# Effects of uniaxial pressure on the spin ice Ho2Ti2O7

R. Edberg  Physics Department, KTH Royal Institute of Technology, Sweden  I.
M. B. Bakke  Centre for Materials Science and Nanotechnology, Department of
Chemistry, University of Oslo, Norway  H. Kondo  Faculty of Engineering,
Kyushu Institute of Technology, Kitakyushu, Fukuoka 804-8550, Japan  L. Ørduk
Sandberg  Niels Bohr Institute, University of Copenhagen, Denmark  M. L.
Haubro  Niels Bohr Institute, University of Copenhagen, Denmark  M. Guthrie
European Spallation Source ERIC, 22363 Lund, Sweden  University of Edinburgh,
School of Physics and Astronomy and Centre for Science at Extreme Conditions
A. T. Holmes  European Spallation Source ERIC, 22363 Lund, Sweden  J. Engqvist
Division of Solid Mechanics, Lund University, P.O. Box 118, SE-221 00 Lund,
Sweden  A. Wildes  Institut Laue-Langevin, 38042 Grenoble, France  K.
Matsuhira  Faculty of Engineering, Kyushu Institute of Technology, Kitakyushu,
Fukuoka 804-8550, Japan  K. Lefmann  Niels Bohr Institute, University of
Copenhagen, Denmark  P. P. Deen  Niels Bohr Institute, University of
Copenhagen, Denmark  European Spallation Source ERIC, 22363 Lund, Sweden  M.
Mito  Faculty of Engineering, Kyushu Institute of Technology, Kitakyushu,
Fukuoka 804-8550, Japan  P. Henelius  Physics Department, KTH Royal Institute
of Technology, Sweden  Faculty of Science and Engineering, Åbo Akademi
University, Åbo, Finland

###### Abstract

The spin ice materials Ho2Ti2O7 and Dy2Ti2O7 are experimental and theoretical
exemplars of highly frustrated magnetic materials. However, the effects of an
applied uniaxial pressure are not well studied, and here we report
magnetization measurements of Ho2Ti2O7 under uniaxial pressure applied in the
$[001]$, $[111]$ and $[110]$ crystalline directions. The basic features are
captured by an extension of the dipolar spin ice model. We find a good match
between our model and measurements with pressures applied along two of the
three directions, and extend the framework to discuss the influence of crystal
misalignment for the third direction. The parameters determined from the
magnetization measurements reproduce neutron scattering measurements we
perform under uniaxial pressure applied along the $[110]$ crystalline
direction. In the detailed analysis we include the recently verified
susceptibility dependence of the demagnetizing factor. Our work demonstrates
the application of a moderate applied pressure to modify the magnetic
interaction parameters. The knowledge can be used to predict critical
pressures needed to induce new phases and transitions in frustrated materials,
and in the case of Ho2Ti2O7 we expect a transition to a ferromagnetic ground
state for uniaxial pressures above $3.3\textup{ GPa}$.

###### pacs:

Valid PACS appear here

##  I Introduction

Higly frustrated magnets display a rich variety of exotic groundstates and
excitationsLacroix  _et al._ (2011); Castelnovo  _et al._ (2008); Brooks-
Bartlett  _et al._ (2014); Henley (2005). A primary reason for this diversity
is that the physical properties of the system are frequently determined by a
delicate balance of weaker interactions of similar strength. Following
Anderson’s original classificationAnderson (1987) there are two major classes
of frustrated systems; one in which the lattice plays a dominant role in
frustrating the system, and another where frustration arises due to competing
interactions. Recently it was realized that even in the first case, competing
interactions may unexpectedly refrustrate the system, as was found for the
spin ice materials Ho2Ti2O7 (HTO) and Dy2Ti2O7 (DTO)Henelius  _et al._ (2016).
Therefore, it is of interest to find physical realizations of a wide variety
of Hamiltonians, since even small alterations of the interactions may yield
novel physical properties and phases.

In order to explore the vast parameter space of frustration there is a
constant drive to synthesize new and promising frustrated materialsLacroix
_et al._ (2011); Wiebe and Hallas (2015); Biesner and Uykur (2020), each
representing a unique set of interaction coefficients determined by, for
example, crystal fields, ionic magnetic moments, inter-ion distances and
atomic overlaps. A different approach to probing frustration is to alter the
interactions of a given material. One way to do so is the application of
external pressure, which alters the position and thereby also the dipolar
interactions and atomic overlaps of the ions of the material. While there are
many observations of novel states and phenomena induced by pressureMirebeau
_et al._ (2002); Kim  _et al._ (2017); Zvyagin  _et al._ (2019), progress is
hampered by a number of experimental challenges that are exacerbated when
studying phenomena at cryogenic temperatures. On the theoretical side there
are open questions, since it is not easy to predict the effects of external
pressure on, for example, exchange interaction parameters.

In the present study, we apply uniaxial external pressure to increase the
parameter space we can explore using the parent material HTO. The physical
properties of the multiaxial Ising materials HTO and DTO show a strong
directional dependence on, for example an applied magnetic field Isakov  _et
al._ (2004); Fennell  _et al._ (2005). Therefore we choose to use uniaxial
pressure rather than isotropic hydrostatic pressure in this study. Our
starting point are relatively straight forward magnetization measurements to
determine the evolution of the exchange parameters under pressure and then we
validate our results by comparing the derived model to neutron scattering
experiments. In this study, we demonstrate the feasibility and limitations of
describing the measurements of HTO under pressure using an effective model
with a single pressure dependent parameter. While our application of
relatively low pressure did not induce any changes in the order of the
material, it shows the feasibility of exploratory experimental studies using
magnetization measurements. Such measurements can be used to determine the
evolution the model parameters and enable theoretical predictions of pressures
necessary to alter the delicate balance of interactions in frustrated
materials enough to cross phase boundaries and induce new phases.

Over the last 15 years HTO and DTO have become model compounds for studying
classically frustrated systems featuring residual ground state entropyRamirez
_et al._ (1999), topological magnetic monopole excitationsCastelnovo  _et al._
(2008), and slow low-temperature dynamicsJaubert and Holdsworth (2009); Revell
_et al._ (2013). The theoretical description in terms of the dipolar spin-ice
model (DSM)den Hertog and Gingras (2000a); Siddharthan  _et al._ (1999);
Yavors’kii  _et al._ (2008) captures the experimental features at a
quantitative level and DTO and HTO are now some of the best characterized
frustrated materials. This makes them an ideal starting point for a systematic
exploration of the evolution of frustrated systems under pressure. In an
earlier study, measurements of the pressure induced changes in the
magnetization of DTO were reportedMito  _et al._ (2007). These results were
first modeled theoretically using a nearest-neighbor exchange interaction
modelJaubert  _et al._ (2010). Recently, the effects of the dipolar
interactions were included in the theory, and the measurements were modeled
using a single parameter in the DSMEdberg  _et al._ (2019). Since DTO has a
large neutron absorption cross section, neutron scattering measurements were
performed on HTO, rather than DTO, but the observed changes in scattering
intensity were modeled within the same framework.

One shortcoming of the previous studyEdberg  _et al._ (2019) was that
magnetization measurements of HTO under pressure were missing, and it was not
possible to connect the parameters determined from the evolution of the
magnetization, measured for DTO, with the changes in the scattering intensity,
determined for HTOEdberg  _et al._ (2019). In this study, we remedy the
situation and report magnetization measurements on HTO, which we can use to
independently determine the pressure dependent parameter in the theory for
HTO. Our main result is that the independent analysis of both the
magnetization and the neutron scattering measurements lead to the very similar
values for the pressure dependent parameter. We therefore verify the
feasibility of our approach to use more straight forward magnetization
measurements to determine the explicit evolution of model parameters to
predict possible phase transitions before performing the more demanding
neutron scattering experiments under pressure.

##  II Experimental Method and Results

###  II.1 Magnetization measurements

High-quality samples are essential in high-pressure studies, and we
synthesized HTO single crystals using the floating zone technique at the image
furnace at Lund University. The crystals were grown from the bottom up with a
growth rate of $3\textup{ mm/hr}$. Visual inspection of the as-grown crystals
revealed no color change along the growth direction as seen by others.Ghasemi
_et al._ (2018)

Figure 1: Sample magnetic moment per holmium ion as function of external
magnetic field $H_{\textup{ext}}$, without demagnetization corrections.
Pressure and magnetic field along the (a) $[001]$, (b) $[111]$ and (c) $[110]$
crystalline direction. The different curves indicate different pressure
applied to the sample, all at sample temperature $1.83\textup{ K}$. The insets
show the changes in magnetic moment at small and moderate field.

We ascertained the high quality of the single crystals using x-ray and neutron
diffraction Deen  _et al._ (2019a) and will publish the details of the growth
and resultant crystalline state elsewhere. To ensure an optimum shape for the
application of uniaxial pressure, the single crystals were cut into cylinders
of diameter $2.0\pm 0.05\textup{ mm}$ and height $3.0\pm 0.05\textup{ mm}$.
The crystals were cut with the major axis of the cylinders along the
crystalline $[001]$, $[110]$ and $[111]$ directions. The quality after cutting
was asserted using x-ray diffraction. The crystalline direction of the major
axis was ascertained after cutting the crystals. For each crystal, we applied
uniaxial pressure and magnetic field along the cylinder major axis. We
measured the magnetic moment at $p_{001}=p_{111}=0,\;1.0\pm 0.2$ and $1.5\pm
0.3\textup{ GPa}$ for the $[001]$ and $[111]$ crystalline direction and at
$p_{110}=0,\;0.5\pm 0.1,\;1.0\pm 0.2$ and $1.5\pm 0.3$ GPa for the $[110]$
crystalline direction. We also measured the DC susceptibility at pressures
$p_{001}=p_{111}=0,\;0.5\pm 0.1,\;1.0\pm 0.2$ and $1.3\pm 0.2\textup{ GPa}$
using a probing field of $0.01\textup{ T}$ along the cylinder major axis.

Using an epoxy resin (Stycast 1266, Ablestick Japan Co., Ltd.) we placed the
crystals into a piston- cylinder type of pressure cell (CR-PSC-KY05-1, Kyowa-
Seisakusho Co., Ltd.), which can be inserted into a superconducting quantum
interference device (SQUID) magnetometer (Quantum Design MPMS-XL). The
combination of Stycast and the above uniaxial pressure enables the Poisson
effect to be excluded and thus the shrinkage ratio in the direction
perpendicular to the load can be ignoredGuo  _et al._ (2000). The quoted value
of the pressure is at liquid helium temperature, after considering the thermal
shrinkage. The pressure value at liquid helium temperature was estimated from
the shift of the superconducting transition temperature of lead under applied
pressureMito (2007). Due to the symmetric design of the upper and lower part
of the sample chamber the magnetic background was negligible for the large
magnetic signal of HTO. The measurements at different pressure were performed
on the same respective crystal for each direction and a maximum field of
$H_{\textup{ext}}=7\textup{ T}$ was used.

Figure 2: Susceptibility per holmium ion as function of temperature for field
and pressure along the (a) $[001]$ and (b) $[111]$ crystalline directions. The
field strength used to probe the susceptibility was $0.01\textup{ T}$. The
insets show the change in susceptibility in the high and low temperature
limit.

###  II.2 Neutron scattering

Magnetic neutron diffraction experiments were performed at the Institut Laue
Langevin (ILL) using the polarized diffuse scattering instrument, D7 Stewart
_et al._ (2008), with nominal incident wavelength $\lambda$ = $4.86\pm 0.1$ Å.
We recorded neutron diffraction profiles for HTO in the $(h,-h,l)$ scattering
plane under uniaxial pressure along the $[110]$ crystalline direction and with
incident neutron polarization along the $[110]$ crystalline direction. For
this experimental study, we cut our synthesized HTO crystals into cylinders of
diameter $3.0\pm 0.05\textup{ mm}$ and height $2.0\pm 0.05\textup{ mm}$, still
with major axis along $[110]$ but with larger diameter and smaller height than
the crystals used in the magnetization measurement. Pressure was applied using
a CuBe anvil-type pressure cell with a CuBe window of minimal thickness of
$2.5\textup{ mm}$ for the entire scattering plane. Further details of the
pressure cell are provided in appendix C and a detailed overview of the
pressure cell will be published elsewhere. The force was calibrated prior to
the experiment using an in situ transducer for deviations experienced at
cryogenic temperatures. The pressure was deduced from the force, allowing for
some uncertainty via friction of the piston. We measured both the neutron
spin-flip and non-spin-flip scattering as a function of the sample rotation
about the major axis of the cylinder ($[110]$ crystalline direction). The data
were corrected for detector and polarization analyzer efficiencies using
standard samples of vanadium and amorphous silica, respectively Stewart  _et
al._ (2008). Background measurements were performed at $300\textup{ K}$ with
an Al sample, closely matched in dimensions to the HTO sample, to determine
scattering from the pressure cell. The background was entirely symmetric in
the sample holder rotation angle and the rotation-averaged background at
ambient pressure was subtracted from the measured signal.

###  II.3 Experimental results

Figures 1 and 2 show the measured magnetic moment and susceptibility at
different pressures for the indicated directions. While the physics of spin
ice in an applied field is a fascinating topic with features such as chain
ordered statesFennell  _et al._ (2005), Kagome iceHigashinaka  _et al._ (2003)
and Kasteleyn transitionsFennell  _et al._ (2007), our aim is to focus on the
effects of pressure. We see a change in the magnetic moment on the order of a
few percent under the application of pressure. For the $[001]$ measurement,
Fig. 1(a), the magnetic moment is monotonically reduced for all fields when
pressure is applied. For the $[111]$ measurement, Fig. 1(b), the magnetic
moment is reduced both in the high and the low field limits, but for an
intermediate field around $1-2\textup{ T}$ the magnetic moment increases as
pressure is applied. For the $[110]$ measurement, shown in Fig. 1(c), there is
an overall decrease when pressure is applied, but the decrease is not
monotonic at high fields, as it is for the other directions. There is also a
large jump between the $p_{110}=0$ measurement and the $p_{110}>0$
measurements in the high and low field regions. We will discuss the
implications of these features further in section IV.1.

Figure 2 shows the susceptibility, $\chi_{\textup{ext}}$, measured with field
and pressure in the $[001]$ and $[111]$ crystalline directions. There is a
noticeable change in the susceptibility when pressure is applied to the
sample, but compared to the magnetic moments in Fig. 1 the effects are much
less clear. For example, in our measurements using a small applied field of
$0.01\textup{ T}$, the recorded change in the susceptibility is not generally
monotonic with respect to the applied pressure.

Figure 3: $[110]$ Spin-flip neutron scattering profile in the $(h,-h,l)$ plane
at (a) $0\textup{ GPa}$ and (b) $0.35\textup{ GPa}$Deen  _et al._ (2019b). The
pressure is applied along the $[110]$ crystalline direction, perpendicular to
the scattering plane. The data displayed have been symmetrized according to
the crystal symmetry. Both measurements were conducted at $1.5\textup{ K}$.
The shaded area marks the region which we integrate and analyze further in
section IV.2.

Figure 3 shows the spin-flip neutron scattering structure factor,
$S(\textbf{Q})$, at ambient and at an applied pressure of $0.35\textup{ GPa}$
at $T=1.5\textup{ K}$Deen  _et al._ (2019b). The well-established diffuse
scattering mapBramwell  _et al._ (2001) shows some of the defining features of
the spin ice phenomenology including regions of intensity at
(0,0,3),(3/2,3/2,3/2), and equivalent wave vectors, separated by pinch points
at, for example, (0,0,2)Fennell  _et al._ (2009); Twengström  _et al._ (2020).
In this study we focus on the effects of pressure on this scattering pattern.
The noted pressure was applied at $300\textup{ K}$ and the subsequent pressure
change as a result of thermal contraction could not be determined in situ.
However, prior temperature dependent calibration measurements of the pressure
cell provides an indication that the noted pressure is approximately correct.
A uniaxial pressure cell with in situ pressure determination for diffuse
neutron scattering and polarization analysis is under development. In Fig.
3(a,b), the measurements are taken with two different samples, with equivalent
sample dimensions, as the first sample broke when pressure was applied. The
samples were cut from the same single crystal. We note a change in the
intensity of the diagonal satellite peaks ($h\approx\pm 1.5,l\approx\pm 1.5$)
when pressure is applied. To better demonstrate these subtle changes, we have
integrated the shaded regions over $Q$, see Fig 3, and fitted a Gaussian curve
to the integrated data. We find that the intensity of the diagonal satellite
peaks increases by about $4\%$ when pressure is applied, and discuss this
further in section IV.2.

##  III Theoretical modeling

###  III.1 Model and simulation method

A theoretical model for the evolution of the interactions in classical spin
ice under uniaxial pressure was proposed in an earlier investigationEdberg
_et al._ (2019). We adapt this model to the current measurements of HTO and
refer to it as the dipolar spin-ice pressure model (DSPM). The DSPM is an
extension of the standard dipolar spin-ice model (s-DSM)Melko and Gingras
(2004). The Hamiltonian for the DSPM for classical unit Ising spins
$\textbf{S}_{i}$ on the pyrochlore lattice is defined as

$\begin{split}\mathcal{H}=&\sum_{\langle
i,j\rangle}J(i,j)\,\mathbf{S}_{i}\cdot\mathbf{S}_{j}\\\
+&Da^{3}\sum_{i<j}\left(\frac{\mathbf{S}_{i}\cdot\mathbf{S}_{j}}{|\mathbf{r}_{ij}|^{3}}-3\frac{\left(\mathbf{S}_{i}\cdot\mathbf{r}_{ij}\right)\left(\mathbf{S}_{j}\cdot\mathbf{r}_{ij}\right)}{|\mathbf{r}_{ij}|^{5}}\right).\end{split}$
(1)

Figure 4: Illustrations of the DSPMEdberg  _et al._ (2019). Pressure is
applied along the (a)$[001]$, (b)$[111]$, (c)$[110]$ direction and the colors
indicate the different exchange interactions
$J_{\parallel}(\textup{red}),J_{1\parallel}(\textup{green}),J_{2\parallel}(\textup{red}),J_{\perp}$(blue).
The angles of the Ising moments $\theta^{001},\theta^{111},\theta^{110}$ are
assumed to change under application of pressure so that spins keep pointing
toward the center of the compressed tetrahedron.

The strength of the dipolar interactions is taken to be $D=1.41\textup{ K}$den
Hertog and Gingras (2000b), $a$ is the nearest neighbor distance,
$\textbf{r}_{i}$ is the position of spin $\textbf{S}_{i}$ and
$\textbf{r}_{ij}=\textbf{r}_{i}-\textbf{r}_{j}$. The angled brackets in the
first sum denotes summation over nearest neighbors. The antiferromagnetic
nearest neighbor interaction $J(i,j)$ is caused by oxygen-mediated
superexchange. The value of $J$ at ambient pressure is set to
$J_{p=0}=1.56\textup{ K}$Bramwell  _et al._ (2001). In the DSPM it is assumed
that $J_{\perp}=J_{p=0}=1.56\textup{ K}$ if the neighbors $i$ and $j$ are in a
lattice plane perpendicular to the direction of applied pressure and
$J(i,j)=J_{\parallel}$ if they are not, ($J_{1\parallel}$,$J_{2\parallel}$ for
pressure in the $[110]$ direction as there is less symmetry) see Fig. 4. The
motivation for this assumption is that the distance between neighboring ions,
and hence the exchange integral, will change when pressure is applied.
Compression of the crystal is modeled with zero Poisson ratio, which is also
the case in the current experimental setup for the measurements of the
magnetic momentGuo  _et al._ (2000). As the lattice is compressed, the local
Ising axis of the spins is modeled to keep pointing towards the centers of the
compressed tetrahedra. The dipolar interaction becomes stronger along the
direction of compression, growing as $|\textbf{r}_{ij}|^{-3}$ with decreasing
distance. We define the lattice compression $\kappa$ as the relative length
contraction along the direction of pressure. At high fields, the saturated
magnetic moment is dependent only on $\kappa$, due to the change in angles of
the Ising moments. The high field saturated value of the magnetic moment
therefore sets the lattice compression, and we use the low field data to
determine exchange interactions.

At this point, a discussion of the crystal field effects in HTO is necessary.
The ionic state of the Ho3+ ion depends on the local electric fields of the
surrounding ion. Shifting their position by applying pressure could affect the
ionic ground state, on which our present analysis is based. Under ambient
pressure, the electronic ${}^{5}I_{8}$ ground state of a single Ho3+ is lifted
by the trigonal field of the surrounding oxygen atoms. Fitting the crystal
field parameters to susceptibility and inelastic neutron measurements yields
an almost pure $|J,M_{J}\rangle=|8,\pm 8\rangle$ non-Kramers ground state
doublet, with an excitation gap of more than 200K, and an almost full magnetic
moment of about 10 $\mu_{B}$Jana and Ghosh (2000); Rosenkranz  _et al._
(2000). These are the two crucial points for our study. Although the ground
state doublet in HTO is not protected by Kramers’ theorem, like it is for DTO,
it is unlikely that the small shift in the ionic positions which we induce by
applying pressure would significantly affect the nature of the ground state
doublet leading to our current Ising description. Considering a perturbation
theory approach, it is not unlikely that the magnetic moment of the Holmium
ion could shift slightly due to crystal field effects, since this is a first-
order effect. However, the changes in the ground state itself are expected to
be very small, since this second-order effect is suppressed by the very
significant energy gap, and by the fact that the ground state has a very small
overlap with states other than $|J,M_{J}\rangle=|8,\pm 8\rangle$.To completely
rule out crystal field effects would require a significant experimental and
theoretical undertaking, but we believe the above perturbative argument to be
strong enough to proceed with our analysis for HTO. We also note that pressure
induced changes in the crystal field parameters in general can be expected to
be more significant. For example, in the related compound Tb2Ti2O7 (TTO), the
energy gap is ten times smaller than in HTO, and the admixture of the ground
state doublet is significantPrincep  _et al._ (2015). This also makes TTO an
interesting case study for application of pressure, but the theoretical
analysis would require a very different approach from the one presented here.

Monte Carlo (MC) simulations using the Metropolis–Hastings algorithm and
single spin flip updates are used to investigate a number of different system
sizes with periodic boundary conditions. Since the lowest temperature used in
the experiment was $1.5\textup{ K}$, monopole excitations are still prevalent
and loop flipsMelko  _et al._ (2001) are not needed. We use the 16-particle
standard cubic unit cell. All supercells are cubic of size $L^{3}$ unit cells,
$L\in[1,...,8]$. Ewald summation is used to effectively account for the long-
range conditionally convergent dipolar contributionsFrenkel and Smit (2001).

###  III.2 Theoretical misalignment effects

The DSPM describes the changes in the magnetic moment observed in
measurementsMito  _et al._ (2007) of DTO under uniaxial pressureEdberg  _et
al._ (2019) quite well. Particularly when pressure and field is applied along
the $[001]$ or $[111]$ crystalline directions the model works very well.
However, when pressure and field is applied in the $[110]$ crystalline
direction for DTO, anomalous behavior is observed which cannot be explained by
the DSPMMito  _et al._ (2007); Edberg  _et al._ (2019), since the pressure-
induced change in the magnetic moment does not saturate at high fieldsMito
_et al._ (2007); Edberg  _et al._ (2019).

In this study, we attempt to explain this previously noted discrepancy for the
measurement in the $[110]$ crystalline direction and speculate that it can be
an experimental artefact due to misalignment of the crystal. Next, we outline
a brief discussion of what effects might be expected from crystal
misalignment.

When measuring the magnetic moment of a single crystal under uniaxial
pressure, there are three quantities that need to be aligned: the crystalline
direction, the direction of the uniaxial pressure and the direction of the
magnetic field.

In order to investigate the influence of misalignment, we consider the
possibility that the crystalline direction matches the direction of pressure
perfectly, and that the magnetic field is misaligned with respect to the
direction of pressure. This assumption minimizes the number of new parameters
introduced to the theoretical model, and we believe that it still captures the
essential effects of misalignment. In the experiment, on the other hand, we
may expect the misalignment between the crystal direction and the applied
pressure to be more significant. Since the crystal is surrounded by more
compressible Stycast it is also possible that this misalignment may change
during a series of compressions. See appendix A for further discussion.

Figure 5: View along $[001]$ of a single tetrahedron with pressure along
$[110]$. (a) Perfect alignment. (b) Field misalignment perpendicular to
$[001]$. We assume that the misalignment angle $\delta\varphi$ can vary with
application of pressure. For a field in this direction there are two
degenerate ground states for perfect alignment (double headed arrows in (a)),
and only one ground state when the field is misaligned (b). We denote the
misalignment at ambient and applied pressure by $\delta\varphi^{0}$ and
$\delta\varphi^{p}$ respectively.

Misalignment has particularly strong effects when the measurement is conducted
with pressure along the $[110]$ directionFukazawa  _et al._ (2002). This is
because half of the spins in the material have a local Ising axis
perpendicular to the $[110]$ direction, so that at high fields, already a
small misalignment will strongly affect the dynamics of these. We therefore
adjust the applied field so that it is misaligned by an angle $\delta\varphi$
with respect to the $[110]$ direction and remains perpendicular to $[001]$, as
this is the simplest way for the field to couple to the previously unaffected
spins. Fig. 5 illustrates the scenario. The magnetic moment is measured along
the direction of the field, and when the field is misaligned we start to
observe the dynamics of the previously perpendicular spins. This is not the
case when we measure along the $[001]$ and $[111]$ crystalline directions,
since in that case, all spins have already a large component along the field.
Hence we expect that the high field behavior will be less affected by
misalignment for measurements in the $[001]$ and $[111]$ crystalline
directions. Our simulations confirm that this is indeed the case. The
saturation fields for the pressure induced changes in magnetic moment in the
$[001]$ and $[111]$ measurements are not affected by misalignment, in contrast
to the $[110]$ direction. The effects of misalignment in the $[110]$ direction
are presented in section IV.1 and the key result is then shown in Fig. 7(c).

###  III.3 Demagnetizing corrections

In order to account for the macroscopic boundary effects, we perform a
demagnetizing transformation. The samples used were cylindrical with an aspect
ratio of $\gamma\equiv\frac{\text{height}}{\text{diameter}}=1.50\pm 0.05$ at
ambient pressure. However, as the the sample is compressed, the aspect ratio
decreases. By fitting $\kappa$ to the change in saturated magnetic moment we
determine $\gamma$. An important aspect, often overlooked, is that the
demagnetization factor $N$ is a function of both $\gamma$ and the internal
volume susceptibility $\chi^{V}_{\textup{int}}$,
$N=N(\gamma,\chi_{\textup{int}}^{V})$ as has been recently calculated and
verifiedChen  _et al._ (2006); Twengström  _et al._ (2017). In high-
susceptibility materials ($\chi_{\textup{int}}>1$ ), such as HTO, the
susceptibility dependence of $N$ is very significant and we include it in the
analysis. We perform a demagnetization transformation according to

$\begin{split}H_{\textup{ext}}&=H_{\textup{int}}+N(\gamma,\chi_{\textup{int}}^{V})M^{V},\\\
\chi_{\textup{ext}}^{V}&=\frac{\chi_{\textup{int}}^{V}}{1+N(\gamma,\chi_{\textup{int}}^{V})\chi_{\textup{int}}^{V}},\end{split}$
(2)

where the intensive quantities $M^{V}$ and $\chi^{V}$ are the magnetization
and volume susceptibility respectively. It is important to differentiate these
from the external extensive quantities $M=M^{V}V(\gamma(p))$ and
$\chi_{\textup{ext}}=\chi_{\textup{ext}}^{V}V(\gamma(p))$, measured in the
experiment. The Monte Carlo calculations, performed using periodic Ewald
boundary conditions, yield internal quantities.

Since the susceptibility of classical spin ice is isotropic the susceptibility
dependence of $N$ calculated in Ref. Chen  _et al._ , 2006 can be used for
HTO, as demonstrated in Ref. Twengström  _et al._ , 2017. Using the tabulated
values of $N(\gamma,\chi_{\textup{int}}^{V})$ we perform an interpolation
using cubic splinesChen  _et al._ (2006). As an example, we show the
temperature dependence of $N$ for an HTO cylinder of aspect ratio $\gamma=1.5$
in Fig. 6, where we use the experimentally measured intrinsic susceptibility
for HTOBovo  _et al._ (2013) at zero field to determine the temperature
dependence.

Figure 6: Demagnetizing factor $N$ as a function of $T$ at zero field for a
HTO cylinder of aspect ratio $\gamma=1.5$.

###  III.4 Calculation of susceptibility

For the susceptibility measurements, we define the relative change in the
susceptibility with respect to the applied pressure as
$\Delta\chi_{\textup{ext}}/\chi_{\textup{ext}}\equiv(\chi_{\textup{ext}}^{p}-\chi^{0}_{\textup{ext}})\chi^{0}_{\textup{ext}}$.
We measured the relative change
$\Delta\chi_{\textup{ext}}/\chi_{\textup{ext}}$ as a function of temperature.
This quantity is highly sensitive to changes in $J$ and $\kappa$, and in order
to gain some physical insight, we consider the high-temperature limit,
$T\to\infty$. In this non-interacting limit the relative change depends only
on $\kappa$. For the crystalline directions along which measurements were
performed we find that

$\begin{split}\lim_{T\to\infty}\frac{\Delta\chi^{111}_{\textup{ext}}}{\chi^{111}_{\textup{ext}}}&=\frac{9\,{\left(1-\kappa\right)}^{2}}{4\,{\left(1-\kappa\right)}^{2}+32}-\frac{1}{4},\\\
\lim_{T\to\infty}\frac{\Delta\chi^{001}_{\textup{ext}}}{\chi^{001}_{\textup{ext}}}&=\frac{3(1-\kappa)^{2}}{2+(1-\kappa)^{2}}-1,\end{split}$
(3)

where the superscripts denote the crystalline direction of pressure and field.

From these formulae, we see that in both directions, the susceptibility at
high temperature is reduced under application of pressure. This is a purely
geometrical effect stemming from the fact that the Ising moments tilt away
from the axis of pressure when the lattice is compressed.

###  III.5 Calculation of magnetic structure factor

We calculate theoretical predictions for the spin-flip magnetic structure
factor according to

$\begin{split}&S(\textbf{Q})=\frac{[f(|\textbf{Q}|)]^{2}}{\mathcal{N}}\\\
&\times\sum_{ij}\left\langle\textbf{S}^{\perp}_{i}\cdot\textbf{S}^{\perp}_{j}-(\textbf{S}_{i}\cdot\textbf{P})(\textbf{S}_{j}\cdot\textbf{P})\right\rangle
e^{i\textbf{Q}\cdot\textbf{r}_{ij}},\end{split}$ (4)

where the scattering wavevector is denoted by Q, the normalized polarization
vector of the incident neutron beam is given by $\textbf{P}\perp\textbf{Q}$.
The component of the spin perpendicular to the wave vector is defined as
${\textbf{S}_{i}^{\perp}=\textbf{S}_{i}-\textbf{S}_{i}\cdot\textbf{Q}/|\textbf{Q}|^{2}\textbf{Q}}$.
$\mathcal{N}$ is the number of particles in the supercell and
$f(|\textbf{Q}|)$ is the magnetic form factor for Ho3+. The angled brackets
$\langle\cdots\rangle$ denote the thermal average which is calculated using
the MC method.

Figure 7: Relative change in sample magnetic moment, $\Delta
M/M=(M^{p}-M^{0})/M^{0}$, as a function of magnetic field at sample
temperature $1.83\textup{ K}$. The pressure and field are along the
(a)$[001]$, (b)$[111]$, (c)$[110]$ crystalline direction. MC results are drawn
as solid lines and performed on cubic systems of $8192$ particles, $L=8$. For
the $[110]$ measurement (c) we show the model prediction for
$J_{1\parallel}=J_{2\parallel}=1.2\textup{ K},\kappa=1\%$ both for perfect
alignment(dashed line) and for a small misalignment when pressure was
applied(solid line).

##  IV Analysis and discussion

###  IV.1 Magnetization and susceptibility

In Fig. 7 we show the relative change in sample magnetic moment, $\Delta
M/M=(M^{p}-M^{0})/M^{0}$, as function of applied magnetic field. The solid
lines show the best theoretical fit for the different directions of pressure
and field. For the $[001]$ and $[111]$ measurements, shown in Fig. 7(a,b),
$\kappa$ has been set so that the change in calculated saturation magnetic
moment agrees with that of the experiment. With this constraint, there is only
one free parameter, $J_{\parallel}$, which we determine from the best fit for
each case. We find that for the $[001]$ and $[111]$ measurement,
$J_{\parallel}$ increases monotonically with pressure. The change in
$J_{\parallel}$ is similar in both directions and the model accommodates the
basic features in the experimental data: the upturn at $1.5\textup{ T}$ in the
$[111]$ measurement as well as the drop at $0.5\textup{ T}$ in the $[001]$
direction, upon decreasing field. For the $[001]$ measurement we have
excellent agreement between theory and experiment. We note that the observed
decrease in the magnetic moment for HTO at low fields contrasts to
measurements made on DTO, where an increase of $\Delta M/M=4\%$ was
observedMito  _et al._ (2007). Although $J_{\parallel}$ and $\kappa$ increase
similarly for both materials, the ratio $J/D$ is smaller for HTO, and the
effects of the dipolar interactions dominate, leading to a decrease in the
magnetic moment. In the $[111]$ direction the experimentally observed minimum
centered at $\mu_{0}H_{\textup{ext}}=3.5\textup{ T}$ is not accounted for in
the model, and the shape of the maximum around $1.8\textup{ T}$ differs, but
the model captures the qualitative experimental features. Furthermore, in all
measurements, the change in magnetic moment is strongly dependent on the
field. This is due to competition between different effects. At large fields,
the moment is saturated and since spins tilt away from the direction of
pressure, it decreases. At low fields, the thermal fluctuations determine the
magnetic moment. In the case of HTO, the magnetization decreases when pressure
is applied, while for DTO in the $[001]$ direction, the preference for a
ferromagnetic ground state is indicated by an upturn in the magnetization.

Figure 7(c) shows the change in magnetic moment for the $[110]$ measurement.
We note that the change has hardly saturated as a function of field.
Seemingly, we need a field of about $7\textup{ T}$ to saturate the change in
the $[110]$ direction, higher than the $5.5\textup{ T}$ needed for the $[111]$
direction. This feature is not possible to reproduce in the DSPM in which the
field needed for $[111]$ saturation is higher than that needed for $[110]$
saturation, since the saturated $[111]$ state requires that we break the two-
in two-out ground state of spin iceHarris  _et al._ (1997).

As discussed in section III.2, we speculate that the high-field unsaturated
behavior can be due to misalignment of the crystal, since measurements are
particularly sensitive in the $[110]$ direction. In Fig. 7(c) we illustrate
the influence of misalignment. We first fit a curve with $\kappa=1\%$, as an
estimate of the compression based on the other directions at $1\textup{ GPa}$
($0.8\%$ and $1.5\%$ for $[001]$ and $[111]$ respectively). In the DSPM, the
change in magnetic moment will saturate at about $1\textup{ T}$ (same order of
magnitude as in the $[001]$ case) regardless of the parameters used. For
simplicity we assume that $J_{1\parallel}=J_{2\parallel}=J_{\parallel}$ which
gives a rough value of $J_{\parallel}=1.2\textup{ K}$ when we fit this single
parameter to match the sub $1\textup{ T}$ experimental results. This DSPM fit
is shown as the dashed line in Fig. 7(c). For all fields above $1\textup{ T}$,
the DSPM gives a constant saturated change in magnetic moment dependent only
on $\kappa$. Clearly this is not what we see in the experiment, where the
change keeps varying up to fields of about $7\textup{ T}$.

If we introduce crystal misalignment according to section III.2, we can
partially reproduce the high-field trend observed in the measurement. Within
this framework, we have two additional parameters
$\delta\varphi^{0},\delta\varphi^{p}$ as discussed in section III.2 for the
misalignment of $H$ with respect to the $[110]$ direction, at zero($0$) and
applied pressure($p$) respectively. To get a decrease in magnetic moment when
pressure is applied, we find it necessary that the field couples stronger to
the perpendicular spins in the reference measurement. Hence, to keep the model
as simple as possible, we set $\delta\varphi^{p}=0^{\circ}$. We then adjust
$\delta\varphi^{0}$ in order to fit the experimental curve above $1\textup{
T}$. We find that $\delta\varphi^{0}=0.75^{\circ}$ gives the best fit. We do
not get the exact same features as those observed, but we do demonstrate that
the decrease in magnetization above $1\textup{ T}$ could be due to
misalignment. For further discussion of more elaborate misalignment models
under pressure we refer to appendix A.

Figure 8: Relative change in external susceptibility for pressure and field
along the (a) $[001]$ (b) $[111]$, both for probing external field of strength
$0.01\textup{ T}$. Solid lines show the theoretical MC prediction from the
parameters fitted from the magnetic moment measurements at $1.0\textup{ GPa}$.
Dashed lines indicate the $T\rightarrow\infty$ analytical limiting value, Eq.
(3). MC simulation for $2000$ particles, $L=5$.

Finally, we use our obtained values of $J_{\parallel}$ and $\kappa$ to predict
the change in susceptibility as a function of temperature. In Fig. 8 we show
the measured relative change in external susceptibility plotted against
temperature for an applied pressure of $1\textup{ GPa}$. The theoretical
predictions are shown for the values derived from the $1.0\textup{ GPa}$
$[001]$ and $[111]$ magnetic moment fits, Fig. 7(a,b), respectively. In
contrast to the measurements of the magnetic moment, the susceptibility
measurements are more challenging due to the low field used and a higher
sensitivity to uncertainties in the demagnetizing factors.

Due to the fluctuations in the $[001]$ measurement, Fig. 8(a), we include a
trend line by fitting rational polynomials to the susceptibility. The
qualitative curve shape is the same for both theory and experiment, but they
differ by an overall shift along the vertical axis. At the lowest temperature
$T=1.83\textup{ K}$ we expect that the relative change in susceptibility
should coincide with the $H_{\textup{ext}}\to 0$ limit in the relative change
in the magnetic moment Fig. 7(a), which for $1\textup{ GPa}$ is $-2\%$. There
is therefore a discrepancy between the two experimental measurements. From
theory we also expect that the susceptibility should be reduced at high
temperature when pressure is applied, see Eq. (3). Since the susceptibility
measurements in Fig. 2 were nonmonotonic in pressure we suspect a systematic
error in the experimental data and note that if we shift the susceptibility
measurement performed under pressure by $-2\%$ the data matches the
theoretical prediction as well as the magnetization measurement performed at
$T=1.83\textup{ K}$.

For the $[111]$ direction there is a more significant mismatch between theory
and experiment. In particular, theory would predict a maximum in
$\Delta\chi/\chi$ near $5\textup{ K}$, which is not present in the experiment.
We would also expect from theory that the relative change should saturate to a
lower value. Indeed, with the assumption of Ising spins it directly follows
from Eq. (3) that $\Delta\chi/\chi$ must reach a negative value at high
temperature. The discrepancy is most likely due to insufficient accuracy in
the present susceptibility measurements.

###  IV.2 Neutron scattering

Figure 9: (left) Relative change in the $[001]$ spin-flip neutron scattering
profile predicted from parameters found in the measurement of the magnetic
moment, Fig. 7, at $1.5\text{ GPa}$ along $[001]$ and $1.5\textup{ K}$
($J_{\parallel}=1.80,\kappa=1.6\%$). (right) Relative change in experimental
$[001]$ spin-flip neutron scattering dataEdberg  _et al._ (2019); Deen  _et
al._ (2018) for HTO at $2.2\textup{ GPa}$ pressure along $[001]$, $1.5\textup{
K}$. Circles mark regions of increased intensity and are plotted at symmetry
equivalent regions in the theoretical prediction. In previous
investigationsEdberg  _et al._ (2019), the DSPM parameters were estimated
based directly on this experimental scattering profile. The current
measurements of the magnetic moment show consistency with this estimate.

The parameters extrapolated from the measurements of the magnetic moment can
be used to compute the thermal spin-spin correlation function and hence the
magnetic structure factor, Eq. (4) within the DSPM. For pressure along the
$[001]$ crystalline axis theoretical and experimental results match well, see
Fig. 7(a), and we use the parameters from this fit to predict the $[001]$
spin-flip structure factor under pressure. The resulting change in
$S(\textbf{Q})$, $\Delta
S(\textbf{Q})/S(\textbf{Q})\equiv(S(\textbf{Q})_{p}-S(\textbf{Q})_{0})/S(\textbf{Q})_{0},$
is shown in the left half of Fig. 9, simulated at $T=1.5\textup{ K}$ and
$p=1.5\textup{ GPa}$. In the right half we show the change in the
experimentalDeen  _et al._ (2018) scattering at $p=2.2\textup{ GPa}$ and
$T=1.5\textup{ K}$ measured in previous workEdberg  _et al._ (2019). In the
outer region we see eight patches of increased intensity which have been
marked and coincide with the theory prediction. In the inner region there are
two regions of increased intensity not seen in the the experiment, but we
note, as mentioned in previous work, that the experimentally measured signal
was poorly sampled at wave vectors shorter than $1.5\textup{
\AA\textsuperscript{-1}}$ due to windows in the scattering plane of the
previous CuBe pressure cellEdberg  _et al._ (2019) used in that neutron
scattering experiment.

Figure 10: $[110]$ spin-flip $S(\textbf{Q})$ line-cut average over
$2.22<Q<2.87$ (reduced units), for polar angle $\phi=0^{\circ}\rightarrow
90^{\circ}$ in the neutron scattering data (shaded area in Fig.3)Deen  _et
al._ (2019b). Gaussian fit to the experimental data (solid). Theoretical
prediction at zero and $1.0\textup{ GPa}$ uniaxial pressure for the parameters
found in the measurements of the magnetic moment, $J_{\parallel}=1.2\textup{
K},\kappa=1\%$, Fig. 7(c) (dashed). MC simulation for $8192$ particles $L=8$.

A key result of this study is therefore that the scattering profile calculated
from the parameters obtained from the fit to the magnetic moment, Fig. 7(a),
matches the best profile that can be obtained by freely adjusting
$J_{\parallel}$ to the experimental neutron data, as was done in the previous
studyEdberg  _et al._ (2019). Furthermore, we use the DSPM to calculate the
spin-flip structure factor in the $(h,-h,l)$ plane with pressure along
$[110]$, shown in Fig. 3. We take the parameters fitted to the magnetic
moment, Fig. 7(c), and calculate $S(\textbf{Q})$ from Eq. (4). In order to
better show the changes in scattering intensity, we integrate the signal
around one of the satellite peaks as shown by the shaded region in Fig. 3. In
Fig. 10 we see Gaussian fits to the experimentalDeen  _et al._ (2019b) data
(solid lines). The theoretical estimate from the fit to the magnetic moment
($J_{\parallel}=1.2\textup{ K}$,$\kappa=1\%$) is shown together with the
theoretical prediction at zero pressure (dashed lines). Although the
experimental error bars, set by the neutron exposure time, are of the same
order as the observed increase under applied pressure, we note that the theory
captures the increased peak intensity of about 4 $\%$ and conclude that the
model describes the observed phenomena both for magnetization and neutron
scattering measurements in several crystalline directions.

###  IV.3 Evolution of the model parameters

Using the straight forward magnetization measurements we can obtain the model
parameter dependence on pressure. In Fig. 11 we depict the pressure dependence
of $J_{\parallel}$ and $\kappa$ in all three direction. We include the current
data points at pressures of $1.0\textup{ GPa}$ and $p=1.5\textup{ GPa}$, as
well as the result of the previous neutron study at an applied pressure of
$2.2\textup{ GPa}$Edberg  _et al._ (2019). The three data points along $[001]$
show a near-linear dependence on pressure. Analysing the phase diagram using a
combination of MC and direct comparison of the state energiesEdberg  _et al._
(2019) we find that all three points lie on a curve in the
$(J_{\parallel},\kappa)$-space for which the exchange interaction evolves as
to cancel the changes in the dipolar dynamics originating from lattice
compression, and the system is on the border between two different types of
dipolar chainMelko and Gingras (2004) ground statesEdberg  _et al._ (2019).
This suggests that the ground state will remain a dipolar chain state under
application of uniaxial pressure along the $[001]$ direction. This result
contrasts to the predictions for DTO, where at sufficiently high pressure a
ferromagnetic ground state is expectedJaubert  _et al._ (2010); Edberg  _et
al._ (2019).

Figure 11: Evolution of the model parameter $J_{\parallel}$ and crystal
compression $\kappa$ under applied uniaxial pressure along the $[001]$
crystalline axis.

In order to extrapolate our results to higher pressure we use linear
extrapolations for the parameters in the $[111]$ and $[110]$ directions:

$\begin{split}J_{\parallel}&=J_{\parallel}^{p=0}+(J_{\parallel}^{p^{*}}-J_{\parallel}^{p=0})p,\\\
\kappa_{\parallel}&=\kappa^{p=0}+(\kappa^{p^{*}}-\kappa^{p=0})p.\end{split}$
(5)

For $\kappa$ this is motivated by Hooks law, while the true pressure
dependence on $J_{\parallel}$ may deviate from our linear model, but we
proceed with $p^{*}$ as the highest measured pressure for the respective
direction.

For the $[111]$ direction, we find that the dipolar chain ground state is
stable up to pressures of at least $4\textup{ GPa}$. In the $[110]$ direction,
on the other hand, we find a transition to the previously mentioned
ferromagnetic state at a critical pressure of $3.3\textup{ GPa}$. In DTO this
state is expected when applying pressure in excess of $3.4\textup{ GPa}$ in
the $[001]$ directionJaubert  _et al._ (2010); Edberg  _et al._ (2019) We
therefore conclude that further uniaxial high pressure studies should be
conducted either on DTO in the $[001]$ direction, or on HTO in the $[110]$
direction. At such high pressures the crystals are likely to break. However,
using techniques like submerging the crystals in epoxy resin as we did in this
study, or using some other type of support material, we believe that such
experiments can be realized. We summarize the fitted and estimated parameter
values in table 1.

Direction | $p=0\textup{ GPa}$ | $p=1\textup{ GPa}$ | $p=1.5\textup{GPa}$ | $p_{c}\textup{ [GPa]}$
---|---|---|---|---
$J\textup{ [K]}$ | $J_{\parallel}\textup{ [K]}$ | $\kappa\textup{ [\%]}$ | $J_{\parallel}\textup{ [K]}$ | $\kappa\textup{ [\%]}$
[001] | 1.56 | 1.69 | 0.8 | 1.80 | 1.6 | -
[111] | 1.56 | 1.63 | 1.5 | 1.75 | 2.8 | $>4$
[110] | 1.56 | 1.2 | 1 | - | - | $3.3$
Table 1: Summary of the fitted and estimated DSPM parameters found in this
work. $p_{c}$ is the critical pressure above which a linear extrapolation
predicts a ferromagneticJaubert  _et al._ (2010) ground state. We estimate the
error to be less than $5\%$ of the fitted parameter value. However, for the
[110] direction this might be as much as $20\%$ due to misalignment having
more prominent effects.

##  V Conclusions

We have performed measurements of the field-induced magnetic moment, magnetic
susceptibility and neutron structure factor of HTO under applied uniaxial
pressure. Through extensive MC calculations we demonstrate that a dipolar spin
ice model, with a pressure-tuned nearest neighbor interaction is able to
capture the most essential features of the measurements of HTO. The framework
is extended to include effects of misalignment, and we have found that
misalignment can, to some extent, describe the anomalous effects observed in
the magnetic moment for the $[110]$ direction in both HTO and DTO.

The $T$-dependent pressure induced changes in the susceptibility turns out to
be a more sensitive quantity to both measure and model than the field-induced
magnetic moment. The low field used experimentally results in a weak signal
which is easily overshadowed. Theoretically, the susceptibility of spin ice
has proven to be a sensitive function of the intrinsic competing
interactionsBovo  _et al._ (2018), and the measurements are sensitive to
sample shapeTwengström  _et al._ (2017), with the optimal sample size probably
spherical. To perform susceptibility measurements on a spherical sample under
pressure is highly challenging, and we therefore have to contend with our
present results. Still, the qualitative curve shape for the susceptibility in
the $[001]$ direction is captured rather well by our model, but there are
significant discrepancies in the $[111]$ measurement.

Our main result is that the model parameters derived from the measurement of
the magnetization also captures the most salient features of the pressure
induced change in the neutron scattering structure factor. That the same model
describes both bulk properties and spin-spin correlation functions lends
credibility to the theory. Therefore we hope that using relatively straight
forward magnetization measurements to determine the pressure dependence of
interaction parameters can prove useful also when it comes to other classes of
frustrated materials. Increased theoretical predictive power supporting
demanding neutron experiments under high pressure would benefit many
investigators in the field. With new, and more intense, neutron sources under
construction we expect this to be a research topic of increasing importance in
the near future.

##  Appendix A Misalignment

Figure 12: Misalignment. (a) Field misalignment, discussed in main text. (b)
Crystalline axis misalignment. The view is along the [001] direction in both
cases. More symmetries are broken in (b) as the tetrahedron is compressed
along the $p$ axis.

The main text discusses the influence of misalignment of the applied field. Of
even greater importance is probably a misalignment of the crystalline axis
with respect to the applied pressure and field. Our basic analysis for the
magnetization measurements suggests that the $[001]$ and $[111]$ directions
are not sensitive to a slight misalignment. The $[110]$ direction, on the
other hand, is much more sensitive due to the large subset of spins that are
perpendicular to the field under perfect alignment. We suspect that this
result holds also in more elaborate models of crystalline axis misalignment. A
more general description than provided in the main text introduces up to six
independent exchange parameters since there are six different types of nearest
neighbor bonds in a tetrahedron. Figure 12 illustrates the two different kinds
of misalignment. The field misalignment depicted in Fig. 12(a), was discussed
in the main text. In this case, there are only three distinct exchange
couplings ($J_{\perp}$,$J_{1\parallel}$,$J_{2\parallel}$) due to the symmetry
of the tetrahedron. In the case of misalignment of the the crystalline axis
the $J_{2\parallel}$ couplings will no longer be equal. Figure 12(b)
illustrates a case where all six distances between different corners of the
compressed tetrahedron are different. In order to reduce the number of free
parameters, we could linearize the the distance dependence of $J$,
$J=J_{p=0}+K(r_{0}-r)$, where $r_{0}$ is the unperturbed distance and $K$ is a
free parameter. This would give the same number of free parameters as in the
basic model presented in the main text. However, $J$ does not necessarily
depend on the distance between the ions in the same way for all nearest
neighbor bonds. Therefore, we contend with the model of the main text, and
believe that it sufficiently demonstrates that the features above $1\textup{
T}$ in the $[110]$ relative change in magnetic moment can be an artefact due
to misalignment.

##  Appendix B Crystals

The crystal quality is particularly important when working with external
pressure. The crystals must withstand high pressure without cracking, and due
to the demagnetizing effects it is important that the crystals are cut with
accuracy. Figure 13(a) shows a picture of one of the crystals used in the
magnetization experiment. The crystals have been aligned to the uniaxial
pressure axis. However as the crystals were submerged in stycast which is much
softer than HTO, a small shift in the alignment may arise as pressure is
applied due to plastic deformation.

In the neutron scattering experiment for the larger $3\textup{ mm}$ diameter
crystals, the nuclear Bragg peaks observed on D7 are consistent with the
alignment. However, the instrumental parameters of D7: incident beam size,
divergence and detection resolution, do not provide a very accurate
determination of the absolute alignment. The nature of the neutron scattering
experiment also prevents the use of stycast to mitigate against the Poisson
expansion. To include such expansion of the sample in the simulation would
introduces additional unknown model parameters, which from these few
measurements is hard to determine. We therefore chose not to include this
effect, and we find that this approximation still gives valid prediction for
the neutron scattering intensity. However, we mention that the physics that
can be reached by including this effect can also be reached by allowing
$J_{\perp}$ to vary, and it will not introduce any new effects, apart from
having more free parameters allowing for an easier fit to data.

(a)

(b)

Figure 13: (13(a)) One of the HTO crystals ($\diameter~{}2\textup{ mm}$) used
for the magnetization measurement. The picture shows the accuracy of the
cutting procedure. (13(b)) One of the crystals used in the neutron scattering
experiment ($\diameter~{}3\textup{ mm}$), placed on top of the steel anvil
which was used for applying uniaxial pressure.

##  Appendix C Uniaxial pressure cell

The neutron scattering uniaxial pressure cell has been developed to enable
neutron scattering experiments in the cold energy spectra range,
$2\lesssim\lambda~{}\lesssim 20~{}\textup{\AA}$, for diffuse magnetic
scattering profiles, inclusive of weak inelastic scattering features, with
polarisation analysis. The uniaxial pressure cell should be able to provide
pressures up to $2\textup{ GPa}$ at cryogenic temperatures. As such, the
requirements include a scattering window that covers a wide angular range with
a very clean background profile and a non-magnetic cell that enables
polarisation analysis. These requirements place stringent restrictions on the
materials for the manufacturing of the cell and lead us to focus on the
optimization of an anvil type cell. An engineering overview of the cell
employed during the D7 experiment is shown in Fig. 14. The main body is
manufactured out of CuBe with non-magnetic stainless steel anvils. Force is
applied at room temperature and recalibrated using a calibration profile
developed within our team. Pressure is deduced from the known contact area.
The calibration profiles are not perfect and may lead to some uncertainty in
the exact pressure applied at the lowest temperatures. Complete details of the
cell will be published elsewhere and an uniaxial pressure cell with in situ
pressure determination is under development ensuring we maintain the
aforementioned parameters.

Figure 14: Engineering drawings and overview of the uniaxial pressure cell.

###### Acknowledgements.

We thank Steven Bramwell for insightful discussions. The neutron scattering
experiments were performed at the Paul Scherrer Institute and the Institute
Laue-Langevin. We thank the sample environment group of the ILL for
preparatory help and access to facilities. The simulations were performed on
resources provided by the Swedish National Infrastructure for Computing (SNIC)
at the Center for High Performance Computing (PDC) at the Royal Institute of
Technology (KTH). The project was supported by Nordforsk through the program
NNSP (Project No. 82248) and by the Danish Agency for Research and Innovation
through DANSCATT.

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