PHYSICAL REVIEW B
101
, 220404(R) (2020)
Rapid Communications
Editors’ Suggestion
Approaching the quantum critical point in a highly correlated all-in–all-out antiferromagnet
Yishu Wang,
1
,
2
T. F. Rosenbaum,
1
D. Prabhakaran,
3
A. T. Boothroyd,
3
and Yejun Feng
1
,
4
,
*
1
Division of Physics, Mathematics, and Astronomy, California Institute of Technology, Pasadena, California 91125, USA
2
The Institute for Quantum Matter and Department of Physics and Astronomy, The Johns Hopkins University,
Baltimore, Maryland 21218, USA
3
Department of Physics, University of Oxford, Clarendon Laboratory, Oxford OX1 3PU, United Kingdom
4
Okinawa Institute of Science and Technology Graduate University, Onna, Okinawa 904-0495, Japan
(Received 11 October 2019; revised manuscript received 13 May 2020; accepted 15 May 2020;
published 4 June 2020)
Continuous quantum phase transition involving all-in–all-out (AIAO) antiferromagnetic order in strongly spin-
orbit-coupled 5
d
compounds could give rise to various exotic electronic phases and strongly-coupled quantum
critical phenomena. Here we experimentally trace the AIAO spin order in Sm
2
Ir
2
O
7
using direct resonant x-ray
magnetic diffraction techniques under high pressure. The magnetic order is suppressed at a critical pressure
P
c
=
6
.
30 GPa, while the lattice symmetry remains in the cubic
Fd
-3
m
space group across the quantum critical point.
Comparing pressure tuning and the chemical series
R
2
Ir
2
O
7
reveals that the approach to the AIAO quantum phase
transition is characterized by contrasting evolutions of the pyrochlore lattice constant
a
and the trigonal distortion
surrounding individual Ir moments, which affects the 5
d
bandwidth and the Ising anisotropy, respectively. We
posit that the opposite effects of pressure and chemical tuning lead to spin fluctuations with different Ising
and Heisenberg character in the quantum critical region. Finally, the observed low pressure scale of the AIAO
quantum phase transition in Sm
2
Ir
2
O
7
identifies a circumscribed region of
P-T
space for investigating the putative
magnetic Weyl semimetal state.
DOI:
10.1103/PhysRevB.101.220404
The mix of magnetic interactions, electron correlations,
and spin-orbit coupling informs the competition between
different quantum ground states and ordering mechanisms,
ranging from Mott to Slater antiferromagnetic insulators [
1
,
2
],
phonon to spin-fluctuation-mediated superconductivity [
3
,
4
],
and Kondo screening to Ruderman-Kittel-Kasuya-Yosida ex-
change in heavy fermion materials [
5
]. For 5
d
pyrochlores
such as
R
2
Ir
2
O
7
(
R
=
Y, Eu, Sm, Nd), the interplay between
intermediate electron correlations and strong spin-orbit cou-
pling leads to all-in–all-out (AIAO) antiferromagnetic order
and, potentially, nontrivial topological band structure, com-
monly known as a Weyl semimetal of broken time-reversal
symmetry [
6
–
10
]. Conversely, without electron correlation,
strong spin-orbit coupling could induce a different topological
Weyl state of broken inversion symmetry, as proposed in
nonmagnetic pyrochlores with a breathing lattice [
11
].
AIAO spin order exists in a pyrochlore lattice when all
spins are aligned along the local (1,1,1) axis either towards
(all-in) or away from (all-out) the center of the tetrahe-
dron [
6
–
10
,
12
–
17
]. In 5
d
pyrochlores, it has been verified
experimentally in
R
2
Ir
2
O
7
(
R
=
Lu, Yb, Tb, Eu, Sm, Nd)
and Cd
2
Os
2
O
7
[
12
–
17
]. However, recent angle-resolved pho-
toemission measurements on both magnetic Nd
2
Ir
2
O
7
and
nonmagnetic Pr
2
Ir
2
O
7
as the end member of the
R
2
Ir
2
O
7
series demonstrate parabolic nodal structures [
2
,
18
] that raise
questions about the existence of magnetic Weyl semimetal
*
yejun@oist.jp
phases in these compounds at ambient pressure. Under suit-
able tuning processes, such as pressure, exotic states may yet
emerge over an adjustable parameter space spanned by the
Coulomb interaction
U
and spin-orbit coupling
λ
, normalized
to the hopping integral
t
[
18
,
19
]. Given that the presence of
AIAO magnetic order serves as a gauge of electron correla-
tions, its quantum critical point, where the magnetic order is
suppressed to zero at zero temperature, could identify some of
the most intriguing regions of intermediate to strong coupling
physics in 5
d
compounds [
6
–
10
]. For example, it has been
suggested that Pr
2
Ir
2
O
7
develops a two-in–two-out spin-ice
configuration that melts into a metallic spin liquid at
T
<
0.4 K
in the proximity of its AIAO quantum critical point [
20
].
The iridate pyrochlores
R
2
Ir
2
O
7
provide a series of model
systems susceptible to continuous tuning, with an approxi-
mately local
J
eff
=
1
/
2 moment from the Ir
4
+
ions of the
5
dt
2
g
band [
15
] and a number of germane theoretical cal-
culations [
6
–
10
]. Sm
2
Ir
2
O
7
, with proven AIAO order [
14
],
negative pressure dependence of its insulating phase [
21
], and
available high-quality single crystals, is a particularly promis-
ing experimental choice. Here, using resonant x-ray magnetic
diffraction (Refs. [
12
–
14
,
22
–
25
], Supplemental Material), we
demonstrate that the AIAO spin order in Sm
2
Ir
2
O
7
expe-
riences a continuous quantum phase transition at a modest
critical pressure
P
c
=
6
.
30 GPa with preserved lattice space
group, exemplifying a directly tracked AIAO quantum crit-
ical point in iridates under pressure. Moreover, the pressure
evolution of Sm
2
Ir
2
O
7
follows a different pathway across the
U
/
t
-
λ/
t
phase space compared with the
R
2
Ir
2
O
7
chemical
2469-9950/2020/101(22)/220404(6)
220404-1
©2020 American Physical Society
YISHU WANG
et al.
PHYSICAL REVIEW B
101
, 220404(R) (2020)
01020
0
0.002
0.004
P (GPa)
I
(220)
/ I
(440)
01020
P (GPa)
Lattice constant (Å)
10.7
10.9
2
(deg)
17.5
17.7
17.9
(1, 1, 1)
P (GPa):
0.1
2.5
6.3
14.1
20.9
(2, 2, 0)
Sm
2
Ir
2
O
7
T=4K
2
(deg)
E = 11.215 keV
T = 4K
0.31
0.33
0.35
coordinate x
11.19
11.22
11.25
0
5000
10000
E (keV)
Counts / s
0.9
2.9
5.6
7.2
(2, 4, 0)
-
~ 0
P (GPa):
(a)
(b)
(c)
(d)
10.0
10.1
10.2
10.3
Simulation
I
(220)
/ I
(440)
FIG. 1. (a) Lattice constant is fit to a two-parameter Birch equation, with an isothermal bulk modulus
B
0
=
215
.
6
±
4
.
8 GPa, and its
pressure derivative
B
=
3
.
9
±
0
.
5. (Inset) Representative longitudinal (
θ
-2
θ
) scans of the (2,2,0) and (1,1,1) reflections showing single peaks
with minimal traces of lattice stress. Our measured
B
is much smaller than that of the silver manometer, in sharp contrast to several accounts
of large
B
values of iridate pyrochlores in the literature [
30
]. (b) Diffraction intensities of (2,2,0) reflection normalized by those of (4,4,0)
reflection as a function of pressure. Different symbols (circle, square, and diamond) in (b) and Figs.
4(a)
and
4(b)
represent each of three
individual samples studied. (c) Simulation of the relative intensity
I
(220)
/
(440)
as a function of
x
. (d) The ATS resonance at four pressures and
4 K. The spectral shape has no azimuthal dependence at (2,4,0) and (0,
−
2,4) reflections with
ψ
∼
0
◦
and
∼
35°, respectively, up to 21 GPa
(not shown). More details are available in Supplemental Material.
series, providing further clues to the nature of intermediate
to strong coupling physics in this model system.
The pyrochlore structure in the
Fd
-3
m
space group is
fully characterized by two parameters, the lattice constant
a
and the coordinate parameter
x
[
9
]. From the single-peaked
(1,1,1) and (2,2,0) diffraction profiles, Sm
2
Ir
2
O
7
remains in
a cubic structure to at least 21 GPa, and the lattice con-
stant
a
(
P
) evolves continuously at 4 K without any visi-
ble sign of a phase transition (Fig.
1
). The simple lattice
evolution strongly suggests a continuous AIAO quantum
phase transition, motivating a full polarization analysis of
resonantly scattered x-ray diffraction signals, in pursuit of
the most unambiguous and comprehensive understanding of
lattice, orbital, and magnetic behavior (Figs.
2
–
4
, Supple-
mental Material, Refs. [
12
–
14
,
22
,
24
,
25
]). The cubic space
group under pressure is illustrated by measured (0,0,6) and
(0,
−
2,4) diffraction intensities in the polarization-preserving
π
-
π
channel, which are minimal and constant at 6
×
10
−
6
level of the main (0,0,4) reflection through 21 GPa [Fig.
4(b)
].
This rules out
F
4-3
m
and
F
4
1
32 as the potential high-pressure
space group, and the
Fd
-3
m
space group likely persists at 4 K
to 21 GPa. The exclusion of these two space groups, especially
a breathing lattice instability of the
F
-43
m
type as observed
in Cd
2
Os
2
O
7
[
24
], indicates that a potential Weyl semimetal
state would unlikely be of a broken inversion-symmetry
type.
The coordinate parameter
x
varies from 0.3125 to 0.375,
with the limits indicating whether the oxygen atoms on the
48
f
sites form a perfect octahedron surrounding the Ir site
or a perfect cube surrounding the
R
site of
R
2
Ir
2
O
7
,re-
spectively. Within the
Fd
-3
m
space group, both Sm and Ir
ions in Sm
2
Ir
2
O
7
do not contribute to diffraction intensity
of the (2,2,0) reflection. Instead, this intensity arises solely
from oxygen ions, allowing
x
to be measured with high
sensitivity [
24
]. For single crystals under high pressure, where
a full structure refinement is not practical due to time and
geometry constraints, measurements of the (2,2,0) reflection
are especially suitable to reveal the evolution of
x
[
24
]. The
normalized diffraction intensities increase slightly under pres-
sure [Fig.
1(b)
], indicating
x
varies by
x
∼
0
.
008
±
0
.
006
220404-2
APPROACHING THE QUANTUM CRITICAL POINT IN A ...
PHYSICAL REVIEW B
101
, 220404(R) (2020)
18
19
20
0
100
200
300
139.4
140.4
141.3
142.3
11.19
11.22
11.25
azimuthal
(deg):
Sm
2
Ir
2
O
7
T = 4 K
P = 2.9 GPa
18
19
20
0
100
200
137.6
138.5
139.5
140.3
141.4
11.19
11.22
11.25
P = 5.6 GPa
18
19
20
0
50
100
Counts / s
137.0
138.4
139.4
141.2
142.6
11.19
11.22
11.25
P = 6.3 GPa
17
18
19
0
20
40
136.1
136.6
137.0
137.9
138.4
139.4
11.19
11.22
11.25
P = 6.7 GPa
18
19
20
21
0
20
40
(deg)
136.6
138.0
139.3
141.3
143.1
144.0
11.19
11.22
11.25
E (keV)
P = 7.2 GPa
(a)
(b)
(c)
(d)
(e)
E = 11.215 keV
(0, 0, 6)
-
18
19
20
0
100
200
300
400
136.6
138.4
140.3
141.3
142.2
11.19
11.22
11.25
P = 0.9 GPa
(f)
FIG. 2. Raw x-ray magnetic diffraction profiles of both mosaic
and energy resonance at (a)–(d) below, and (e), (f) above the critical
pressure from one sample. The common spectral weight (marked
by solid red circles) between energy spectra at all azimuthal angles,
defines the resonance profile. All counting rates are normalized to a
100 mA synchrotron ring current in this figure and Fig.
3
.
from ambient pressure to 21 GPa when the change of intensity
is compared to simulations [Fig.
1(c)
].
The anisotropic tensor susceptibility (ATS) resonance
[
12
–
14
,
22
,
24
] is sensitive to the individual
t
2g
and
e
g
bands
of Ir 5
d
states. Measured at the (2, 4, 0) reflection in the
polarization-switching
π
-
σ
channel [Fig.
1(d)
], the ATS res-
onance demonstrates a constant shape under pressure. In
R
2
Ir
2
O
7
, the ATS resonance profile differs in shape from
that of the magnetic resonance (Fig.
2
and Refs. [
13
,
14
]),
indicating that the magnetic electrons are confined in the
lower
t
2g
band. Our result in Fig.
1(d)
demonstrates that
both bands experience no significant energy shift over this
Sm
2
Ir
2
O
7
P = 6.3GPa
010203040
0
1
2
T (K)
I
(006)
/ I
(004)
x1e-6
(a)
(b)
E(keV):
11.215
(0, 0, 6)
-
11.19
11.22
11.25
E (keV)
17.6
18.1
0
50
(deg)
Counts / s
4
10
13
16
18
20
21
22
23
24
25
26
27
28
30
32
35
40
T (K):
0
20
2
(deg)
Counts/s
4K
(c)
37.9
38.0
0
10
0
2
23 K
27 K
(0, 0, 6)
-
(0, 0, 6)
-
01020
10.1954
10.1964
T(K)
a
(Å)
FIG. 3. Temperature evolution of the resonant x-ray magnetic
diffraction at 6.26 GPa. (a) Mosaic scans and energy profiles mea-
sured at the mosaic position indicated by the arrow. Above 27 K,
the magnetic resonance below the Ir
L
3
edge [
13
] fully disappears
and the mosaic profile no longer varies. Given the similar shapes
of three mosaic profiles at azimuthal
ψ
=
137
◦
∼
140
◦
in Fig.
2(d)
,
the residual mosaic form is likely due to dislocations and voids,
instead of multiple scattering. For resonance profiles, the residual
spectral weight above
T
N
at 11.225 keV is due to enhanced mul-
tiple scattering above the absorption edge. (b) Integrated mosaic
intensity in (a) vs. temperature, fit to a power law plus a con-
stant. (Inset) Lattice expansion at low temperature demonstrates a
noticeable magnetostriction effect. (c) Longitudinal scans of the
(0,0,6) reflection, fit to resolution-limited pseudo-Voigt line shapes
at 4 and 23 K, and a Lorentzian shape plus a linear background
at 27 K.
pressure range. A constant ATS resonance is also observed
in Cd
2
Os
2
O
7
under pressure [
24
].
Through many resonant x-ray diffraction studies in the
recent decade [
12
–
14
,
22
,
24
], it has been clarified that the
magnetic order parameter of the AIAO type of antiferromag-
netism can be extracted from the (0,0,6) reflection intensity
in a polarization-switching channel under specific azimuthal
conditions that would totally suppress the ATS contribution
(Supplemental Material [
25
]). Both the sample mosaic and
energy resonance of the (0,0,6) reflection were followed at
4 K across the magnetic quantum phase transition (Fig.
2
).
Through careful studies of the resonance shapes at finely
stepped azimuthal positions, one can distinguish the influence
of multiple scattering, extract the true magnetic behavior, and
verify the magnetic phase boundary. The effort of minimizing
the multiple scattering during the measurement time further
220404-3
YISHU WANG
et al.
PHYSICAL REVIEW B
101
, 220404(R) (2020)
FIG. 4. (a) Integrated magnetic (0,0,6) reflection intensity in the
π
-
σ
channel, normalized by the (0,0,4) reflection intensity in the
π
-
π
channel. A power-law fit plus a constant (solid line) reveals
P
c
=
6
.
30 GPa at
T
=
4
.
0
±
0
.
3 K. (b) Integrated (0,
−
2,4) and (0,0,6) reflection
intensities in the
π
-
π
channel, normalized by the (0,0,4) reflection intensity, indicate the
Fd
-3
m
space group persists to 21 GPa. (c) The
projected AIAO phase of Sm
2
Ir
2
O
7
(gray area) is scaled from the power-law fit of the magnetic intensity in (a), with magnetic
P-T
phase
boundary points measured on our samples (orange circles), and metal-insulator transition points (green circles) from Ref. [
21
]. (d) Correlation
between pyrochlore lattice parameters
x
and
a
in
R
2
Ir
2
O
7
for elements
R
=
Gd (green down triangles), Eu (orange diamonds), Sm (purple
squares), Nd (blue circles), and Pr (red up triangles), from seven independent research groups (Supplemental Table, Refs. [
21
,
25
,
26
,
31
,
34
–
37
]).
(
x
,
a
) values of the same group but of different element
R
are connected by linear segments, highlighting the correlation between
x
and
a
(gray
dashed line) despite systematic variations between different crystal growers. (e) Evolution of the AIAO order in the three-dimensional
T-a-x
phase space as a function of
P
and
R
, including Sm
2
Ir
2
O
7
under pressure (red circles) and the
R
series (blue squares) following the
x-a
correlation in (d). The opposing branches of doping and pressure point to the different role played by correlations.
allows a detailed study of the temperature evolution of the
magnetic (0,0,6) reflection in Sm
2
Ir
2
O
7
at a fixed azimuthal
position. At
P
=
6
.
26 GPa, just below
P
c
, the mosaic pro-
file is measured from 4 to 40 K, with the energy reso-
nance profile measured at selected temperatures in between
[Fig.
3(a)
]. The integrated diffraction intensity continuously
approaches a constant beyond 27 K [Fig.
3(b)
], demon-
strating a second-order thermal AIAO phase transition at
6.26 GPa.
High-resolution longitudinal scans of the (0,0,6) reflection
at 6.26 GPa and both 4 and 23 K indicate that Sm
2
Ir
2
O
7
still
has long-range AIAO order, as the diffraction line shapes are
instrument resolution limited with a spin correlation length of
at least 1450 Å. At 27 K, the line shape broadens to a diffu-
sive shape, indicating a shortened spin correlation length of
∼
450 Å at the magnetic transition. Furthermore, our high-
resolution 2
θ
value of the (0,0,6) diffraction reveals that the
lattice constant shrinks with increasing temperature from 4 to
27 K due to a decreasing staggered moment
M
[Fig.
3(b)
,
inset]. This anomalous
a
(
T
) reflects the overall magnetostric-
tion, which was also observed in antiferromagnetic Nd
2
Ir
2
O
7
and NiS
2
at ambient pressure [
26
,
27
]. As
a
(
T
)evolvessim-
ilarly to the diffraction intensity in Fig.
3(b)
, there is a con-
sistent relationship of
a
(
T
)
∼
I
(006)
(
T
)
∼
M
2
. The mag-
netostriction
a
/
a
∼
5
×
10
−
4
in Sm
2
Ir
2
O
7
(
T
N
=
26
.
8K)
is comparable in size to that in Nd
2
Ir
2
O
7
(
T
N
=
33 K) [
26
].
At
P
=
6
.
7 GPa, above
P
c
, there is no observed magnetic
resonance [Fig.
2(d)
], and a similar study of the mosaic profile
generates no temperature dependence up to 20 K.
We fit both the thermal and pressure evolution [Fig.
3(b)
and Fig.
4(a)
] of the resonant magnetic diffraction intensity
to critical power-law forms:
I
(006)
∼
(
T
c
-
T
)
2
β
and
I
(006)
∼
(
P
c
-
P
)
2
γ
. We find
T
c
=
26
.
8
±
0
.
3 K and
β
=
0
.
41
±
0
.
05
at
P
=
6
.
26 GPa, and
P
c
=
6
.
30
±
0
.
05 GPa and
γ
=
0
.
15
±
0
.
03 at
T
=
4
.
0 K. The exponent
β
is between the mean-field
expectation of 0.5 and three-dimensional Heisenberg spin
fluctuations of 0.37, but the order parameter evolves more
rapidly under pressure with a small
γ
.
For the
P-T
phase diagram of AIAO order in Sm
2
Ir
2
O
7
,
we scale
T
N
(
P
) by the magnetic diffraction intensity
I
(006)
in
220404-4
APPROACHING THE QUANTUM CRITICAL POINT IN A ...
PHYSICAL REVIEW B
101
, 220404(R) (2020)
Fig.
4(a)
. This mapping of magnetic intensity to the phase
boundary is justified by the consideration that within this
small pressure range (
a
∼
0
.
10 Å or
a
/
a
∼
1
.
0%), the
order parameter
M
should connect to the energy scale of
T
N
as
I
(006)
(
P
)
∼
M
2
∼
T
N
(
P
), similarly demonstrated in
several antiferromagnets under pressure [
4
,
24
]. The projected
phase boundary
T
N
(
P
) is consistent with the three observed
phase points [Fig.
4(c)
], identified through magnetization
M
(
T
) at ambient
P
, the temperature dependence of
I
(006)
(
T
)
at
P
=
6
.
26 GPa, and the pressure dependence of
I
(006)
(
P
)
at
T
=
4 K. At ambient pressure, Sm moments in Sm
2
Ir
2
O
7
have an estimated size of 0
.
1
μ
B
/
Sm
3
+
, and order at
T
∼
10 K [
28
]. Both are much smaller than the Ir moment size of
0
.
3
μ
B
/
Ir
4
+
and the ordering temperature
T
N
∼
110 K [
28
].
By comparison to several pyrochlore iridates with large
A
-site
moments (2
.
6
−
9
μ
B
per Nd
3
+
,Er
3
+
,orTb
3
+
)[
29
,
30
], the
magnetic coupling strength between Sm
3
+
4
f
moments is
likely much below 0.1 meV [
29
], and Sm
3
+
ordering relies on
the assistance of the Ir
4
+
molecular field, making it parasitic to
the Ir AIAO order. We thus consider Sm
3
+
ions as disordered
at
P
c
=
6
.
30 GPa.
Although both pressure and chemical variation of
R
(from
Eu to Pr) in
R
2
Ir
2
O
7
[
19
] are effective in suppressing
T
N
to
zero while the lattice persists in the cubic
Fd
-3
m
symmetry,
there exist microscopic differences between these two tuning
mechanisms. For the two structural parameters
a
and
x
of
the pyrochlore lattice,
a
decreases
∼
1.0% at
P
c
in Sm
2
Ir
2
O
7
,
but increases
∼
1.5% in the chemical series from Eu to Pr
[Fig.
4(d)
]. The parameter
x
indicates compressive trigonal
distortion of the octahedron IrO
6
.ForEu
2
Ir
2
O
7
and Pr
2
Ir
2
O
7
at ambient pressure,
x
reduces from 0.339 to 0.330 [Fig.
4(d)
].
However,
x
increases by
∼
0.0025
±
0.0017 in Sm
2
Ir
2
O
7
at
P
c
(Fig.
1
). In other 5
d
AIAO pyrochlores,
x
increases
from 0.330 to 0.335 in Eu
2
Ir
2
O
7
over 17 GPa at 295 K [
31
],
and from 0.319 to about 0.325 in Cd
2
Os
2
O
7
over 40 GPa at
4K[
24
].
AIAO order exists in pyrochlore lattices due to both the
electron correlation,
U
/
t,
and local Ising spin anisotropy. From
symmetry considerations, the pyrochlore structure naturally
hosts the Dzyaloshinskii-Moriya (DM) interaction, and the
direct
DM interaction leads to the AIAO order [
32
]. As the
DM strength is proportional to the spin-orbit coupling
λ
,
AIAO order has been discovered in many 5
d
pyrochlores. The
trigonal distortion of local crystal fields is of similar strength
to
λ
[
14
,
33
], and varies the Ir-O-Ir bond angle to affect all
ranks of the superexchange interaction. The increasing
x
in
the chemical series sharpens the Ir-O-Ir bond angle from
∼
131
.
3
◦
at
x
=
0
.
330 in Pr
2
Ir
2
O
7
to
∼
126
.
7
◦
at
x
=
0
.
339
in Eu
2
Ir
2
O
7
. By contrast, a shrinking
a
(
P
) directly increases
the 5
d
bandwidth or equivalently the hopping strength
t
.A
3% volume reduction at 6.7 GPa would inject
∼
600 meV
energy into each unit cell [
22
], presumably distributed among
all valence electrons at the Fermi surface, with half to broaden
the Ir 5
d
band. The increased spatial extent of Ir 5
d
orbitals
under pressure reduces
U
/
t
. Conversely, the lattice expansion
in the chemical series
R
2
Ir
2
O
7
reduces the hopping strength
t
towards the paramagnetic metal. We thus expect
t
to be
predominantly affected by
a
, while the spin anisotropy is
controlled by
x
.
The contrasting effects of pressure and chemical tuning
now become clear and are captured in Fig.
4(e)
. Pressure
maintains the axial nature of the local spin anisotropy, but
increases the hopping integral
t
and reduces
U
to suppress the
long-range order. Chemical tuning of
R
2
Ir
2
O
7
suppresses the
AIAO state by reducing the axial spin anisotropy towards a
more isotropic Heisenberg state, mainly through a weakened
DM interaction from a more obtuse Ir-O-Ir angle [
7
,
10
].
There are likely two separate pathways across the quantum
phase boundary in
U
/
t
-
λ/
t
parameter space [Fig.
4(e)
][
19
],
accompanied by different types of spin fluctuations at the
respective critical points.
A metal-insulator transition runs concurrently with the
AIAO magnetic order in
R
2
Ir
2
O
7
(
R
=
Eu, Sm, and Nd)
at ambient pressure, and was measured in pelleted poly-
crystalline Sm
2
Ir
2
O
7
up to 2.2 GPa [
21
]. It demonstrates a
negative pressure dependence similar to our measured AIAO
magnetic phase boundary [Fig.
4(c)
]. Several theoretical
simulations [
6
,
7
,
9
] have suggested the existence of a Weyl
semimetallic AIAO phase between the AIAO Mott insulator
and the paramagnetic metal. While the Coulomb interaction
U
varies from 0.5 to 2 eV in various models, there seems
to be agreement that the Weyl semimetal phase spans a
finite width of
U
∼
0
.
2 eV. As the pressure-driven AIAO
quantum phase transition happens within
∼
0
.
3 eV change in
t
from the ambient condition, a span of
U
∼
0
.
2 eV would
likely cover the whole pressure range of AIAO order evolution
in Sm
2
Ir
2
O
7
. Nevertheless, given the small critical exponent
γ
and the strongly convex shape of the
P-T
phase diagram, the
electronic structure might well only demonstrate topological
features very close to the pressure phase boundary (if at all).
Our direct magnetic phase diagram pinpoints such a narrow
phase space of interest.
Sm
2
Ir
2
O
7
represents one of the cleanest systems to explore
the AIAO type of antiferromagnetic quantum criticality with
large Ir moments maintaining local Ising anisotropy under
pressure. With no breaking of inversion symmetry through the
quantum critical point, it provides a fascinating comparison
to the strongly coupled AIAO quantum phase transition in
Cd
2
Os
2
O
7
[
24
]. Although the electronic evolution through
P
c
remains to be resolved, theoretical simulations agree in
general that a metallic paramagnetic phase exists beyond the
AIAO order. Whether or not there exists a magnetic Weyl
semimetal phase, a strong electronic evolution likely exists
close to the AIAO quantum critical point, providing the
quantum critical region strong-coupling characteristics with
intertwined spin and charge fluctuations.
We are grateful to Y. Ren for discussions. Y.F. acknowl-
edges support from Okinawa Institute of Science and Tech-
nology Graduate University with subsidy funding from the
Cabinet Office, Government of Japan. The work at Caltech
was supported by National Science Foundation Grant No.
DMR-1606858. The work in Oxford was supported by U.K.
Engineering and Physical Sciences Research Council Grant
No. EP/N034872/1. The work at the Advanced Photon Source
of Argonne National Laboratory was supported by the U.S.
Department of Energy Basic Energy Sciences under Contract
No. DE-AC02-06CH11357.
220404-5
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