# Dipolar Spin Ice Under Uniaxial Pressure

R. Edberg  Physics Department, KTH Royal Institute of Technology, Sweden  L.
Ørduk Sandberg  Nanoscience Center, Niels Bohr Institute, University of
Copenhagen, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark  I. M. Bergh
Bakke  University of Oslo, Centre for Materials Science and Nanotechnology,
NO-0315, Oslo, Norway  M. L. Haubro  Nanoscience Center, Niels Bohr Institute,
University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark
L. C. Folkers  Department of Chemistry, University of Lund, Sweden  L. Mangin-
Thro  A. Wildes  Institut Laue-Langevin, 38042 Grenoble, France  O. Zaharko
Laboratory for Neutron Scattering, Paul Scherrer Institute, 5232 Villigen-PSI,
Switzerland  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  M. H. Sørby  Department for Neutron Materials
Characterization, Institute for Energy Technology, P.O. Box 40, NO-2027,
Kjeller, Norway  K. Lefmann  Nanoscience Center, Niels Bohr Institute,
University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark
P. P. Deen  Nanoscience Center, Niels Bohr Institute, University of
Copenhagen, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark  European
Spallation Source ERIC, 22363 Lund, Sweden  P. Henelius  Physics Department,
KTH Royal Institute of Technology, Sweden  Faculty of Science and Engineering,
Åbo Akademi University, Åbo, Finland

###### Abstract

The magnetically frustrated spin ice family of materials is host to numerous
exotic phenomena such as magnetic monopole excitations and macroscopic
residual entropy extending to low temperature. A finite-temperature ordering
transition in the absence of applied fields has not been experimentally
observed in the classical spin ice materials Dy2Ti2O7 and Ho2Ti2O7. Such a
transition could be induced by the application of pressure, and in this work
we consider the effects of uniaxial pressure on classical spin ice systems.
Theoretically we find that the pressure induced ordering transition in
Dy2Ti2O7 is strongly affected by the dipolar interaction. We also report
measurements on the neutron structure factor of Ho2Ti2O7 under pressure, and
compare the experimental results to the predictions of our theoretical model.

###### pacs:

Valid PACS appear here

##  I Introduction

In a frustrated magnet a unique ground state can be difficult to reach, since
the primary interactions are obstructed by competition between interactions of
similar strength or a special geometry of the lattice, leading to a strong
ground state degeneracy in the unperturbed caseLacroix  _et al._ (2011).
Therefore, an eventual low-temperature ordering transition and resolution of
the third law of thermodynamics is often driven by weak, perturbative
interactions whose effects are typically hidden in non-frustrated systems. In
recent years, the geometrically frustrated spin ice compounds have revealed
many unusual phenomena such as magnetic monopole excitations Castelnovo  _et
al._ (2008), fractionalization of the magnetic moment Brooks-Bartlett  _et
al._ (2014) and a low-temperature Coulomb phase Henley (2005). However, a low-
temperature ordering transition below the Coulomb phase in the absence of
applied fields has not been experimentally observed in the classical spin ice
compounds Dy2Ti2O7 (DTO) and Ho2Ti2O7 (HTO) Pomaranski  _et al._ (2013);
Henelius  _et al._ (2016). Theoretically there are predictions for a dipolar-
driven low-temperature transition in the $0.2\textup{ K}$ temperature range
Melko and Gingras (2004), but experimentally the measurements are challenging
due to extremely slow relaxation effects Pomaranski  _et al._ (2013), and the
nature of such a transition is not known.

One way to increase the possible transition temperature into a more
experimentally accessible range is to apply a suitable perturbation, such as
pressure, to the system. Previous experimental studies detect a clear uniaxial
pressure-induced signature in the susceptibility of DTO Mito  _et al._ (2007),
while hydrostatic pressure has no noticeable effect on HTO Mirebeau and
Goncharenko (2004). A previous theoretical study predicts that uniaxial
pressure should give rise to an unusual infinite order phase transition in
DTO, with features of both a continuous and discontinuous transition, to a
ferromagnetic state Jaubert  _et al._ (2010). In this study, we include the
long-range dipolar interaction in the theoretical model, and find that it does
not favor the ferromagnetic state. We also perform spin-polarized neutron
scattering experiments on HTO under uniaxial pressure and compare the results
to our theoretical predictions. While our model captures the qualitative
features of the experiment, further measurements are needed in order to fully
understand the effects of pressure on the low-temperature spin ice state in
DTO and HTO.

##  II Model, Simulation and Experimental methods

###  II.1 Model

Existing literature presents a large number of spin ice models, ranging from
effective monopole models Castelnovo  _et al._ (2008), electrolyte theories
Bramwell  _et al._ (2009) to microscopic models incorporating for example
dipolar, exchange and even hyperfine interations den Hertog and Gingras
(2000a); Yavors’kii  _et al._ (2008); Henelius  _et al._ (2016). Different
models emphasize different aspects of the physical properties of spin ices. We
aim to select the simplest model that captures the most relevant aspects of
the pressure dependence of the system. In the previous theoretical study on
the effects of pressure in the classical spin ice materials, a nearest
neighbor model was used Jaubert  _et al._ (2010). Due to a self-screening of
the long-range part of the dipolar interaction, this relatively simple model
gives a surprisingly good description of classical spin ice behaviour den
Hertog and Gingras (2000b); Isakov  _et al._ (2005). It can be analyzed
analytically using a Husimi tree solution Jaubert  _et al._ (2013), and the
low-temperature properties are similar to those of the monopole model
Castelnovo  _et al._ (2008). However, to capture the experimental low-
temperature behavior of DTO and HTO, dipolar interactions have proven to be
important den Hertog and Gingras (2000a), and we therefore consider the
dipolar spin ice model.

The Hamiltonian for normalized spins $\mathbf{S}_{i}$ residing on the sites
$\textbf{r}_{i}$ of the pyrochlore lattice with nearest neighbor distance $a$,
is given by a nearest neighbor antiferromagnetic exchange interaction $J$ and
a long-range dipolar interaction $D$,

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

Figure 1: Model for uniaxial pressure applied in the [001](a), [111](b) and
[110](c) directions. $J_{\perp}$ remains fixed while $J_{(k)\parallel}$ vary
with pressure. Colors indicate the different exchange interactions. The angle
$\theta$ between the Ising-axis and the side of the tetrahedron is altered to
$\theta^{001}$,$\theta^{111}$ and $\theta^{110}$ when the crystal lattice is
compressed in the respective direction.

Here, the angle brackets denote summation over nearest neighbors and
$\textbf{r}_{ij}=\textbf{r}_{i}-\textbf{r}_{j}$. The antiferromagnetic
interaction $J$ is caused by oxygen-mediated superexchange between adjacent
magnetic ions. Due to the strong crystal field, the spins are oriented along
the local $[111]$-axis connecting the center points of adjacent tetrahedra. To
capture the detailed behavior of a number of physical properties for DTO,
exchange interactions up to third nearest neighbors have been determined
Yavors’kii  _et al._ (2008). For simplicity, we do not investigate the effects
of pressure on weaker further neighbor interactions, and truncate the exchange
(but not the dipolar) interaction at the first nearest neighbor. We therefore
use as our starting point, the so-called s-DSM model with $J=3.72\textup{ K}$,
$D=1.41\textup{ K}$ Melko and Gingras (2004) for DTO and $J=1.56\textup{ K}$,
$D=1.41\textup{ K}$ for HTO Jaubert and Holdsworth (2011).

We model compression of the crystal with zero Poisson ratio, as suggested by
experiment Mito  _et al._ (2007). When atoms move closer, the wave function
overlap and corresponding exchange parameters are expected to change. We make
the assumption that $J(i,j)=J_{\perp}=J_{P=0}$ if the neighbors $i$ and $j$
are in a lattice plane perpendicular to the direction of applied pressure and
$J(i,j)=J_{k\parallel}$ if they are not. $J_{P=0}$ is the ambient pressure
value of the s-DSM model. $k$ labels the $6$ different directions of nearest
neighbor bonds and we make no a priori assumptions on the magnitude of
$J_{k\parallel}$. The dipolar interactions become stronger along the direction
of compression, growing as $|\textbf{r}_{ij}|^{-3}$, when distances decrease.
Furthermore, we make the assumption that the spins continue to point towards
the centers of the tetrahedra as the crystal deforms, altering the local
$[111]$-axis. This change of spin directions leads to a change in both
exchange and dipolar interactions, due to the dot products in Eq. (1).

Motivated by experiments Mito  _et al._ (2007), we model pressure applied
along the $[001]$, $[111]$ and $[110]$ directions. By symmetry, for the first
two directions, it suffices to model the nearest neighbor interactions with
two parameters, $J_{\perp}$ and $J_{\parallel}$. For the $[110]$ direction,
however, the lower symmetry of the lattice requires us to use three different
exchange parameters, $J_{\perp}$, $J_{1\parallel}$ and $J_{2\parallel}$. We
illustrate these different exchange interactions for a single tetrahedron in
Fig. 1 for the three respective directions of pressure. ($J_{(k)\parallel}$
need not be the same for the different directions of pressure.)

Demagnetizing effects in susceptibility measurements can, under ambient
pressure, be minimized by using needle-shaped samples. However, such samples
are likely to break under pressure, and less elongated samples are used in
practice. It is therefore important to consider the demagnetizing
transformation

$B_{\textup{ext}}=B_{\textup{int}}+NM.$ (2)

which gives the relationship between the internal field, $B_{\textup{int}}$,
the external field, $B_{\textup{ext}}$, the magnetization, $M$, and the
demagnetizing factor of the sample, $N$. Note that $N$ depends on the shape of
the sample and therefore changes upon compression. We find that this effect
cannot be ignored in an accurate description of the experiments Mito  _et al._
(2007). Hence the uniaxial pressure directly alters both microscopic
properties; such as spin directions and interaction parameters, and
macroscopic properties; such as the magnetization and shape of the sample.

In previous magnetization measurements, it was found that the signatures of
pressure is largest when pressure is applied along the $[001]$ direction Mito
_et al._ (2007). Earlier theoretical work predicts a low-temperature
transition into an ordered phase carrying net magnetization along this axis
Jaubert  _et al._ (2010). This motivates us to make an extended study of the
zero temperature phases for our model in the $[001]$ direction to investigate
the system at low temperature.

###  II.2 Simulation method and demagnetization corrections

Monte Carlo simulations using single spin flip and loop updates are used to
investigate a number of different system sizes with periodic boundary
conditions according to the Metropolis-Hastings algorithm. We use the 16
particle standard cubic unit cell Melko and Gingras (2004). All super-cells
used are cubic of size $L^{3}$, $L\in[1,\cdots,8]$. Ewald summation is used to
effectively account for the long range conditionally convergent dipolar
contributions Frenkel and Smit (2001). The geometrical compression is
introduced by an appropriate linear coordinate transformation depending on the
direction of applied pressure. In particular, the Ewald summation is done
using sheared lattice vectors for the reciprocal space contribution.

In order to account for the macroscopic boundary effects, demagnetization
transformations are made by modeling the sample as a prolate ellipsoid. The
major axes of the ellipsoid are given by the dimensions of the samples used in
the experiment. For the DTO experimentMito  _et al._ (2007), only the shape of
the $[001]$ and $[110]$ crystals could be recovered; $1.77\times 1.80\times
1.61\textup{ mm}^{3}$ ($[001]$ pressure along the side of length $1.61\textup{
mm}$), and $1.28\times 0.96\times 1.63\textup{ mm}^{3}$ ($[110]$ pressure
along the side of length $1.63\textup{ mm}$) Mito (2018). These dimensions
give $N_{001}=0.319$ and $N_{110}=0.426$ respectively in the high-temperature
limit A Osborn (1945), and we have not taken the temperature dependence of the
demagnetization factor into account Twengström  _et al._ (2017). The sample
dimensions are deformed in accordance with the microscopic compression of the
lattice along the direction of pressure. The dimensions for the sample
measured in the $[111]$ direction were unavailable and we assume the shape to
be a perfect sphere with $N_{111}=1/3$ under ambient pressure.

###  II.3 Experimental Methods

Magnetic neutron diffraction experiments were performed at the Institut Laue
Langevin (ILL) using the polarized diffuse scattering instrument, D7 Stewart
_et al._ (2008); Deen  _et al._ (2018). Elastic neutron diffraction profiles
with uniaxial polarization analysis (neutron polarization along the vertical
axis) were measured for HTO under uniaxial pressure with an incident
wavelength $\lambda$ = 4.86 Å. Polarisation analysis on D7 measures spin flip
and non-spin flip contributions. In the limit of negligible background and
nuclear spin incoherent scattering, the spin flip scattering will have only
the magnetic scattering given by Eq. (4). The data have been corrected for
detector and polarization analyzer efficiencies using standard samples of
vanadium and amorphous silica respectively Stewart  _et al._ (2008).

Uniaxial pressure was applied to HTO crystals using a Cu-Be anvil-type
pressure cell optimized for uniaxial pressure (details of the cell will be
published elsewhere L. Ø. Sandberg  _et al._ (2019)). Temperature control was
achieved using a liquid helium cryostat. The samples were cylindrical shaped
single crystals, $V=10$ mm3, cut from the larger image-furnace-grown crystals,
with a surface alignment better than $0.1^{\circ}$. The pressure applied was
determined via force gauge calibrations considering low temperature changes to
the cell with a maximum pressure of 2.2(0.5) GPa.

The crystals were synthezised using an image furnace located at the Univerity
of Lund with ambient oxygen conditions. Structural neutron diffraction
measurements on ZEBRA of the Paul Scherrer Institute, Switzerland, showed an
absence of structural diffuse scattering, which would have indicated oxygen
defect clusters Sala  _et al._ (2014).

##  III Theoretical and Experimental Results

###  III.1 Magnetization

The magnetization of DTO single crystals under applied uniaxial pressure and
magnetic field has earlier been measured at $T=1.7\textup{ K}$ Mito  _et al._
(2007). The magnetic field ranged up to $3\textup{ T}$ and was parallel to the
axis of pressure. Measurements were performed with pressure along the $[001]$,
$[111]$ and $[110]$ directions. We adjust the parameters of the proposed
model, Eq. (1), to match the experimental data. The best match between
experiment and model is shown in Fig. 2, and next we describe the details of
the modeling procedure.

Figure 2: Relative increase in the sample magnetization $\Delta
M/M=(M_{P>0}-M_{P=0})/M_{P=0}$ when field and pressure are both applied in the
$[001]$, $[111]$ and $[110]$ directions at $1.7\textup{ K}$. Experimental
resultMito  _et al._ (2007) (dotted) and our fit by MC simulation for a system
of 8192 spins (solid).

We assume that the magnetic moments retain their Ising symmetry under
pressure, and that a sufficiently strong field aligns the moments. For such a
field, changes in the demagnetization factor or in the interaction energies
will have no effect on the magnetization. The difference in saturation
magnetization observed when applying pressure in different directions is
purely due to lattice compression which changes the angle of the local
[111]-axis. The components of the magnetic moments parallel to the direction
of the applied field decreases, which reduces the saturation magnetization
along the relevant direction. From the value of the saturation magnetization
we can therefore determine the compression $\kappa$. We find that the crystals
deform by $\kappa^{001}=1.34\%$, $\kappa^{111}=3.3\%$ and $\kappa^{110}=1.0\%$
for the $[001]$, $[111]$ and $[110]$ directions respectively. This gives the
demagnetization factors $N_{001}^{\kappa}=0.322$, $N_{110}^{\kappa}=0.433$ and
$N_{111}^{\kappa}=0.341$ under compression, for the respective directions.

The data for the $[001]$ direction appears saturated already at about $1$ T,
while we assume that the data for the $[111]$ direction has saturated at 3 T.
For the $[110]$ direction the experimental data has not saturated at high
field. We find this result peculiar since the configuration that minimizes the
Zeeman energy can be reached without leaving the spin ice manifold for fields
along $[110]$ and $[001]$, while this is not so for the $[111]$ direction. We
would expect that a stronger field is needed to create the necessary monopole
excitations for full $[111]$ saturation, but this is not reflected by the
experiment. Due to this anomaly in the high field behavior we take the value
at $1.5\textup{ T}$, which would firmly saturate the crystal in the $[001]$
direction, to be the saturation value for the $[110]$ direction, resulting in
a compression of $1\%$.

Figure 3: Simulated values of magnetization plotted against temperature for
1024 spins. Ambient pressure (black) together with uniaxial pressure along
$[110]$ ($1.20$ GPa, green), $[001]$ ($1.05$ GPa, red) and $[111]$ ($1.30$
GPa, blue).

With the compression and demagnetization factor fixed, the $J_{\parallel}$
exchange parameter is adjusted in order to give the best fit for the measured
magnetization as a function of external field; these data are shown as
continuous lines in Fig. 2. It was found that for both the $[001]$ and $[111]$
directions, the $J_{\parallel}$ coupling increases from the original value of
$J_{\parallel}=3.72$ K. Our best estimate gives
$J^{001}_{\parallel}=3.91\textup{ K}$ and $J^{111}_{\parallel}=4.31\textup{
K}$. For the $[110]$ direction, the parameters decrease and it is the
difference $J^{110}_{1\parallel}-J^{110}_{2\parallel}$ that determines the
zero field limit in magnetization change. Our best fit gives
$J^{110}_{1\parallel}=3.29\textup{ K}$ and $J^{110}_{2\parallel}=3.35\textup{
K}$. Assuming that the exchange parameters are independent of temperature we
find that, for all directions, the ground state of our model is a dipolar spin
chain state with zero net magnetization, which is also the expected ground
state of the dipolar spin ice model under ambient pressureMelko and Gingras
(2004); Henelius  _et al._ (2016).

This is illustrated in Fig. 3, which shows how the finite size absolute
magnetization drops to zero in a first order transition. The transition
temperature increases with pressure, especially when pressure is applied in
the $[001]$ and $[110]$ directions. The twelve-fold degeneracy of the ground
state is lifted due to the change of symmetry in the deformed unit cell. With
pressure along $[001]$, the ground state manifold is split into two
submanifolds. One with chains perpendicular to the $[001]$ direction,
$\textup{M}_{\perp}$, and one with chains having a component along $[001]$,
$\textup{M}_{\parallel}$, see Fig. 4. Our fit to magnetic susceptibility data
for DTO, finds that the $\textup{M}_{\parallel}$ state has the lowest energy
of the two possible states.

Figure 4: The degeneracy is lifted under application of pressure. For the
$[001]$ direction, there exists two submanifolds of ground states; (a)
$M_{\perp}$ and (b) $M_{\parallel}$. Red and blue bonds corresponds to
$J_{\parallel}$ and $J_{\perp}$ respectively. Only up-tetrahedrons are drawn
for clarity. Grey lines indicate the directions of the spin chains.

###  III.2 Ground states for pressure along $[001]$

In Fig. 5, we illustrate schematically the ground states for our spin ice
model, Eq. (1), when varying $J_{\parallel}$ and lattice compression for
pressure applied in the $[001]$ direction. We find that there are four
possible ground states: (a) the all-in-all-out state (AFM), (b),(c) The spin
chain states ($\textup{M}_{\perp}$, $\textup{M}_{\parallel}$) and (d) the
earlier proposed ferromagnetic state (FM). We introduce the critical values
$J_{c1},J_{c2},J_{c3}$ and $J_{c1}^{\kappa},J_{c2}^{\kappa},J_{c3}^{\kappa}$
denoting the phase boundaries without and with compression respectively. For
DTO without compression, the state boundaries are at $J_{c1}=9.02\textup{ K}$,
$J_{c2}=3.72\textup{ K}$, $J_{c3}=3.33\textup{ K}$. For DTO at $1.05\textup{
GPa}$, inducing $\kappa^{001}=1.34\%$ compression, we have
$J_{c1}^{\kappa}=9.21\textup{ K}$, $J_{c2}^{\kappa}=4.02\textup{ K}$,
$J_{c3}^{\kappa}=3.64\textup{ K}$.

Figure 5: Ground states of the model schematically plotted with varying
$J_{\parallel}$ in the $[001]$ direction. The values $J_{c1},J_{c2},J_{c3}$
and $J_{c1}^{\kappa},J_{c2}^{\kappa},J_{c3}^{\kappa}$ denote the phase
boundaries without and with compression respectively. We plot instances of the
ground states in the leftmost column for the 16 particle unit cell. Red and
blue bonds corresponds to $J_{\parallel}$ and $J_{\perp}$ respectively. Only
up-tetrahedrons are drawn for clarity.

In previous work Jaubert  _et al._ (2010), spin ice was modeled for pressure
in the $[001]$ direction with effective nearest-neighbor interactions,
$\mathcal{H}=-\sum_{\langle
i,j\rangle}J_{\textup{eff}}(i,j)\,\mathbf{S}_{i}\cdot\mathbf{S}_{j}$, but
without dipolar interactions and no lattice compression in the Hamiltonian. It
was found by fitting to the experimental data that
$J_{\parallel\textup{eff}}>J_{\perp\textup{eff}}$, and that this results in a
ferromagnetic phase transition at low temperatures.

(a) $\kappa^{001}=0$

(b) $\kappa^{001}=1.34\%$

Figure 6: Ground state phase diagram in terms of normalized interactions,
$J/D$. Results are shown for both zero and $1.34\%$ compression; the value
suggested by the DTO magnetization data. Blue, light red, and grey regions
represent the $M_{\perp}$, AFM, and FM states, respectively. The
$M_{\parallel}$ region is very narrow and shown in dark red close to the
$J_{\perp}=J_{\parallel}$ line. The phase boundaries shift with compression
and $J_{\parallel}$ increases with pressure. We include also the ambient
pressure value for HTO.

In this work, we find that with the inclusion of dipolar interactions the
picture is more complicated, since the $M_{\parallel}$ state will compete with
the FM state. The outcome of this is that a larger difference between
$J_{\parallel\textup{eff}}$ and $J_{\perp\textup{eff}}$ is needed for the
ferromagnetic transition to occur. At the nearest neighbor distance the
interaction in the s-DSM model is $J_{\textup{eff}}=5D-J$, and we note that in
order for $J_{\textup{eff}}$ to increase, $J$ must decrease. We find that the
ground state of the system is FM only when
$J_{\parallel\textup{eff}}>5D-J_{c3}>5D-J_{\perp}=J_{\perp\textup{eff}}$,
instead of immediately when $J_{\parallel\textup{eff}}>J_{\perp\textup{eff}}$.
Furthermore, with the inclusion of lattice compression and the associated
change of the local $[111]$-axis, we find, when fitting to the magnetization
measurements, that $J_{\parallel}$ increases as a function of pressure, rather
than decreases. This change seemingly moves away from, rather than towards,
the FM boundary, as illustrated in the left hand side of Fig. 5. However, a
further complication is that the critical value for the phase boundary
$J^{\kappa}_{c3}$ also increases with lattice compression (right hand side of
Fig. 5). So it may still be possible for the system to display the
ferromagnetic transition, but this would require much higher pressure than
what has been applied in the current measurements. By linear extrapolation,
$J_{\parallel}$ for DTO would cross the zone boundary at $3.4\textup{ GPa}$
and the FM transition would occur at low temperatures.

In Fig. 6 we show the general ground state phase diagram as a function of
normalized $J_{\perp}$ and $J_{\parallel}$ interactions. Fig. 6(a) depicts the
ground states of the system with no lattice compression. In this case
$J_{\parallel}=J_{\perp}$ defines the boundary between the
$\textup{M}_{\perp}$ and $\textup{M}_{\parallel}$ phases. When adding
compression, the phase boundaries are shifted, which is illustrated as we plot
the phase diagram for the experimentally relevant compression in Fig. 6(b). In
particular, the $\textup{M}_{\parallel}$ region moves, breaking the
$M_{\perp}-M_{\parallel}$ degeneracy for $J_{\parallel}=J_{\perp}$ in favor of
the $\textup{M}_{\parallel}$ states. The overall behavior is the same, with an
AFM ground state for large $J_{\parallel}$ and a FM ground state for small
$J_{\parallel}$.

###  III.3 Prediction of neutron diffraction results for DTO

Neutron scattering is an ideal and unique tool to extract the spin-spin
correlations in materials Marshall and Lovesey (1971). This enables a close
comparison of experimental measurements and theoretically calculated
correlations, a combination which often plays a crucial role in the
understanding of magnetic materials at a microscopic level.

Using parameter values relevant for DTO, we sample the thermal average of the
simulated spin-spin correlations and calculate the magnetic structure factor
for neutron diffraction, $S_{\textup{mag}}$, according to

$S_{\textup{mag}}(\textbf{Q})=\frac{[f(|\textbf{Q}|)]^{2}}{N}\sum_{ij}\left\langle\textbf{S}^{\perp}_{i}\cdot\textbf{S}^{\perp}_{j}\right\rangle
e^{i\textbf{Q}\cdot\textbf{r}_{ij}},$ (3)

where Q is the scattering wave-vector, $N$ is the number of spins in the
simulation cell, $f(|\textbf{Q}|)$ is the magnetic form factor, and the spin
perpendicular component is given by
$\textbf{S}_{i}^{\perp}=\textbf{S}_{i}-\textbf{S}_{i}\cdot\textbf{Q}/|\textbf{Q}|$.
We calculate also the structure factor for magnetic spin flip scattering,
relevant for experiments with neutron polarization analysis $S_{\textup{SF}}$
according to

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

where P is the normalized polarization direction of the incident neutron beam,
and $\textbf{P}\perp\textbf{Q}$ Blume (1963). In our experiment, the neutron
polarization direction is parallel to the direction of applied pressure and we
calculate the neutron scattering profiles for the plane in reciprocal space
perpendicular to the direction of the applied pressure.

Based on the model parameters determined from the magnetization measurements
in the previous section, our prediction for the relative increases in
$S(\textbf{Q})$, $\Delta
S(\textbf{q})/S(\textbf{q})=(S_{P>0}(\textbf{q})-S_{P=0}(\textbf{q}))/S_{P=0}(\textbf{q})$,
at $T=1.7\textup{ K}$ and pressures, $P_{001}=1.05\text{ GPa}$,
$P_{110}=1.20\text{ GPa}$, $P_{111}=1.30\text{ GPa}$ relevant for DTO are
shown in Fig. 7. We find significant variations in the relative scattering
intensities for the different scattering planes both for the magnetic and
polarized spin flip cross sections. The largest variation can be seen for
pressure along the $[001]$ direction with a relative increase of about $8\%$
in the $[001]$ spin flip channel of the $(h,k,0)$ plane, found for scattering
vectors close to (2,0,0) and symmetry related points in reciprocal space.

(a) $[001]$ Magnetic.

(b) $[001]$ SF.

(c) $[110]$ Magnetic.

(d) $[110]$ SF.

(e) $[111]$ Magnetic.

(f) $[111]$ SF.

Figure 7: Predictions for the relative difference in the total magnetic
scattering and the spin flip scattering by MC simulation on 8192 spins.
Exchange parameters and lattice compression are taken from the fit to
magnetization measurements on DTO Fig. 2. In all cases, the reciprocal plane
is perpendicular to the direction of pressure and polarization is parallel
with the direction of pressure. (a),(b) pressure along $[001]$ ($1.05$ GPa),
(c),(d) pressure along $[110]$ ($1.20$ GPa), and (e),(f) pressure along
$[111]$ ($1.30$ GPa), all at zero field and $T=1.7\text{ K}$.

###  III.4 Neutron Diffraction Experiment on HTO

In order to make a close comparison with the theoretical predictions of Fig. 7
we would ideally like to perform neutron scattering measurements on DTO
crystals. However, the high neutron absorption cross section of natural
dysprosium, cost of isotopically enriched samples, and the concurrent high
probability of crystals cracking during the application of uniaxial pressure
make these experiments inordinately expensive and challenging. Instead, we
have chosen to perform neutron scattering measurements on crystals of HTO,
which share many low temperature properties with DTOLacroix  _et al._ (2011).

Motivated by the theoretical results, uniaxial pressure was applied along the
$[001]$ crystalline axis with $[001]$-polarized neutrons used to probe the
$(h,k,0)$ plane. The axis perpendicular to the applied pressure is left
unconstrained and therefore can give rise to the Poisson effect, in contrast
to the previous magnetization measurementsMito  _et al._ (2007). However,
within the $Q$-resolution of the instrument Fennell  _et al._ (2017), no
indicative change in the lattice parameters perpendicular to the pressure
could be observed.

Figure 8: Relative increases in the $[001]$ spin flip channel when pressure is
applied along the $[001]$ direction. Left half: Estimate based on MC
simulations of the model for 8192 spins ($J_{\perp}=1.56\textup{ K}$,
$J_{\parallel}=1.975\textup{ K}$, $\kappa=2.8\%$). Right half: Experimental
neutron scattering data after cell correction. Data taken at $1.5\textup{ K}$
and $2.2\textup{ GPa}$. The circles drawn mark regions of increased intensity
for high $Q$ in the experimental data and are plotted in symmetry equivalent
places for the theoretical estimate.

We construct a theoretical estimate of the change in scattering intensity.
Since we do not know the pressure dependent susceptibility or magnetization
data for HTO we are not able to make a detailed determination of the evolution
of $J_{\parallel}$ and $\kappa$ with pressure in HTO (in contrast to DTO).
However, since the two materials are chemically similar, and in order to make
a rough estimate, we assume that the compression is linear in the applied
pressure and that their compressibilities are identical. Using the fitted
value for DTO ($\kappa=1.34\%$ at $1.05\textup{ GPa}$) then gives a
compression of $\kappa=2.8\%$ for HTO at $2.2\textup{ GPa}$. The magnetic ions
in HTO and DTO are significantly different, and it would be unreasonable to
use the similarity argument also for the exchange interaction. Instead we
adjust the single parameter $J_{\parallel}$ for HTO to give the best possible
match with experiment.

A comparison between the theoretical prediction and the experimental results
for the relative increase in the magnetic structure factor under $2.2\textup{
GPa}$ pressure, is shown in Fig. 8. A fourfold rotational averaging,
consistent with the crystalline symmetry, has been performed on the
experimental data. The experimental data have been corrected for a strong
background contribution from the pressure cell L. Ø. Sandberg  _et al._
(2019). This background dominates the low angular region of reciprocal space,
leaving this region rather poorly sampled.

There is a reasonable correspondence between the measurement and the
theoretical prediction in the region corresponding to wave-vector transfers
$Q>1.5\textup{ \r{A}}^{-1}$, where the high-intensity regions are marked by
circles. Further experimental activity will be required to improve the
background from the pressure cell and reliably access the $Q<1.5\textup{
\r{A}}^{-1}$ region. In particular it is of interest to gather more statistics
and to improve the construction and geometry of the cell to get a cleaner
signal and better control of the pressure. The measurement of further
crystallographic directions would also be of great value.

We note that the data from HTO has weaker features than those predicted for
DTO, Fig. 7(b). This can be accounted for by the tuning of the $J_{\parallel}$
parameter. Both changes in $\kappa$ and changes in $J_{\parallel}$ give rise
to a cross/square pattern in the relative increases in the scattering, which
are not prominent in the experimental data. The $J_{\parallel}$ parameter can
be tuned such that these features are cancelled to a large degree. We choose
it to get a profile as similar to the experiment as possible. Fig. 9 shows the
evolution of the relative increases in the scattering (left half) when varying
$J_{\parallel}$, together with the experimental data (right half). Note that
the intensity in the cross/square pattern inverts as we increase
$J_{\parallel}$ by only $3\%$, from $1.96\textup{ K}$ to $2.00\textup{ K}$.

(a) $J_{\parallel}=1.94\textup{ K}$, $\kappa=2.8\%$.

(b) $J_{\parallel}=1.96\textup{ K}$, $\kappa=2.8\%$.

(c) $J_{\parallel}=1.98\textup{ K}$, $\kappa=2.8\%$.

(d) $J_{\parallel}=2.00\textup{ K}$, $\kappa=2.8\%$.

Figure 9: Evolution of theoretical predictions for the relative increases in
the $[001]$ spin flip channel when pressure is applied along the $[001]$
direction. Left half: MC simulation of the model for 8192 spins with varying
$J_{\parallel}$ and $\kappa$ fixed. Right half: Experimental data (constant).
We see that the cross/square pattern is inverted when we increase
$J_{\parallel}$ from $1.94\textup{ K}$ to $2.00\textup{ K}$. The circles drawn
mark regions of increased intensity for high $Q$ in the experimental data and
are plotted in symmetry equivalent places for the theoretical estimate.

We determine that the closest fit has $J_{\parallel}=1.975\sim 1.98\textup{
K}$, about $27\%$ larger than $J_{\perp}$, in contrast to
$J_{\parallel}/J_{\perp}-1=5\%$ seen for DTO. The fact that the experimental
profile does not have a strong cross/square pattern implies, within our model,
that the ground state of HTO lies close to the border between the $M_{\perp}$
and $M_{\parallel}$ states. This is because the inversion point of the
cross/square pattern is at the border between the $M_{\parallel}$ and
$M_{\perp}$ phases in the ground state phase diagram, Fig. 6.

##  IV Discussion and Conclusion

Using a dipolar spin ice model, we have accurately modeled the experimental
changes in magnetization of DTO under the application of pressure in multiple
crystalline directions. Using the model parameters derived from the
magnetization measurements we have calculated the correspondingly expected
changes in neutron scattering cross section for DTO. Neutron scattering
experiments were performed on HTO under applied pressure, and we found the
response to be weaker than what we predicted for DTO. One possible reason for
the difference is that HTO may be located at the boundary between two
different chain states, as discussed in the previous section. Another is that
the compressibility of HTO may be smaller than that of DTO, and that the
associated pressure induced changes in the exchange parameters are also
smaller. Furthermore, we have not considered changes to exchange parameters
past the nearest neighbors, which certainly could influence the results
Yavors’kii  _et al._ (2008); Henelius  _et al._ (2016). In conclusion, we note
that in order to determine which of these scenarios are relevant for HTO, more
experiments are necessary. In particular similar magnetization measurements of
HTO under pressure, as previously were performed on DTO, would be highly
useful.

###### Acknowledgements.

We thank M. Mito and K. Matsuhira for useful discussions and sharing their
data. The neutron scattering experiments were performed at the Paul Scherrer
Institute and the Institute Laue-Langevin. The project was supported by
Nordforsk through the program NNSP and by the Danish Agency for Research and
Innovation through DANSCATT.

## References

  * Lacroix  _et al._ (2011) C. Lacroix, P. Mendels, and F. Mila, eds.,  _Highly Frustrated Magnetism_ , Springer Series in Solid-State Sciences, Vol. 164 (Springer, 2011).
  * Castelnovo  _et al._ (2008) C. Castelnovo, R. Moessner, and S. L. Sondhi, Nature 451, 42 (2008).
  * Brooks-Bartlett  _et al._ (2014) M. E. Brooks-Bartlett, S. T. Banks, L. D. C. Jaubert, A. Harman-Clarke, and P. C. W. Holdsworth, Phys. Rev. X 4, 011007 (2014).
  * Henley (2005) C. L. Henley, Phys. Rev. B 71, 014424 (2005).
  * Pomaranski  _et al._ (2013) D. Pomaranski, L. R. Yaraskavitch, S. Meng, K. A. Ross, H. M. L. Noad, H. A. Dabkowska, B. D. Gaulin, and J. B. Kycia, Nature Physics 9, 353 (2013).
  * Henelius  _et al._ (2016) P. Henelius, T. Lin, M. Enjalran, Z. Hao, J. G. Rau, J. Altosaar, F. Flicker, T. Yavors’kii, and M. J. P. Gingras, Phys. Rev. B 93, 024402 (2016).
  * Melko and Gingras (2004) R. G. Melko and M. J. P. Gingras, Journal of Physics: Condensed Matter 16, R1277 (2004).
  * Mito  _et al._ (2007) M. Mito, S. Kuwabara, K. Matsuhira, H. Deguchi, S. Takagi, and Z. Hiroi, Journal of Magnetism and Magnetic Materials 310, e432 (2007), proceedings of the 17th International Conference on Magnetism.
  * Mirebeau and Goncharenko (2004) I. Mirebeau and I. Goncharenko, Journal of Physics: Condensed Matter 16, S653 (2004).
  * Jaubert  _et al._ (2010) L. D. C. Jaubert, J. T. Chalker, P. C. W. Holdsworth, and R. Moessner, Phys. Rev. Lett. 105, 087201 (2010).
  * Bramwell  _et al._ (2009) S. T. Bramwell, S. R. Giblin, S. Calder, R. Aldus, D. Prabhakaran, and T. Fennell, Nature 461, 956 (2009).
  * den Hertog and Gingras (2000a) B. C. den Hertog and M. J. P. Gingras, Phys. Rev. Lett. 84, 3430 (2000a).
  * Yavors’kii  _et al._ (2008) T. Yavors’kii, T. Fennell, M. J. P. Gingras, and S. T. Bramwell, Phys. Rev. Lett. 101, 037204 (2008).
  * den Hertog and Gingras (2000b) B. C. den Hertog and M. J. P. Gingras, Phys. Rev. Lett. 84, 3430 (2000b).
  * Isakov  _et al._ (2005) S. V. Isakov, R. Moessner, and S. L. Sondhi, Phys. Rev. Lett. 95, 217201 (2005).
  * Jaubert  _et al._ (2013) L. D. C. Jaubert, M. J. Harris, T. Fennell, R. G. Melko, S. T. Bramwell, and P. C. W. Holdsworth, Phys. Rev. X 3, 011014 (2013).
  * Jaubert and Holdsworth (2011) L. D. C. Jaubert and P. C. W. Holdsworth, Journal of Physics: Condensed Matter 23, 164222 (2011).
  * Frenkel and Smit (2001) D. Frenkel and B. Smit,  _Understanding Molecular Simulation: From Algorithms to Applications_, Computational science (Elsevier Science, 2001).
  * Mito (2018) M. Mito, Private Communication (2018).
  * A Osborn (1945) J. A Osborn, Physical Review 67, 351 (1945).
  * Twengström  _et al._ (2017) M. Twengström, L. Bovo, M. J. P. Gingras, S. T. Bramwell, and P. Henelius, Phys. Rev. Materials 1, 044406 (2017).
  * Stewart  _et al._ (2008) J. R. Stewart, P. P. Deen, K. H. Andersen, H. Schober, J.-F. Barthélémy, J. M. Hillier, A. P. Murani, T. Hayes, and B. Lindenau, Journal of Applied Crystallography 42, 69 (2008).
  * Deen  _et al._ (2018) P. Deen, I.M.Bakke, R. Edberg, H. Fjellvåg, K. Leffmann, L. Ø. Sandberg, and A. Wildes, ILL (2018), URL https://doi.ill.fr/10.5291/ILL-DATA.5-32-856.
  * L. Ø. Sandberg  _et al._ (2019) L. Ø. Sandberg, M. L. Haubro, M. A. Olsen, M. Guthrie, more, K. Lefmann, and P. P. Deen, Review of Scientific Instruments (in preparation) (2019).
  * Sala  _et al._ (2014) G. Sala, M. J. Gutmann, D. Prabhakaran, D. Pomaranski, C. Mitchelitis, J. B. Kycia, D. G. Porter, C. Castelnovo, and J. P. Goff, Nature Materials 13, 488 (2014).
  * Marshall and Lovesey (1971) W. Marshall and S. W. Lovesey,  _Theory of thermal neutron scattering: the use of neutrons for the investigation of condensed matter_ (Clarendon Press Oxford, 1971).
  * Blume (1963) M. Blume, Phys. Rev. 130, 1670 (1963).
  * Fennell  _et al._ (2017) T. Fennell, L. Mangin-Thro, H. Mutka, G. Nilsen, and A. Wildes, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 857 (2017), 10.1016/j.nima.2017.03.024.