciprocal acoustic properties in this system
[23, 24, 25]. For the Faraday geometries (${\bf B}\parallel{\bf k}$), we
performed ultrasound experiments for $\pm{\bf B}$ and compared the results. Up
to an ultrasound frequency of 500 MHz, we did not observe reproducible
difference with the experimental resolution of $\Delta c/c\sim 10^{-5}$.

###  III.3 Field dependence of dielectric constants

Figure 5:  Dielectric constant along the [100] direction
$\varepsilon_{[100]}$. The magnetic fields are applied along the (a) [001] and
(b) [100] directions. The inset shows the results obtained under static fields
along the [010] direction at selected temperatures.

Figures 5(a) and 5(b) show the magnetic-field dependences of the dielectric
constant for electric fields applied along the [100] axis
$\varepsilon_{[100]}$. For ${\bf B}\parallel[001]$ [Fig. 5(a)],
$\varepsilon_{[100]}$ shows two broad peak structures at 18 and 28 T. The
hysteresis for field-up and -down sweeps indicates that the temperature
changes due to a MCE. The ME coupling in this system is dominated by the
exchange-striction mechanism, where the induced polarization is proportional
to $\mathbf{S}_{i}\cdot\mathbf{S}_{j}$. The increase of $\varepsilon_{[100]}$
suggests that the induced polarizations are not canceled between the cupolas
in a single magnetic layer. The second peak appears with a drastic change at
$B_{\mathrm{c}1}^{[001]}=28$ T, which is consistent with the ultrasound
results [Fig. 4(a)]. At $B_{\mathrm{c}2}^{[001]}=29.5$ T, no anomaly was
resolved.

For ${\bf B}\parallel[100]$ [Fig. 5(b)], $\varepsilon_{[100]}$ shows a sharp
peak at $B_{\mathrm{c}1}^{[100]}$ and a broad hump at 39 T which corresponds
to the saturation field. The inset figure shows the results for ${\bf
B}\parallel[010]$ at selected temperatures. In addition to the sharp anomaly
at $B_{\mathrm{c}1}^{[100]}=13.8$ T, step-like anomalies are observed at
$B_{\mathrm{c}2}^{[100]}=20$–24 T depending on the temperature. The relatively
weak anomaly at $B_{\mathrm{c}2}^{[100]}$ is consistent with the ultrasound
and magnetization results.

###  III.4 Magnetocaloric effect

Figure 6:  MCE curves obtained for the field directions (a, d) ${\bf
B}\parallel[001]$, (b, e) ${\bf B}\parallel[100]$, and (c, f) ${\bf
B}\parallel[110]$. (a-c) Dashed (solid) lines represent the data for field up
(down) sweeps. (d-f) Down-sweep results are shown with the anomalies detected
in the elastic and dielectric constants. Symbols are summarized in Fig. 6(f).
The dotted lines show guides for the phase boundary and crossover line (see
text for details).

Figures 6(a)–6(c) show the MCE data for the field directions along [001],
[100], and [110] axes, respectively. The up-sweep (down-sweep) data are shown
by dotted (solid) curves. Here, the temperature change during the adiabatic
magnetization is measured from different initial temperatures $T_{0}$. For
$T_{0}>5$ K, the MCE curves are reversible, indicating the adiabatic condition
and good thermal contact between the sample and the thermometer. In contrast,
for $T_{0}<5$ K, the MCE curves show irreversible behavior with hysteresis.
The irreversible temperature change indicates dissipation such as domain-wall
motions or dynamical effects [38, 39]. For an overview, only the down-sweep
data are presented in Figs. 6(d)–6(f) with the transition fields determined by
the ultrasound and dielectric measurements.

First, the results for ${\bf B}\parallel[001]$ [Figs. 6(a) and 6(d)] are
discussed. When increasing field, the temperature decreases up to
$B_{\mathrm{c}1}^{[001]}$. The temperature decrease indicates an increase of
magnetic entropy towards the field-induced transition. At
$B<B_{\mathrm{c}1}^{[001]}$, irreversibilities are is clearly observed. Above
$B_{\mathrm{c}1}^{[001]}$, the MCE curves are reversible and show kinks around
the saturation field $B_{\mathrm{s}}^{[001]}$. No anomaly is observed at
$B_{\mathrm{c}2}^{[001]}$, indicating that the entropy does not change at this
phase transition. Such a transition without entropy change is observed as well
in the frustrated magnet CdCr2O4, where also a clear anomaly is detected in
the elastic properties [42]. The low-temperature MCE curves merge at $T\approx
2$ K at $28<B<36$ T, which suggests a discontinuous change of entropy like a
first-order phase boundary in this region. This might indicate that another
phase exists in the low-temperature region, where one needs to perform
experiments in static magnetic fields, for instance, with a 3He cryostat.

Second, we discuss the results for ${\bf B}\parallel[100]$ [Figs. 6(b) and
6(e)]. For this field direction, three anomalies are observed at
$B_{\mathrm{c}1}^{[100]}$, $B_{\mathrm{c}2}^{[100]}$, and
$B_{\mathrm{s}}^{[100]}$, consistent with the ultrasound results. The MCE
curves are irreversible below $B_{\mathrm{s}}^{[100]}$. Noteworthy, the
anomaly at $B_{\mathrm{c}2}^{[100]}$ evidences a first-order phase transition.
At first-order transitions, a reversible temperature change $\Delta
T_{\mathrm{rev}}$ and irreversible heating $\Delta T_{\mathrm{irr}}>0$ are
involved [38, 39, 40, 41]. At around $T_{0}=5$ K, the phase transition at
$B_{\mathrm{c}2}^{[100]}$ is detected only for the down sweep. At the lowest
temperature of $T_{0}=2$ K [black curve in Fig. 6(b)], in contrast, a
temperature increase is observed both for the up and down sweep. This
indicates that $\Delta T_{\mathrm{irr}}$ is larger than $\Delta
T_{\mathrm{rev}}$ in this temperature range. Usually, $\Delta
T_{\mathrm{irr}}$ and the hysteresis become smaller at higher temperatures
because the energy barrier between two phases can be overcome by thermal
fluctuations [38]. At around $T_{0}=5$ K (cyan and green curves), the
contributions of $\Delta T_{\mathrm{rev}}$ and $\Delta T_{\mathrm{irr}}$ seem
to be cancelled out for the up sweep ($\Delta T_{\mathrm{irr}}>0$ and $\Delta
T_{\mathrm{rev}}<0$), while summed up for the down sweep ($\Delta
T_{\mathrm{irr}}>0$ and $\Delta T_{\mathrm{rev}}>0$).

Third, we discuss the results for ${\bf B}\parallel[110]$ [Figs. 6(c) and
6(f)]. At least, two anomalies are observed at $B_{\mathrm{c}1}^{[110]}$ and
$B_{\mathrm{s}}^{[110]}$. Hysteresis appears below $B_{\mathrm{s}}^{[110]}$
similar to ${\bf B}\parallel[100]$. At around 35 T, below the saturation, tiny
kinks are observed. These anomalies are not seen in the ultrasound
measurements. Apparently, these anomalies depend on the peak field (i.e.,
different $dB/dt$ rate) and are not well reproduced. They might be related to
another phase transition, however, we leave it for a future work.

##  IV Theoretical phase diagram

Our experimental results show two additional features at
$B_{\mathrm{c}2}^{[001]}$ and $B_{\mathrm{c}2}^{[100]}$, which are not
predicted in previous theoretical calculations [13]. One possible origin is a
chiral twist of the cupola [Fig. 1(b)], which tilts the DM vectors as well. In
this section, we present a theoretical phase diagram for a twist angle of
$\phi=1^{\circ}$ and discuss how the phase diagram changes from the case
$\phi=0^{\circ}$ [13]. For representative spin orientations, see Appendix A.

Figures 7(a)–7(c) show the results for $\phi=1^{\circ}$ for ${\bf
B}\parallel[001]$, ${\bf B}\parallel[100]$, and ${\bf B}\parallel[110]$. We
find that all the phases seen for $\phi=0^{\circ}$ are also stable for
$\phi=1^{\circ}$, namely, I, II, III, Y, Z, Y’, and Z’ [12, 13]. These phases
are identified by ${\bf m}_{\rm AF}$. In each phase, ${\bf m}_{\rm AF}$
behaves qualitatively the same as for the case $\phi=0^{\circ}$, except for
the phase III where the direction of ${\bf m}_{\rm AF}$ deviates slightly from
the $[100]$ direction due to the nonzero $\phi$. The obtained phase diagrams
are very similar for the field directions ${\bf B}\parallel[001]$ and ${\bf
B}\parallel[110]$, while the high-field Y phase is largely affected for ${\bf
B}\parallel[100]$: the phase boundary on the high-field side of the Y phase
shifts significantly to low fields, leaving a clear crossover (cyan dashed
line) at the original phase boundary.

Figure 7:  Theoretical magnetic-field-temperature phase diagram for (a) ${\bf
B}\parallel[001]$, (b) ${\bf B}\parallel[100]$, and (c) ${\bf
B}\parallel[110]$. Here, the results for the effective Hamiltonian [Eq. (1)]
with the parameters $J_{1}=0.6$, $J_{2}=1/6$, $J^{\prime}=1/2$,
$J^{\prime\prime}=1/100$, $D=0.7$, $\theta=90^{\circ}$, and $\phi=1^{\circ}$
are shown. The cyan dashed line in (b) shows the crossover field extracted
from an inflection point of $dm/dB$. For the representative spin structures of
the proposed phases, see Appendix A.

##  V Discussions

Here, we compare the theoretical and experimental results. Based on the
calculated entropy and elastic constants, we discuss the MCE and acoustic
results in the field-induced phases.

###  V.1 Entropy landscape

Figure 8:  Contour plots of the theoretically calculated magnetic entropy for
(a) ${\bf B}\parallel[001]$, (b) ${\bf B}\parallel[100]$, and (c) ${\bf
B}\parallel[110]$. The used parameters are the same as in Fig. 7. The white
solid lines and the cyan dashed line show the phase boundaries and crossover
extracted from the phase diagram (Fig. 7), respectively.

Figures 8(a)–8(c) show the results of calculations of the entropy landscape in
the phase diagram. The entropy at each field-temperature point is calculated
from the temperature dependence of the internal energy as explained in Sec.
II.2. Our MCE results [Figs. 6(a)–6(c)], which follow the isentropic curves
[black lines in Figs. 8(a)–8(c)], agree quite well with the calculations. For
${\bf B}\parallel[001]$, the entropy decreases toward the phases II and III,
and increases above $B_{\mathrm{s}}$. The phase III-II boundary is different
between the experiment and theory, where the phase III is only stable at
finite temperatures. This discrepancy is discussed in the next subsection. For
${\bf B}\parallel[100]$, the entropy decreases toward $B_{\mathrm{s}}$ with
slight discontinuous anomalies at $B_{\mathrm{c1}}$ and $B_{\mathrm{c2}}$. For
${\bf B}\parallel[110]$, the entropy decreases toward $B_{\mathrm{s}}$ with a
discontinuous anomaly at $B_{\mathrm{c1}}$. The discontinuous entropy changes
indicate first-order phase boundaries, which are also consistent with the
experimental results (Figs. 4–6).

###  V.2 Elastic anomalies

Figure 9:  Second derivatives of the magnetic energy with respect to strain as
a function of the magnetic field along (a) ${\bf B}\parallel[001]$, (b) ${\bf
B}\parallel[100]$, and (c) ${\bf B}\parallel[110]$. The derivatives are taken
at $T=0$ for the strains $\epsilon_{xy}$ (upper panel) and $\epsilon_{xx}$
(lower panel). For details, see the main text.  Figure 10:  Second derivatives
of the magnetic energy near the phase III-II boundary ($T=0.23$) for ${\bf
B}\parallel[001]$. The derivatives are taken for the strains $\epsilon_{xy}$
(upper panel) and $\epsilon_{xx}$ (lower panel).

Here, we discuss the elastic constants of the field-induced phases based on
the CMF theory. Figures 9(a)–9(c) present the calculated second derivatives of
the magnetic free energy with respect to strain, $\Delta_{r}^{(2)}$ and
$\Delta_{j^{\prime}}^{(2)}$, for three magnetic field directions. The upper
and lower panels show the results for the $\epsilon_{xy}$ ($c_{66}$ mode) and
$\epsilon_{xx}$ ($c_{11}$ mode) strains, respectively. We note that the
derivatives for different strains (acoustic modes) clearly show different
responses, reflecting the symmetry of the magnetic structures. Because of this
acoustic-mode dependence, ultrasound results provide a powerful basis to
discuss the magnetic phases.

Compared with the experimental ultrasound results [Figs. 4(a)–4(c)], the
calculated results are in quantitative agreement for ${\bf B}\parallel[100]$
and ${\bf B}\parallel[110]$. The characteristic curvature changes at
$B_{\mathrm{c}1}^{[100]}$, $B_{\mathrm{s}}^{[100]}$,
$B_{\mathrm{c}1}^{[110]}$, and $B_{\mathrm{s}}^{[110]}$ are well captured by
the calculation. The small anomaly at $B_{\mathrm{c}2}^{[100]}$ is also
reproduced in the calculation. The relatively broad anomaly at
$B_{\mathrm{s}}^{[100]}$, which is predicted to be a crossover, is also
reproduced [Figs. 4(b) and 9(b)]. These agreements support the magnetic
structures predicted by our theoretical model (Z, Y, Z’, and Y’ phases).

For the case ${\bf B}\parallel[001]$, we need a careful consideration. The
calculation predicts that the phase III only appears at finite temperatures.
Thus, at the lowest temperature, only two phase transitions are predicted,
while our ultrasound experiments detect three phase transitions
($B_{\mathrm{c}1}^{[001]}$, $B_{\mathrm{c}2}^{[001]}$,
$B_{\mathrm{s}}^{[001]}$) below 4 K. Our ultrasound results show a
characteristic anomaly at the field range from $B_{\mathrm{c}1}^{[001]}$ to
$B_{\mathrm{c}2}^{[001]}$, where only the $c_{66}$ mode exhibits a drastic
response, while the $c_{11}$ mode only shows a small anomaly. Figures 9(a) and
10 display the derivatives of the free energy across the phases I-II, and
I-III-II, respectively. Only across the phase III, the drastic response in the
$c_{66}$ mode is reproduced. For the case of the phase II, the $c_{11}$ mode
shows even larger response near the saturation field. Therefore, the results
suggest that the phase transitions at $B_{\mathrm{c}1}^{[001]}$ and
$B_{\mathrm{c}2}^{[001]}$ correspond to the I-III and III-II transitions,
respectively. One discrepancy, the hardening of $c_{66}$ (experiment) and the
decrease of $\Delta_{r}^{(2)}$ (theory) in the phase III (Fig. 10) might be
due to the crystal deformation in this phase. The ultrasound results [Fig.
4(a)] show the drastic change of the elastic constants in the phase III with
large hysteresis, indicating that the crystal structure changes in this phase.
When the magnetic structure changes, the crystal structure can also change due
to magneto-elastic coupling, which modifies the exchange coupling to stabilize
the magnetic structure at the expense of the elastic energy [43, 44]. Thus,
the exchange parameters used in the CMF calculations might change under
magnetic field. Another partial discrepancy, the theoretical prediction of the
phase III stabilized only at finite temperatures, might also be due to the
modified exchange parameters under magnetic field. The stability of the phase
III is sensitive to the angle of the DM vector $\theta$ [12], namely the O2-
anion position, which can be modulated via the magneto-elastic coupling. By
including this coupling term in the model Hamiltonian, the agreement between
the experiment and theory might further improve.

##  VI Conclusions

We present high-field results on SrTCPO using ultrasound, dielectric, and
magnetocaloric-effect measurements. Compared to previous studies, we find new
phase boundaries at $B_{\mathrm{c}2}^{[001]}$ and $B_{\mathrm{c}2}^{[100]}$.
Both phase transitions have not been detected by magnetization measurement.
Based on CMF calculations, we discuss the origin of these phases. By
considering a chiral twist effect of the cupolas, the $B$–$T$ phase diagram is
reasonably well reproduced for three magnetic field directions. The
assignments of the field-induced phases are carefully done by comparing the
entropy and elastic response obtained by experiments and calculations. We
propose that the phase between
$B_{\mathrm{c}1}^{[001]}$–$B_{\mathrm{c}2}^{[001]}$ is the phase III predicted
by the CMF theory based on the drastic response of the $c_{66}$ mode. The
reasonable agreement between the experiment and theory indicates that the CMF
theory well captures the nature of this quantum spin system, where the
competing exchange interactions and the chirality play an important role in
the emergence of the rich phase diagram.

We acknowledge the support of the HLD at HZDR, member of the European Magnetic
Field Laboratory (EMFL), and the DFG through the Collaborative Research Center
SFB 1143 (Project No. 247310070) and the Würzburg-Dresden Cluster of
Excellence ct.qmat (EXC 2147, Project No. 39085490). This work was partly
supported by the JSPS KAKENHI, Grants-In-Aid for Scientific Research (Nos.
JP19K23421, JP20K14403, JP22H00104, JP22K03509) and JSPS Bi-lateral Joint
Research Projects (JPJSBP120193507) This work was partly performed at the High
Field Laboratory for Superconducting Materials, Institute for Materials
Research, Tohoku University (Project No. 18H0014). T. N. was supported by the
JSPS through a Grant-in-Aid for JSPS Fellows. K. K. was supported by the MEXT
Leading Initiative for Excellent Young Researchers (LEADER).

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## Appendix A: Spin configurations

Figure 11 presents the representative spin configurations obtained by the CMF
theory including the chiral twist effect. The field and temperature parameters
are given in the caption.

Figure 11:  Spin configurations with the following parameters: (a) $B=0.0$,
$T=0.00$, (b) ${\bf B}||[001]$ $B=1.0$, $T=0.00$, (c) ${\bf B}||[001]$
$B=1.8$, $T=0.00$, (d) ${\bf B}||[001]$ $B=1.3$, $T=0.23$, (e) ${\bf
B}||[100]$ $B=0.5$, $T=0.00$, (f) ${\bf B}||[100]$ $B=1.5$, $T=0.00$, (g)
${\bf B}||[100]$ $B=2.2$, $T=0.00$, (h) ${\bf B}||[110]$ $B=0.5$, $T=0.00$,
and (i) ${\bf B}||[110]$ $B=1.5$, $T=0.00$.