# Magnetotransport of Sm2Ir2O7 across the pressure-induced quantum-critical
phase boundary

M.J. Coak<EMAIL_ADDRESS>Department of Physics, University of Warwick,
Gibbet Hill Road, Coventry CV4 7AL, United Kingdom  London Centre for
Nanotechnology, University College London, Gordon St, London WC1H 0AH, United
Kingdom  School of Physics and Astronomy, University of Birmingham, Edgbaston,
Birmingham B15 2TT, United Kingdom  K. Götze  Department of Physics,
University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, United Kingdom
Deutsches Elektronen-Synchrotron (DESY), 22607 Hamburg, Germany  T. Northam De
La Fuente  Department of Physics, Royal Holloway, University of London, Egham
TW20 0EX, United Kingdom  Institut Laue-Langevin, CS 20156, 38042 Grenoble
Cedex 9, France  C. Castelnovo  Cavendish Laboratory, University of Cambridge,
JJ Thomson Ave, Cambridge CB3 0HE, United Kingdom  J. P. Tidey  Department of
Chemistry, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, United
Kingdom  J. Singleton  National High Magnetic Field Laboratory, Los Alamos
National Laboratory, MS-E536, Los Alamos, NM 87545, United States of America
A.T. Boothroyd  Department of Physics, University of Oxford, Clarendon
Laboratory, Oxford, OX1 3PU, United Kingdom  D. Prabhakaran  Department of
Physics, University of Oxford, Clarendon Laboratory, Oxford, OX1 3PU, United
Kingdom  P.A. Goddard<EMAIL_ADDRESS>Department of Physics,
University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, United Kingdom

###### Abstract

Rare-earth pyrochlore iridates host two interlocking magnetic sublattices of
corner-sharing tetrahedra and can harbour a unique combination of frustrated
moments, exotic excitations and highly correlated electrons. They are also the
first systems predicted to display both topological Weyl semimetal and axion
insulator phases [1]. We have measured the transport and magnetotransport
properties of single-crystal Sm2Ir2O7 up to and beyond the pressure-induced
quantum critical point (QCP) for all-in-all-out (AIAO) Ir order at $p_{{\rm
c}}$ = 63 kbar previously identified by resonant X-ray scattering [2]. Our
findings overturn the accepted expectation that the suppression of AIAO order
should lead to metallic conduction persisting down to zero temperature.
Instead, the resistivity-minimum temperature, which tracks the decrease in the
AIAO ordering temperature for pressures up to 30 kbar, begins to increase
under further application of pressure, pointing to the presence of a second
as-yet unidentified mechanism leading to non-metallic behavior. The
magnetotransport does track the suppression of Ir magnetism, however, with a
strong hysteresis observed only within the AIAO phase boundary, similar to
that found for Ho2Ir2O7 and attributed to plastic deformation of Ir domains
[3]. Around $p_{{\rm c}}$ we find the emergence of a new type of electronic
phase, characterized by a negative magnetoresistance with small hysteresis at
the lowest temperatures, and hysteresis-free positive magnetoresistance above
5 K. A Weyl semimetal phase is predicted to occur in the vicinity of the QCP.
The temperature dependence of our low-temperature transport data, however, are
found to be best described by a model consistent with a Weyl semimetal across
the entire pressure range.

The magnetic pyrochlore lattice, with spins arranged into a network of corner-
sharing tetrahedra, has a strong history of yielding new physical states,
discoveries and insights. The seminal observation of a spin-ice state in
Ho2Ti2O7 and Dy2Ti2O7 [4, 5] and the associated magnetic monopole picture for
its low-energy excitations [6, 7, 8, 9] have led to lasting interest in this
family of materials. The pyrochlore iridates add to the richness of these
compounds by introducing a second inter-penetrating magnetic lattice with its
own order, and interactions between the two. This complex magnetic behavior,
their highly correlated electronic states, and predictions of topological
properties, make pyrochlore iridates fascinating systems for investigating
exotic quantum phenomena.

The family of lanthanide pyrochlore iridates $Ln_{2}$Ir2O7, where $Ln$ is a
magnetic trivalent lanthanide ion, provide an ideal environment for
investigating fundamental magnetism and correlated-electron physics. They
share the crystal structure shown in Fig 1(a) — a cubic pyrochlore structure
with space group $Fd$–$3m$ [12, 10, 11]. There are two interwoven sub-lattices
of magnetic ions: a network of corner-sharing $Ln^{3+}$ tetrahedra with an
oxygen site in the center, and another of corner-sharing Ir4+ tetrahedra. Each
Ir site sits in the center of an oxygen octahedron. While Pr2Ir2O7 remains
paramagnetic to low temperatures, the Ir sublattices in most (if not all) of
the other members of this family undergo a transition to all-in-all-out (AIAO)
antiferromagnetic long-range order [13, 11, 14, 2, 15], as illustrated in Fig.
1(b). In this configuration, all the spins on one Ir tetrahedron point
radially outwards, and all those on its neighboring tetrahedra point inwards,
giving zero total moment.

Figure 1: (a) Ambient-pressure crystal structure of Sm2Ir2O7, based on
structural data from Refs [10, 11]. The crystal forms a cubic pyrochlore
structure, with space group $Fd$–$3m$. The Sm atoms form a network of corner-
sharing tetrahedra. Between these is a second network of corner-sharing
tetrahedra of Ir atoms shown in (b). Each Ir is surrounded by an oxygen
octahedron. The black arrows in (b) denote an example of the all-in-all-out Ir
magnetic order. The green arrow shows the direction of one of the
$\left[1\,1\,1\right]$ axes. The lattice parameter $a$ = 10.311(2) Å for
Sm2Ir2O7 at room temperature and ambient pressure.

In addition to the AIAO magnetism of the Ir ions, the rare-earth sublattice in
these materials may also show interesting and correlated behavior. In Dy2Ir2O7
and Ho2Ir2O7, the large rare-earth moments are constrained to lie along the
local $\left[1\,1\,1\right]$ directions and are coupled by both dipolar and
exchange interactions, which are overall ferromagnetic [16, 17, 3]. These
interactions favor the two-in-two-out (2I2O) configuration, promoting spin-ice
physics, akin to that seen in the frustrated pyrochlore titanates, and the
appearance of magnetic monopole quasiparticles [5, 18]. The added effect of
exchange with Ir nearest neighbors means that at sufficiently low temperatures
both Dy2Ir2O7 and Ho2Ir2O7 are expected to undergo a transition to a
fragmented monopole crystal [19, 16, 17]. Experiments on Ho2Ir2O7 at
intermediate temperatures show that the magnetoresistance is sensitive to the
density of magnetic monopoles and that the Ir AIAO domains can be manipulated,
via the Ir–Ho coupling, using an applied magnetic field [3]. The other members
of the family have considerably smaller moments on the rare-earth site and
their magnetic ground state remains the subject of debate.

The materials in this series additionally all share equivalent transport
behavior — metallic high-temperature resistivity and a resistivity minimum
$T_{\mathrm{min}}$ (described in the literature as a metal-insulator
transition temperature) below which the resistivity increases with cooling.
This $T_{\mathrm{min}}$ occurs at the same temperature as the Ir AIAO ordering
and has been attributed to a spin-orbit-coupling assisted Mott transition [20,
21]. The exception is $Ln$ = Pr, in which the AIAO order is absent and which
remains metallic at all temperatures. ARPES measurements have shown that in
the paramagnetic phase the band structure exhibits quadratic-band-touching
points [22, 23]. Perturbations to this electronic structure, such as symmetry
breaking due to magnetic order on the rare-earth sites, are predicted to give
rise to topologically non-trivial electronic states, including Weyl semimetals
(WSMs) and accompanying Fermi arcs [22], and evidence for such phenomena has
been reported extensively [25, 26, 20, 24, 27, 28].

There are two principal means to tune interactions in these materials.
Compressing the lattice parameter, $a$, will predominately strengthen the
hopping term $t$ in the Mott insulator description, while preserving local
spin environments. Chemical substitution of the rare-earth site, on the other
hand, mostly affects the oxygen coordinate parameter $x$. This parameter
describes distortions away from a perfect octahedral environment ($x$=0.3125)
around the Ir sites, towards a cubic environment around the $Ln$ sites
($x$=0.375) as its value increases (and the Ir-O-Ir bond angle decreases) [2].
The reduction in $x$ caused by decreasing $Ln$ ionic radius as one moves from
Gd $\rightarrow$ Eu $\rightarrow$ Sm $\rightarrow$ Nd reduces the local Ir
Ising anisotropy 111Contrary to the Sm magnetism, where the presence of a
strong single-ion anisotropy due to crystal electric field (CEF) effects is
robustly established, the single-ion behavior of the Ir magnetism is less well
understood. Here by ‘Ising anisotropy’ of the Ir moments we refer to the fact
that an AIAO order with well-defined local axes develops at low temperature
and pressure. It remains to be determined whether this is purely an effect of
the (anisotropic) direct Dzyaloshinskii-Moriya interaction combined with
(isotropic) AFM exchange, or whether a single-ion CEF contribution plays a
role., pushing down the AIAO ordering temperature [20, 2, 29, 30, 31]. This
suppresses $T_{\mathrm{min}}$ accordingly and reduces the resistivity towards
metallic behavior. The next compound in this progression, Pr2Ir2O7 [32, 33,
24], is then metallic at all temperatures and does not show any magnetic order
[32, 29, 2]. Reduction of either $a$ or $x$ has the result of suppressing the
Ir AIAO order, but through distinct mechanisms. It is intriguing to ask what
differences may result from the two approaches.

The strong sensitivity of the Ir ordering temperature, and the concomitant
resistivity minimum, to the fine details of the crystal structure, as well as
the many theoretical predictions of the presence of topological electronic
states in the phase diagram [34], means that controlled tuning of the lattice
is expected to be particularly enlightening in these materials. Applied
hydrostatic pressure provides a clean, continuous and reversible method for
achieving this tuning.

There have been several pressure studies reporting the evolution of
resistivity in $Ln_{2}$Ir2O7 samples. Most show a reduction in resistivity
with pressure as they are driven towards metallization. Polycrystalline
Nd2Ir2O7 reproduces the metallic behavior of Pr2Ir2O7 beyond 65 kbar of
applied pressure [20, 24]. The $T_{\mathrm{min}}$ transition is continuously
suppressed with pressure until it disappears at this point [31, 20, 24].
Magnetic field has also been shown to induce metallicity in Nd2Ir2O7 by
flipping the AIAO order to a 2-in-2-out state [35]. In Gd2Ir2O7,
$T_{\mathrm{min}}$ linearly increases with applied pressure [31]. In Eu2Ir2O7
there are reports of a slight increase of $T_{\mathrm{min}}$ under pressures
up to 22 kbar in polycrystalline pellets [31] or very little change up to 120
kbar in single crystals [36].

The differences in this behavior across the family are likely due to the
interplay of the competing changes in $x$ and $a$ lattice parameters induced
in each compound. Sm2Ir2O7, which in the $Ln$ series lies between Eu and Nd
and hence between these two regimes of different pressure dependence, is
therefore an interesting material for investigation.

The poly- or single-crystalline nature of the samples in this family is known
to be important. Polycrystalline samples of $Ln_{2}$Ir2O7 can differ in
transition temperature and form of resistivity to single-crystal samples [37,
26, 17]. These differences likely arise from slight variations in composition,
the effects of the broken symmetry at the surface or the restricted magnetic
domain size in polycrystalline samples, the random orientation of individual
grains, and the unknown transport properties of inter-grain boundaries present
in pressed pellets. For these reasons, conclusions derived from transport
measurements on polycrystalline samples should be treated with caution. Single
crystals are indispensable for true clean tests of the physics in these
systems, and for direction-sensitive magnetotransport measurements, but have
only recently been successfully grown for many of these materials [17, 3, 2,
37].

Single crystals of Sm2Ir2O7 exhibit a high-temperature metallic phase (in
which resistivity drops with decreasing temperature). A transition to AIAO Ir
order occurs around $T_{\mathrm{AIAO}}$ = 115 K at ambient pressure, and the
associated $T_{\mathrm{min}}$ is clearly seen in transport data as a
resistivity minimum below which resistivity rises with decreasing temperature
[29, 31]. There is some evidence from muon-spin relaxation measurements that
the Sm moments undergo ordering, possibly short-range, at temperatures below
about 10 K [38], but the details remain unknown. Recently Wang _et al._
showed, using resonant X-ray diffraction on single crystals, that the Ir AIAO
ordering temperature is reduced to a quantum critical point (QCP) at a
pressure $p_{{\rm c}}$ = 63 kbar [2]. Above $p_{{\rm c}}$ a metallic state is
expected once the magnetic transition responsible for driving insulating
behavior is removed, similar to that seen in Pr2Ir2O7 or in Nd2Ir2O7 in
applied fields or pressures [31, 20, 24]. There are also several predictions
that Sm2Ir2O7 hosts WSM states [1, 21, 39, 11, 2], most likely in proximity to
the quantum-critical region close to where the magnetic order is suppressed
[24, 2]. Initial transport measurements on polycrystalline samples have indeed
shown that $T_{\mathrm{min}}$ drops gradually in Sm2Ir2O7 as pressure is
applied in the low pressure region up to 20 kbar [31]. However, so far no
measurements of electronic properties have been performed in the vicinity of
the QCP, nor have any transport experiments been performed on single crystals
of this material.

In this work, we report magnetotransport measurements on a high-quality
single-crystal sample of Sm2Ir2O7 in fields up to 15 T and pressures up to and
beyond the 63 kbar quantum critical point. Our magnetoresistance data show
that the AIAO Ir order is indeed suppressed at this pressure, in line with the
X-ray measurements [2]. However, while the resistance drops dramatically with
pressure, the $T_{\mathrm{min}}$ resistivity minimum does not fall to zero
temperature with the magnetic order as expected. Instead, for pressures above
about 30 kbar the high-temperature metallic state is disrupted upon cooling by
another process, which becomes more robust as pressure is increased further.
Moreover, the magnetotransport changes significantly in the vicinity of the
QCP, indicative of the emergence of a new low-temperature electronic state.
Finally, a model consistent with Weyl semimetallic behavior is found to best
describe the temperature dependence of the low-temperature resistivity across
the entire pressure range. The temperature window over which the model
effectively reproduces the data is largest at the lowest pressures.

Figure 2: Magnetic susceptibility data (dark and light blue symbols) of a
single crystal of Sm2Ir2O7 in an applied field of 0.1 T parallel to
$\left[1\,1\,1\right]$, measured as a function of temperature after cooling in
zero field (ZFC) and with the field applied (FC). The Ir AIAO ordering
temperature where the ZFC and FC data bifurcate, $T_{\mathrm{AIAO}}$, is
marked with an arrow. The inset shows magnetization data (purple symbols)
against field at 0.9 K for a powder sample measured to 60 T in a pulsed
magnet. In both panels the black line is the result of single-ion calculations
of the Sm3+ moment taking into account the crystal-field environment as
described in the text.

## Results

### Magnetic properties

The main panel of Fig. 2 shows the magnetic susceptibility of Sm2Ir2O7,
measured in a $\left[1\,1\,1\right]$ magnetic field of 0.1 T while warming
after cooling in zero field (ZFC) and after cooling in the same 0.1 T field
(FC). The temperature where the FC and ZFC data bifurcate, $T_{\mathrm{AIAO}}$
=110(2) K, coincides with the onset of the Ir AIAO ordered state, in agreement
with the results of previous magnetic measurements [12], resonant X-ray
diffraction [11, 2] and muon-spin rotation [38] studies. The FC susceptibility
data continue to rise as temperature is reduced further. A hump is seen in the
data at temperatures just below 150 K, and an additional shoulder-like feature
appears in the FC data centered at about 60–70 K. There is no strong evidence
in these data for long-range magnetic order on the Sm sublattice below 10 K,
as suggested by Ref. [38] from features in the muon-spin relaxation rate.
Instead, our data are consistent with the presence of paramagnetic Sm moments
at temperatures down to 2 K.

The inset of Fig. 2 shows the field dependence of the magnetization $M(H)$ at
low temperature. The data were measured on a powder sample using extraction
magnetometry [40] at the National High Magnetic Field Laboratory, Los Alamos,
in pulsed fields up to 60 T and temperatures down to 0.9 K. As the Ir moments
are ordered below 110 K, we expect that these data are dominated by the
paramagnetic Sm moments.

Initial crystalline electric field (CEF) calculations using a point-charge CEF
Hamiltonian and the Stevens’ operator formalism (see Supplemental Appendix,
Section 4, appended to this document), show that the Sm3+ ground-state doublet
wavefunction is approximately $|J,m_{J}\rangle=|5/2,\pm 3/2\rangle$. The
magnetic moment of this approximate ground state is expected to be 0.43
$\mu_{\rm B}$ at $T=0$. Calculations of the single-ion properties for Sm2Ir2O7
can be improved by employing intermediate coupling and allowing mixing of
higher $J$ levels, as implemented in the SPECTRE software [41]. This method
was used in Ref. [43] to obtain a set of CEF parameters for Sm3+ in Sm2Ti2O7.
Assuming the parameters therein for Sm2Ti2O7 are a good approximation for
Sm2Ir2O7, we calculate that at $T=0$ K and $\mu_{0}H=0.5$ T, the moments
exhibit a strong easy-axis anisotropy along the local $\left[1\,1\,1\right]$
axis with $\mu_{\parallel[111]}=0.233\,$ $\mu_{\rm B}$ and $\mu_{\perp[111]}$
= 0.001 $\mu_{\rm B}$ leading to four non-colinear Ising axes in each Sm
tetrahedron. The moment calculated in this way is around half the value of
that from the $|5/2,\pm 3/2\rangle$ doublet alone. The discrepancy is largely
due to a small admixture of terms from higher $J$ levels in the ground state,
especially the $J=7/2$ level [43].

The black line in the main panel of Fig. 2 is the single-ion susceptibility
for $H//[111]$ calculated using the intermediate-coupling wavefunctions. In
spite of the simplicity of the single-ion approximation, which neglects Sm–Sm
as well as Sm–Ir interactions, we see that it captures the leading behavior of
the ZFC susceptibility. In particular, it shows that the low-temperature
Curie-like tail can be accounted for entirely by the Sm sublattice. It is no
surprise that this calculation is unable to reproduce the hump seen below 150
K, as this feature is likely caused by domains of weak ferromagnetism
associated with the Ir AIAO order, which have been proposed in other
pyrochlore iridates [37, 42]. The line in the inset to Fig. 2 is the powder-
averaged single-ion calculation for the magnetization. The calculations
predict that elevated fields mix the CEF-split energy levels, causing the Sm
moment to become less Ising-like and grow significantly beyond its low-field
value, consistent with the experimental data.

While dipolar interactions dominate the behavior of the rare-earth ions in
other spin-ice materials such as holmium or dysprosium titanate, the small
size of the Sm moments suggests that they can be safely disregarded in the
temperature and field regime of interest in our work. The same argument
however does not apply to exchange interactions, and further work beyond the
scope of the present paper is needed to ascertain the effects of Sm-Sm and Sm-
Ir exchange couplings to the behavior of the magnetisation in an applied
field. It is reasonable to expect the Sm ions to prefer to adopt an AIAO order
at sufficiently low temperature, similar to Ho2Ir2O7 and Dy2Ir2O7, but they
may favour a 2I2O configuration, depending on the sign and relative size of
the Sm–Sm exchange interaction compared to the Sm–Ir interaction [16, 17].

### Transport properties under applied pressure

#### Temperature dependence of resistivity

_Low pressure_ — Fig. 3 shows the temperature dependence of the resistivity of
a single crystal of Sm2Ir2O7, measured in a diamond anvil cell as discussed in
the Methods section and at approximately zero applied pressure (0.3(3) kbar).

The resistivity shows metallic behavior (defined as positive ${\rm d}\rho/{\rm
d}T$) at high temperature, but undergoes a transition to a negative ${\rm
d}\rho/{\rm d}T$ regime below $T_{\mathrm{min}}$ = 110(3) K, where the
resistivity exhibits a minimum. This transition is referred to as a metal-
insulator transition in the literature. However, as discussed below, our data
indicate that the low-temperature state cannot be described as a conventional
insulator characterized by a gap at the Fermi energy. To avoid confusion, we
refer to this region as ‘non-metallic’. The resistivity minimum occurs at the
same temperature as the Ir AIAO magnetic transition seen in the magnetization,
in agreement with the literature. Our resistivity data are consistent with
previously reported low-pressure data on polycrystals by Liu  _et al._ [31]
and on pressed pellets by Matsuhira _et al._ [29]. We note that the anomalous
results of Graf  _et al._ [44], also on pressed pellets, who report a much
lower transition temperature are not reproduced in other studies.

Figure 3: Resistivity vs temperature at near-ambient pressure. The temperature
of the resistivity minimum, $T_{\mathrm{min}}$, is marked with an arrow. The
inset shows the result of differentiating the data with respect to
temperature, which reveals a subtle kink or shoulder around 60 K,
$T_{\mathrm{kink}}$.

Figure 4: (a) Resistivity vs temperature for increasing applied pressures,
plotted on a logarithmic y-axis. The temperature position of the resistivity
minimum, $T_{\mathrm{min}}$, is marked with arrows. (b) Values of
$T_{\mathrm{min}}$ plotted against applied pressure, blue open circles. Data
from the work of Liu _et al._ [31] are overlayed as closed black circles. (c)
Resistivity extrapolated (via a quadratic fit below 8 K) to $T$ = 0 K plotted
against pressure. Note that the vertical axis is logarithmic. Dotted lines are
intended as guides to the eye, highlighting a change of behavior around 60
kbar, close to the Ir AIAO QCP.

The temperature derivative of the resistivity reveals a subtle change of
gradient or kink, which we denote $T_{\mathrm{kink}}$, at around 60–70 K in
both the 0.3(3) kbar and 1(1) kbar data sets (Fig. 3 Inset). This feature is
not seen or resolved in our data for $p\geq 30$ kbar, but it is seen, after
differentiation, in the data of Liu  _et al._ across their 0–22 kbar pressure
range, where it increases in temperature with applied pressure (see Fig. S12
in the Supplemental Appendix).

Figure 5: Fits of the low-temperature resistivity. (a) Resistivity data
(points) and fits (black line) to $\rho^{-1}=\rho_{0}^{-1}+(A/T)^{-1}$, as
described in the text. Inset shows $\rho$ with the $\rho_{0}$ contribution
removed versus $1/T$ for $p=0.3(3)$ kbar, yielding a straight line for $T$ up
to 55 K. (b) The same as (a) plotted on a logarithmic vertical axis, to allow
the higher pressure curves to be seen more clearly. Black arrows indicate the
temperatures where each fit diverges from the data by 2%. (c) Pressure
dependence of the fit parameter $A$, the coefficient of the $1/T$ term (red
circles). Dashed lines are guides to the eye (note that the vertical axis is
logarithmic), highlighting a change of character around 60 kbar close to the
QCP. The pressure dependence of $\rho_{0}$ from Fig 4(c) is also shown
overlaid as green squares for comparison.

Fig. 4(a) shows the resistivity of the same Sm2Ir2O7 crystal at each pressure
measured, increasing from 0.3(3) kbar at the top, blue, to 87(2) kbar at the
bottom, red. Applying pressure up to 87 kbar continuously reduces the
magnitude of the resistivity across all temperatures (by a factor of 40 at
room temperature and 190 at 2 K). The absolute value of resistivity is
initially comparatively high ($\sim 10^{6}$ $\mu\Omega$cm) in the metallic
state, suggesting that this material may not be best described as a simple
coherent metal even above $T_{\mathrm{min}}$.

A transition from metallic to non-metallic behavior upon cooling is observed
at all pressures. Liu  _et al._ [31] measured the pressure dependence of the
resistivity of their polycrystal sample up to 22 kbar and reported a linear
decrease in $T_{\mathrm{min}}$ over this range and this fits well with our
results. However, just above the maximum pressure achieved by Liu  _et al._ ,
our data show a change in behavior: the value of $T_{\mathrm{min}}$ begins to
increase again, in a linear trend, from approximately 30 kbar, as shown in
Fig. 4(b). This resistivity minimum is sharp and well defined up to 30(2)
kbar, but becomes markedly broader and smoother at higher pressures, possibly
indicative of a change from a phase transition to a crossover. Also at this
point, the rise in resistivity for $T<T_{\mathrm{min}}$ becomes less
pronounced than that observed at lower pressures.

The nature of the electronic state below $T_{\mathrm{min}}$ in this family
remains the subject of debate — several studies have compared the form of the
low-temperature resistivity to various models of localized transport to
attempt to shed light on this matter, see e.g. Refs [45, 46, 47]. In the
present case, the functional form of the resistivity temperature dependence in
the non-metallic phase below $T_{\mathrm{min}}$ proves challenging to describe
with common simple models. To start, a simple Arrhenius-type activated
exponential behavior falls very far from describing the data, which are much
shallower in slope than an exponential. It is clear from inspection of the
curves in Fig. 4(a) that although the magnitude of the resistivity increases
dramatically at low temperatures, it does not appear to diverge at zero
temperature. Extrapolating the curves to $T=0$ results in a finite $y$-axis
intercept $\rho_{0}$ for each pressure. We show the result of doing just this
in Fig. 4(c), with a quadratic function fit to the data below 8 K and
extrapolated below their lower limit. The extrapolated $\rho_{0}$ value is
seen to drop off rapidly with applied pressure up to around 60 kbar, the
location of the AIAO QCP, after which it decreases at a much slower rate.

As discussed by Savary _et al._ [48], a theoretical treatment of the transport
properties in the pyrochlore iridates is complex and challenging; no clear
predictions have been published at this time and there is as yet no
experimental consensus. In polycrystalline Sm2Ir2O7 [31], and in Gd2Ir2O7
[49], the low-temperature resistivity has been described with a
$\rho\propto\mathrm{exp}\left(T_{0}/T\right)^{1/4}$ variable-range hopping
expression — an activated insulating behavior typical of systems with a high
degree of local disorder [50, 51, 52, 53, 54]. Telang and Singh, in their
study of the effect of Bi doping in Sm2Ir2O7 pressed powders [55], describe
the undoped sample as obeying a $1/T$ temperature dependence typifying WSM
behavior. This then reportedly evolves through a $-T^{1/4}$ power law
dependence tentatively attributed to quantum critical effects as doping
suppresses the resistivity, to a $\mathrm{ln}(T)$ region interpreted as
signifying quadratic-band-touching. In polycrystalline Y2Ir2O7, a $T^{-4}$
model [46] and a $T^{-3/2}$ model [47] were fit to the transport data, and
attributed to a WSM state with thermally screened charged impurities, and with
the presence of impurities and electron–electron correlations, respectively.
Fits to all these models, and more, are shown in the Supplemental Appendix
Section 1 to compare their relative merits with respect to our single-crystal
data. With the exception of the $-T^{1/4}$ function, all of these models
diverge at zero temperature, so it is unsurprising that they are not able to
describe our data beyond highly restricted temperature ranges222A
$\rho(T)\propto-T^{1/3}$ temperature dependence, shown in the Supplemental
Appendix Fig. S3, was found to give a markedly better fit to our data than any
of these models previously employed in the literature. However, there seems to
be no clear physical basis for this functional form and we just note its
ability to fit the data..

The simplest physical model that can reproduce the behavior of our resistivity
data is a combination of two parallel resistance channels, one which diverges
and accounts for the non-metallic nature of the material and a second term
which leads to a finite resistivity at zero temperature. The latter term could
be any function that varies slowly compared to the diverging term, including
$\rho\propto T$ or $T^{2}$, but for simplicity we choose a temperature-
independent value, set to be equal to the $\rho_{0}$ values determined from
data extrapolation (shown in Fig. 4(c)). This term implies a metallic channel
with impurity scattering is present in the sample333The strong, monotonic
pressure dependence of $\rho_{0}$ rules out that it arises from an
experimental artefact, such as an electrical short. that could potentially
arise due to surface states or conduction along magnetic domain-wall
boundaries, as suggested to be present in Nd2Ir2O7 [35].

The best fits (see Supplemental Appendix Section 1D) to the data at all
pressures were found using the model $\rho^{-1}=\rho_{0}^{-1}+(A/T)^{-1}$ and
are shown in Fig. 5(a) and (b). As mentioned earlier, a $\rho\propto 1/T$
divergent term has previously been predicted for WSMs [56] and it describes
our data particularly well at low pressures as can be seen in the inset to
Fig. 5(a). This shows the 0.3(3) kbar $\rho(T)$ data lying on a straight line
for $2<T<55$K when plotted versus $1/T$ after the $\rho_{0}^{-1}$ contribution
has been subtracted. The temperature range over which the model effectively
reproduces the data is largest at the two lowest pressures, farthest from the
QCP. This is indicated by the arrows in Fig. 5(b), which show the upper limit
of the region of validity of the fit (pragmatically set here at 2% divergence
between fit and data). The coefficient $A$ of the $1/T$ term also varies with
pressure, as shown in Fig. 5(c). Similar to $\rho_{0}$, $A$ is dramatically
suppressed with initial pressurization, by two orders of magnitude up to the
vicinity of the QCP at $p_{c}$, where a change of behavior is seen, and
further reductions are much more gradual. Indeed, the pressure dependences of
$\rho_{0}$ and $A$ are found to agree strikingly well, see Fig. 5(c),
suggesting that the two conduction channels may be connected in some way.

In Sm2Ir2O7, the lattice constant is known to decrease linearly with pressure
across the whole range measured here, while the oxygen-coordinate parameter
$x$ changes only a little [2].

The swift drop on increasing pressure up to 60 kbar in the parameters
$\rho_{0}$ and $A$ that quantify the non-metallic behavior at low temperatures
indicates that the degree of localization in the AIAO Ir-ordered phase is
highly sensitive to reductions in the lattice constant and the concomitant
enhancement of the hopping parameter, which is consistent with the ground
state being underpinned by electronic correlations. By contrast, in the state
that emerges above the QCP, the localization of electrons is much reduced as
compared to the behavior seen under ambient conditions, and further changes in
pressure have only a small effect. The relative decoupling of this state from
changes in the lattice constant suggests that correlations play a smaller role
in the electronic ground state in this pressure regime.

#### Hysteresis in magnetoresistance

Figure 6: Magnetoresistance at $T=$1.6(1) K and near-ambient pressure, showing
a clear hysteresis loop. The field was applied along the approximate
$\left[1\,1\,1\right]$ direction, perpendicular to the plane of measurement.
Data are normalized to their zero-field values.The green points denote the
first application of the field up to +15 T after cooling the sample in zero
field, then arrows around the subsequent blue data points show the evolution
of the resistivity as the field is swept to -15 T and then back to +15 T
again, at a rate of 1 Tmin-1. Figure 7: Magnetoresistance $\rho(B)-\rho(0)$ at
fixed temperatures for pressures of (a) 1(1) kbar, (b) 43(2) kbar (c) 61(2)
kbar and (d) 75(2) kbar. The field was applied along the approximate
$\left[1\,1\,1\right]$ direction, perpendicular to the plane of measurement.
By 61 kbar the form of the hysteresis loop is changed, and the
magnetoresistance is negative at low temperatures. Green points in (b)
illustrate examples of binning and symmetrizing the data to obtain effective
single-valued $\rho(B)$ data, separating out the hysteresis effects, as
displayed for all data in Fig. 8. The data all have value 0 at $B=0$, and are
arbitrarily offset vertically for visibility.

At each pressure, we measured the magnetic-field dependence of the resistivity
at various temperatures down to 1.5 K. The field was varied between $\pm 15$ T
in the direction perpendicular to the plane of the resistivity measurement,
which is along the diamond-anvil-cell axis and close to the
$\left[1\,1\,1\right]$ direction. Fig. 6 shows that the low-temperature
$\left[1\,1\,1\right]$ magnetoresistance exhibits a marked and reproducible
hysteresis, which persists for pressures below about 61 kbar. Also shown is
the first field sweep performed on cooling in zero field from high
temperatures (green line) which lies between the up and down sweeps of
subsequent field sweeps. The strong hysteresis remains as temperature is
increased up to about 50 K, but closes gradually at higher temperatures, and
before the AIAO-ordering transition is reached, see Fig. 7. As can be seen in
Fig. 7 and the Supplemental Appendix Section 2B (data for all pressures) and
Fig. S17 (an interpolated colormap plot of hysteresis magnitude across the
entire temperature-pressure range), this trend continues at elevated pressures
until about 61 kbar, at which point the size of the hysteresis reduces
dramatically, disappearing close to the AIAO QCP.

This hysteretic behavior is highly reminiscent of that observed in Ho2Ir2O7.
In that case, a magnetic field close to the $\left[1\,1\,1\right]$ direction
acts to align the large Ho moments in a 3I1O/1I3O monopole-crystal arrangement
which, via the Ho-Ir coupling, energetically favors one Ir AIAO domain over
the other and alters the domain ratio in the sample. On sweeping down, the
domain ratio remains constant until a negative field is applied, whereupon the
second domain is favored and begins to grow preferentially. In Sm2Ir2O7, the
Sm ions have a significantly smaller moment than the Ho ions in Ho2Ir2O7 (=
10$\mu_{\rm B}$). If the same effect is responsible for the hysteresis in both
materials, this would explain why the magnitude of the hysteresis is lower in
Sm2Ir2O7 and also why the hysteresis is seen not to close even at the highest
fields reached. At these temperatures, a field of about 4 T is sufficient to
close the hysteresis in Ho2Ir2O7 and achieve a steady 70:30% domain ratio (a
value determined by the presence of strong pinning sites). By contrast, in
Sm2Ir2O7, the field required to saturate the moments and fully close the
hysteresis loops is far greater than the 15 T achieved in these measurements;
there are no high-field regions where the resistance becomes single-valued.
Indeed, from the ratio of the moment sizes in the two cases we would expect
the field required to close the hysteresis to be roughly ten times larger in
Sm2Ir2O7 than in Ho2Ir2O7. This also explains why the hysteresis loops shown
by Pearce _et al._ appear smoother than found here for Sm2Ir2O7 – the data in
Fig. 6 access only the low-field region of the hysteresis loop, which likely
extends out towards 100 T.

Figure 8: Background magnetoresistance ratio
$\nicefrac{{\rho(B)}}{{\rho(B=0)}}$ at fixed temperatures for pressures of (a)
1(1) kbar, (b) 43(2) kbar (c) 61(2) kbar and (d) 75(2) kbar. By 61 kbar the MR
is markedly negative at low temperature. These data are extracted from Fig. 7
by binning and symmetrizing the data to remove $\rho_{xy}$ components, and
averaging up and down sweeps to separate out the hysteresis effects. The data
all have value 1 at $B=0$, and are arbitrarily offset vertically for
visibility.

The sensitivity of the magnetoresistance in Ho2Ir2O7 to changes in Ir-domain
ratio arises because the monopole density varies differently with field in the
two Ir-domain types, which in turn affects the monopole-quasiparticle
scattering rate via two coupling mechanisms linking: (i) electronic spin and
magnetic monopole charges; and (ii) electronic charge and the electric dipoles
associated with monopoles. The validity of the magnetic monopole description
in Sm2Ir2O7 has not yet been definitively established and so it is not fully
clear if the hysteresis has the same origin as in Ho2Ir2O7; however we can
estimate that mechanism (i) should be smaller for the low-moment Sm ions. On
the other hand, mechanism (ii), which is driven by Sm–Sm interactions, could
be of a similar size in both cases, as we expect the Sm exchange energy to be
similar to or stronger than the equivalent in Ho2Ir2O7.

An alternative, related effect is proposed to account for the hysteresis
observed in Nd2Ir2O7, which also has small rare-earth moments. Ueda _et al._
suggest that the current is preferentially carried through the Ir-domain walls
and so the resistance responds to changes in the size and distribution of
these as the field is swept [35]. Which of these mechanisms is responsible for
the effects seen in Sm2Ir2O7 remains an interesting open question, but what is
clear is that the large hysteresis is characteristic of the AIAO phase and
disappears as the QCP is approached. For pressures above $p_{{\rm c}}$, some
hysteresis is still seen in the magnetoresistance at the lowest temperatures
(see Fig. 7(c,d)), but it is significantly smaller than that observed within
the AIAO phase. Furthermore, the background form of the magnetoresistance
undergoes a marked change at this pressure.

#### Background magnetoresistance

In order to highlight the background form of the magnetoresistance of the
sample separately from the hysteresis effects, one can average the multivalued
resistivity at the same field value, to find an effective single-valued
$\rho(H)$ for each pressure and temperature.

The low-temperature magnetoresistance data at all pressures show an asymmetry
between positive and negative fields (Fig. 6)—an overall linear slope—which we
attribute to a component of the Hall voltage mixing into the measured
longitudinal resistivity. The magnitude of such a Hall signal is inversely
proportional to the carrier density (for a simple single band at least). Here,
the strongest slope is seen at low pressures where resistivity is high. To
reliably extract absolute values of carrier density, a Hall-bar geometry on a
lamellar sample with accurately placed contacts would be required; the small
and irregularly shaped sample in our restrictive diamond anvil cell
measurements precludes this treatment. To remove the Hall asymmetry, the data
were symmetrized by averaging negative and positive field traces. Examples of
the resulting $\rho(H)$ plots are shown in Fig. 7(b) in green and in Fig. 8
(data for all pressures are shown in the SM). With these data we can
systematically describe the form and sign of the $\left[1\,1\,1\right]$
magnetoresistance separately from the large hysteresis seen at low pressures.
As can be seen from Fig. 8, the low-temperature, low-pressure background
magnetoresistance first drops with field before rising as field is increased.
A negative component is expected in the low-temperature magnetoresistance of
materials with locally fluctuating or disordered moments, such as a paramagnet
or a Kondo system. As the field aligns the spins, spin scattering is reduced
and resistance drops. In this material at low pressure and temperature, only
the Sm3+ moments are free to align with the field, as indicated by
magnetometry measurements. The behavior of the magnetoresistance in this
region is reminiscent of that seen in Ho2Ir2O7 under an applied
$\left[1\,1\,1\right]$ field [3]. In that case, there is also a negative
magnetoresistance at low fields, caused by the alignment of the Ho3+ moments,
followed by change to positive magnetoresistance as the effects of enhanced
monopole scattering starts to dominate. As temperature is raised the
magnetoresistance flattens out in both materials in similar way.

As pressure is increased, the background magnetoresistance retains the overall
shape it has at low pressures, until the critical pressure for complete
suppression of the Ir AIAO is approached. Here, the low temperature
magnetoresistance becomes strongly negative across the whole field range (see
Fig. 8) with a small amount of hysteresis (Fig. 7). As temperature is raised,
the negative magnetoresistance diminishes, becomes weakly positive and then
flattens out. The narrow hysteresis closes and disappears at the same
temperature the MR changes sign. This form of the magnetoresistance continues
for all pressures measured above 61(2) kbar.

There are several potential causes for positive magnetoresistive behavior.
Simplest is the case of orbital motion of charge carriers in non-magnetic or
antiferromagnetic (semi)metals with closed Fermi surfaces. Tafti _et al._
attribute the positive magnetoresistance in high-pressure Eu2Ir2O7 to this
effect [36].

Figure 9: Pressure–temperature phase diagram for Sm2Ir2O7. The background
colormap displays interpolated values of the magnetoresistance (MR) ratio and
is indicative of the size of the MR. Temperatures of the resistivity minimum
$T_{\mathrm{min}}$ are plotted as open circles. Previous data from the
literature up to 20 kbar [31] are overlaid as closed circles Above about 30
kbar $T_{\mathrm{min}}$ shows a linear increase with pressure (solid line),
which can be extrapolated back to ambient pressure (dotted line) where it
coincides with the kink feature observed in the gradient of $\rho(\mathrm{T})$
at the lowest pressures. These are plotted (open squares) along with similar
features found in the data of Liu _et al._ [31] up to 20 kbar (closed
squares). Triangles denote the Ir AIAO ordered phase observed by Wang _et al._
[2] and star symbols represent the temperatures at which the magnitude of the
hysteresis in the MR extrapolates to zero. The curved solid line is a guide to
the eye through these points. The green shaded region highlights a regime
around the QCP with a negative background magnetoresistance. The diamond
symbols denote the temperatures where the magnetoresistance switches from
negative to positive, which bound this region.

## Discussion

Figure 9 shows the pressure-temperature phase diagram of both the magnetic and
transport properties of Sm2Ir2O7 constructed by pulling together all of the
features seen in our measurements, as well as prior results in the literature.
The background colormap displays the ratio of the single-valued
magnetoresistance at 15 T and zero field, to distinguish regions of positive
(blue) and negative (green) MR. Regions where the MR is flat, showing no
variation in field, are shown in white on this scale. (See Supplemental
Appendix for full datasets—Fig. 16—and further details of the colormap
—Section 3). The purple stars represent the temperatures at which the
hysteresis in MR closes (see Fig. S17) and are seen to follow closely the AIAO
magnetic transitions at $T_{\mathrm{AIAO}}$ (orange triangles) found in the
resonant X-ray measurements of Wang _et al._ [2]. The curved solid line is a
guide to the eye that connects these points.

The measured transport properties are clearly consistent with the pressure-
induced suppression of the AIAO phase as manifested in three specific trends
in the data. _First_ , the crossover from metallic to non-metallic conduction,
marked by the resistivity minimum at $T$ = $T_{\mathrm{min}}$, as reported by
Liu _et al._ [31] (closed circles) and our low-pressure data (open circles),
closely tracks the decrease in $T_{\mathrm{AIAO}}$ at all pressures up to 30
kbar. This is consistent with the interpretation of $T_{\mathrm{min}}$ as a
Mott transition associated with the Ir ordering [20, 21, 2]. _Second_ , the
magnitude of the resistivity continuously decreases as pressure is raised
(Fig. 4), and the extrapolated finite $\rho_{0}$ values drop rapidly with
pressure up to 60(4) kbar, which is where AIAO Ir order is completely quashed.
As mentioned earlier, the AIAO magnetic order is driven by the combination of
$U/t$ (Coulomb repulsion/hopping strength) and local Ising anisotropy [2]. As
pressure is applied, the lattice parameter decreases significantly, the $5d$
bandwidth and hopping strength increases, and $U/t$ is suppressed [2]. The
combined result of pressure then is to stymy the AIAO ordering temperature and
enhance conductivity. _Third_ , the marked hysteresis seen in the
magnetoresistance for magnetic field applied close to the
$\left[1\,1\,1\right]$ direction is present only within the AIAO ordered
phase. This makes sense if, as discussed earlier, the hysteresis has its
origin in the plastic deformation of the Ir domains via coupling with the Sm3+
moments, equivalent to that reported in Ho2Ir2O7 or Nd2Ir2O7. Once the AIAO
order is suppressed, the domains are no longer present and the large
magnetoresistance hysteresis disappears. All these magnetotransport features
affirm the existence of the same quantum critical point at $p_{{\rm c}}$ = 63
kbar suggested by Wang _et al._ from their X-ray data.

Despite these connections between the Ir AIAO order and the measured transport
properties, an important and clear result of our experiments is that Sm2Ir2O7
remains metallic at high temperatures and non-metallic at low temperatures
below a finite $T_{\mathrm{min}}$ for the entire pressure range measured. In
particular, the metal-insulator transition marked by the resistivity minimum
at $T_{\mathrm{min}}$ is not suppressed to zero temperature along with the Ir
AIAO order, as was naively anticipated from the coincidence of the two
transitions at low pressures and from the pressure dependence of
$T_{\mathrm{min}}$ in Nd2Ir2O7 [31, 20, 24].

As mentioned, $T_{\mathrm{min}}$ initially decreases, following the phase line
of the Ir AIAO order observed by Wang _et al._ [2]. However, above 30 kbar
this trend reverses and $T_{\mathrm{min}}$ increases linearly up to the
highest pressures measured. It seems that some additional mechanism is causing
a resurgence of localization of the charge carriers once the pressure pushes
the AIAO transition below a certain temperature. This new localization
mechanism then strengthens with further application of pressure.
Interestingly, an extrapolation of the high-pressure linear trend of
$T_{\mathrm{min}}(p)$ back to low pressures, (dashed black line) coincides
with the position of the subtle change in $d\rho/dT$ observed in our sample at
$T_{\mathrm{kink}}$. This feature is also present in the low-pressure data of
Liu _et al._ (closed squares, see also Supplemental Appendix Fig. S12) —
albeit at a slightly lower temperature than in our data — and is seen to
increase linearly with pressure at a gradient equal to the high-pressure
dependence of $T_{\mathrm{min}}$. We note that a similar feature is observed
in the resistivity of Eu2Ir2O7, where it also appears in the heat capacity
[37]. There it is shown to be most clear in the purest crystals with smallest
deviations from Eu and Ir stoichiometric values, suggesting in this case that
it is an intrinsic effect that gets smeared out in lower quality samples.

Possible origins for this kink feature include low-temperature structural
and/or magnetic variations, and we note again that a shoulder-like feature
appears in the field-cooled susceptibility data close to $T_{\rm kink}$ (see
Figure 2).

Whatever the cause, it is apparent that $T_{\mathrm{min}}$ and
$T_{\mathrm{AIAO}}$ separate for pressures above 30 kbar, and the Ir AIAO
order ceases to be the driver of the non-metallic state when the temperature
scale associated with the new, as yet unidentified, mechanism begins to
dominate. This explains why the material remains non-metallic once the AIAO
order is suppressed at the $p_{{\rm c}}$ = 63 kbar QCP. For pressures between
30 and 63 kbar, the Ir ordering still takes place below $T_{\mathrm{AIAO}}$.
The main markers of the AIAO ground state in the transport in this
intermediate pressure region are the large magnetoresistive hysteresis
discussed earlier and the strong negative pressure dependence of the
extrapolated $\rho_{0}$ value.

A new form of magnetotransport emerges around the QCP. The large hysteresis
and competition between negative and positive magnetoresistance components,
seen at low temperatures in the AIAO phase, now changes to a region of weakly
hysteretic but strongly negative magnetoresistance that extends from just
below $p_{{\rm c}}$ to beyond the highest pressure measured (green region in
Fig. 9). The negative magnetoresistance evolves slowly to a weakly positive
response as the temperature is raised above about 5 K, disappearing altogether
at elevated temperatures. The open diamond symbols in Fig. 9 illustrate the
temperatures where the MR is zero (see extended data in Supplementary Figs.
S40 and S35–S38). According to Wang _et al._ , while the long-range Ir AIAO
order is suppressed above $p_{{\rm c}}$, the axial anisotropy is preserved,
leaving behind weakly correlated or paramagnetic Ising-like moments [2]. Any
low-temperature magnetic order on the Sm sublattice is also predicted to be
lost once the Ir order is gone. Alignment of these fluctuating spins in an
applied magnetic field could account for the negative magnetoresistance
observed above the QCP. The positive magnetoresistance at higher temperatures
is consistent with orbital effects arising from small Fermi-surface pockets,
such as might be expected for a disordered semimetal and, as mentioned above,
a similar origin was proposed for the high-pressure magnetoresistance seen in
Eu2Ir2O7 [36].

The precise origin of these new electronic properties remains to be
determined. Several theoretical proposals have been made for the electronic
phases that might exist in the vicinity of the AIAO QCP [1, 21, 48, 39, 57].
These include a WSM phase expected to be located in a relatively narrow region
of Coulomb $U$ between the AIAO Mott insulator and the paramagnetic-metal
phase (previously believed to occur once AIAO order is suppressed) [58, 2,
59], and tentatively reported in the results of Telang and Singh [55].
However, as Wang _et al._ point out from their estimates of the shift in
hopping $t$ required to realize the QCP in Sm2Ir2O7, this theoretically
predicted narrow region where the WSM exists could span the entire pressure
range measured [2]. Our analysis of the single-crystal transport data finds
them to be well described by a $1/T$ functional form, which is indeed
consistent with models of WSMs [56], albeit with the inclusion of a parallel
slowly-varying (relative to $1/T$) conduction channel, possibly arising from
surface or domain-wall conduction. The temperature range over which this $1/T$
dependence is valid is found to be largest at the lowest pressure, away from
the QCP.

Fig. 5 shows a clear and significant pressure dependence in the form of
$\rho(T)$, with a marked behavioral change at the critical pressure where AIAO
magnetic order is suppressed.

Another theoretical prediction is that the QCP in Sm2Ir2O7 separates the WSM
from a non-Fermi-liquid semimetal phase distinguished by quadratic-band
touching (QBT) [48]. Indeed, ARPES measurements suggest that this is the
ambient pressure phase exhibited by Pr2Ir2O7 at low temperatures [22] and by
Nd2Ir2O7 in its metallic state at elevated temperatures [23]. It should also
be pointed out that optical conductivity spectra for Sm2Ir2O7 point towards a
semimetal nature [58], and are distinct from the finite gap seen in these data
for Eu2Ir2O7. Recent work on Pr2Ir2O7 has additionally reported topological
Weyl semimetal states under the application of magnetic field [60]. A recent
paper reporting transport measurements on Sm2Ir2O7 speculates that the WSM is
present at ambient pressure and that doping more than 5% of Bi on the Sm site
leads to a transformation to the QBT phase, although the evidence in favor of
this interpretation is limited [55]. Our data also do not provide evidence for
QBT behavior emerging under pressure; however, we highlight again that our
experiments reveal that the paramagnetic metallic phase is not realized once
AIAO order is destroyed and instead another, as yet mysterious, electronic
phase emerges.

Our data share certain similarities with work done by Tafti _et al._ on
Eu2Ir2O7, neighbor to Sm2Ir2O7 in the chemical series of this family [36].
While the $T_{\mathrm{min}}$ in that material does not appear to shift very
much at all with applied pressure, the authors do find that the resistivity
drops significantly as pressure is increased. In the low-pressure region, they
can fit their $\rho(T)$ data to an activated model at low temperatures (which
does not work for Sm2Ir2O7). The size of the insulating gap in Eu2Ir2O7 drops
swiftly towards zero at 80 kbar and then remains low and slowly varying upon
further increase in pressure, reminiscent of the $A$ coefficient we extract
from the $1/T$ fits for our Sm2Ir2O7 data. The authors describe their high-
pressure, low-temperature state as a novel metal with resistivity that
increases on cooling. In contrast to our results, Tafti _et al._ find that the
magnetoresistance of Eu2Ir2O7 is positive down to the lowest temperatures in
the high pressure region [36]. It should be noted that the AIAO phase
transition has not yet been tracked with pressure in the Eu material. Tafti
_et al._ also find that the resistivity at high temperatures and low pressures
in Eu2Ir2O7 increases on cooling. This is in contrast to the more metallic
behavior (i.e. decreasing resistivity on cooling) observed at ambient pressure
in single-crystal measurements of other pyrochlore iridates $Ln_{2}$Ir2O7,
including $Ln$ = Pr [32], Nd [35], Sm (our data), Ho [3] and even in other
studies of the same Eu material considered by Tafti _et al._ [37], as well as
in some pressed-powder measurements, e.g. Pr [29], Nd [29, 62, 20] and Sm
[44]. We suggest that a metallic temperature dependence above
$T_{\mathrm{min}}$ is the true intrinsic behavior of the pyrochlore iridates
in this region of the phase diagram.

In summary, our data show a significant change in the electronic properties of
Sm2Ir2O7 as the AIAO Ir magnetic order is gradually destroyed by pressure.
Although the metallic-to-non-metallic crossover at $T=T_{\mathrm{min}}$ is not
suppressed in tandem with the magnetic transition, the magnitude of the
resistivity, the extrapolated finite $T=0$ resistivity, and the coefficient of
the $1/T$ resistivity component all drop swiftly as the QCP is approached,
then switch over to a much reduced rate of reduction with further increasing
pressure. The $1/T$ temperature dependence, which is predicted for a Weyl
semimetal, fits the data up to 55 K at 0.3 kbar, but this range is reduced at
higher pressures. While it is not currently possible from these data alone to
precisely identify the electronic state that emerges beyond the QCP, we have
measured both transport and magnetotransport data across the entire phase
diagram in order to provide information to guide theoretical studies. Our
results also act as a basis for future experimental efforts, with optical
conductivity and angle-dependent magnetotransport particularly well suited to
shedding light on the possible Weyl semimetal and anisotropic, quadratic-band-
touching phases.

## Methods

Single crystals of Sm2Ir2O7 were grown by the flux method, using KF as the
flux, as described in Ref. [63]. They were characterized and verified to be of
good crystalline quality by X-ray diffraction. Magnetotransport data were
carried out in a diamond anvil cell [64, 65] with non-magnetic BeCu body and
hardened BeCu gasket. The culet of the diamonds used was 1 mm and the gasket
hole was drilled to 0.5 mm. The cell was pressurized at room temperature for
each pressure point, then measured on a home-built probe loaded into a Oxford
Instruments 17 T cryomagnet fitted with a variable temperature insert. A
single crystal was prepared for resistivity measurements in the standard
4-wire geometry with gold wires bonded to the crystal with DuPont 6838 silver-
loaded epoxy. Glycerol was used as a pressure transmitting medium and the
pressure was measured via the fluorescence spectrum of a chip of ruby placed
within the pressure region [66]. Typical pressure uncertainties, as denoted on
values given in the text are estimated from two factors. Firstly There is a
gradient or distribution of pressure within the sample volume, which we
quantify via the broadness of the ruby fluorescence peak and its fitting
uncertainty. Secondly, previous calibrations on identical cells have shown
pressure losses upon cooling, due to thermal expansion. The resulting
uncertainty from these two factors we determine as $\pm$1 kbar below 10 kbar
and $\pm$2 kbar above, noting that the pressure will always be above 0 kbar.

The magnetic field was applied along the cell axis, a direction close to the
crystallographic $\left[1\,1\,1\right]$. The crystal was sat on the diamond
anvil faces and the current applied in the plane of the anvil faces,
perpendicular to the field. Low-field DC magnetization measurements were
carried out on a powder sample in a Quantum Design MPMS SQUID magnetometer
system. The sample was then additionally measured in up to 60 T of applied
field via extraction-coil magnetometry at the NHMFL, Los Alamos. As this
technique does not return absolute values of the magnetization, these were
obtained via calibrating the low-field slope up to 7 T with the MPMS data at
1.8 K. There is no discernible difference between the latter and the pulsed
field data at 1.5K and at 0.9K shown in the manuscript.

###### Acknowledgements.

We thank T. Orton and P. Ruddy at the University of Warwick for technical
assistance. This project has received funding from the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme (grant agreement No. 681260). We acknowledge the Engineering and
Physical Sciences Research Council (EPSRC), UK and the Oxford-ShanghaiTech
collaboration project for financial support. This work was supported by EPSRC
grants No. EP/P034616/1, No. EP/V062654/1 and No. EP/N034872/1. A portion of
this work was performed at the National High Magnetic Field Laboratory
(NHMFL), which is supported by National Science Foundation Cooperative
Agreement No. DMR-1644779 and the Department of Energy (DOE). J. S.
acknowledges support from the DOE BES program “Science at 100 T”, which
permitted the design and construction of the specialized equipment used in the
high-field studies.

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