1
Approaching the quantum critical point in a highly
-
correlated
all
-
in
-
all
-
out
antiferromagnet
Yishu Wang
1
,2
,
T. F. Rosenbaum
1
,
D. Pra
bhakaran
3
, A. T. Boothroyd
3
,
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
*
Correspondence
author. Email:
<
yejun@oist.jp
>.
A
BSTRACT
:
C
ontinuous
quantum
phase transition
s
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
trac
e
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
, w
hile the
lattice
symmetry remains in the
cubic
Fd
-
3m
space group
across the quantum critical point
.
Comparing pressure tuning and
the
chemical series
R
2
Ir
2
O
7
reveals that the
suppression of the AIAO order and the
approach
to the spin
-
disordered state is characterized by contrasting evolutions of
both
the
pyrochlore lattice
constant
a
and
the trigonal distortion
x
.
The former affects the 5
d
bandwidth, the latter the Ising anisotropy, and as such 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
.
The mix of magnetic interactions, electron
correlations, and spin
-
orbit coupling informs
the competition between different quantum ground states and ordering mechanisms
,
rang
in
g
from
Mott
to
Slater
antiferro
magnetic insulators
[
1
-
2
]
,
phonon
-
to
spin
-
fluctuation
-
mediated
superconductivity
[
3
-
4
]
,
and
Kondo screening
to
RKKY exchange 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 correlation
s
and
strong spin
-
orbit coupling
lead
s
to
all
-
in
-
all
-
out (AIAO)
antiferromagnetic order
and
,
potentially
,
non
-
trivial
topological
band structure,
commonly
known
as
a
Weyl semimetal
of
broken time reversal symmetry
[
6
-
10
]
.
Conversely, w
ithout
electron correlation,
strong
spin
-
orbital coupling
could
induce
a
different
topological
Weyl
state
of
broken inversion symmetry
, as
proposed
in
non
-
magnetic
pyrochlore
s
with a breathing
lattice
[
11
].
AIAO spin order
in 5
d
pyrochlores
has been
verified
experimentally
in
both
R
2
Ir
2
O
7
(
R
=
Lu,
Yb,
Tb,
Eu, Sm, Nd)
and Cd
2
Os
2
O
7
[
1
2
-
1
7
]
.
However,
recent angle
-
resolved
photoemission
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 structure
s
[
2,
1
8
]
that
raise
question
s
about
the
existence of
magnetic Weyl
semimetal
phases
in these compounds
at ambient
pressure
.
U
nder suitable tuning process
es
,
such as pressure,
exotic
states
may yet emerge
over
2
an adjustable
parameter space
spanned by
the
Coulomb
interaction
U
and spin
-
orbit coupling
l
, normalized
to
the hopping
integral
t
[
1
8
,
19
].
Given that
the
presence of
AIAO
magnetic
order
serves as a g
auge
of
electron correlation
s
,
its quantum critical point, where the
magnetic
order is suppressed t
o
zero at
zero temperature,
could
identify
some of the most intriguing
region
s
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
develop
s
a two
-
in
-
two
-
out spin ice configuration
that
melts
in
to a
metallic
spin liquid
at
T
< 0.4 K
in the proximity
of
its
AIAO
quantum critical point
[
2
0
].
The
i
ridate pyrochlores
R
2
Ir
2
O
7
provide
a
series
of
model system
s
susceptible to
continuous pressure tuning,
with a
n approximately
local
J
eff
=1/2
moment
from the
Ir
4+
i
ons of
the
5
d
t
2g
band
[
15
]
and
a number of germane
theoretical calcu
l
ations
[
6
-
10
]
.
Sm
2
Ir
2
O
7
,
with
proven
AIAO order [
1
4
]
, negative pressure
dependence of its insulating phase
[
2
1
]
,
and
available high
-
quality single crystals
, is
a particularly promising
experimental
choice
. A
s we
demonstrate below,
the
AIAO
spin
order
in Sm
2
Ir
2
O
7
experiences
a continuous quantum phase
transi
tion
at a modest
critical
pressure
P
c
=6.30 GPa
in the presence of
constant
lattice
symmetry
,
exemplifying the first directly
-
tracked
AIAO
quantum critical point
in
iridates
under pre
s
sure
.
Moreover, the
pressure evolution
of Sm
2
Ir
2
O
7
follows a different pathway
across the
U
/
t
-
l
/
t
phase space
compared
with
the
R
2
Ir
2
O
7
chemical series
,
providing further
clues to the nature of
intermediate
to
strong
-
coupling
physics in this model system.
We directly probe the AIAO
spin
order in Sm
2
Ir
2
O
7
under pressure
using resonant x
-
ray
magnetic diffraction
at beamline 4
-
ID
-
D of the Advanced Photon Source
(
[
2
2
-
2
3
]
and
Supplementary materials)
.
The pyrochlore structur
e
in the
Fd
-
3m
space group
is fully
ch
aracterized by two parameters, the lattice constant
a
and the
coordination parameter
x
[
9
].
From
the
single
-
peaked (
1
,
1
,
1
) and (2, 2, 0)
diffraction
orders,
Sm
2
Ir
2
O
7
remains in a cubic
structure
to at least
21 GPa
, and t
he lattice
constant
a
(
P
)
evolves
contin
u
ous
ly
at
4 K
without
any visible
sign
of
a
phase transition
(Fig. 1)
.
The
simple
lattice
evolution
strongly
suggests a
continuous
AIAO
quantum phase transition,
motivating
further
polarization
-
analysis of
resonant
ly scattered
x
-
ray magnetic diffraction
signals
(
Figs. 2
-
4
)
.
The
cubic
space group
under
pressure
is
illustrated
by
m
easured (
0
,
0,
6
) and (
0
,
-
2,
4
) diffraction intensities
in the
p
-
p
’
channel, which are
minimal and
constant
through 21 GPa
(Fig.
4
b
).
Thus the
Fd
-
3m
space
group persists
at 4
K
,
ruling out a
breathing
lattice
instability of
the
F
-
43m
space group
type
as
observed in Cd
2
Os
2
O
7
[
2
4
]
.
Within the
Fd
-
3m
space group,
both Sm and Ir
ions
in Sm
2
Ir
2
O
7
do not contribute to
diffraction inte
nsities of (1,
1,
1) and (2,
2,
0)
orders
. As the
se
diffraction
intensities
arise
solely
from
oxygen ions, the parameter
x
can be m
easured with high sensitivity
[
24
]
. F
or single
crystals under high pressure
, where a full
structure
refinement is not
practical
due to time
and
geometry
constrain
t
s, measurements o
f
these peaks are especially suitable to reveal the
pressure evolution of
x
[
24
]
.
T
he
normalized diffrac
tion intensities
in the
p
-
p
’ channel
appear
constant under pressure
(Figs.
1
b
,
1
c
), implying
a stable
x
over
21 GPa in Sm
2
Ir
2
O
7
.
The
ATS
resonance
in the
p
-
s
channel
, measured at
the
(
2
,
4
,
0) order
and
y
~
0
o
,
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. 3
and Ref. [
13
-
14
]), indicating
that
the
magnetic electrons are con
fin
ed in
the
lower
t
2g
band.
T
he ATS resonance
is sensitive to
the
individual
t
2g
and
e
g
bands
of
Ir 5
d
states
,
and
our result
in Fig.
1
d
demonstrates
that
both bands
experience
no significant energy shift
over this pressure range.
The
constant
behavior of
ATS
resonance
is also observed in
Cd
2
Os
2
O
7
under pressure
[
24
].
3
W
e
show
in Fig. 2
raw
magnetic diffraction
profiles
in the
p
-
s
channel
of
both
the
sample mosaic
and energy
resonance
of
the
(
0
,
0,
6
) order
at
pressures
across
the magnetic
quantum
phase transition
.
In addition to the evolution
with
P
at fixed
T
=
4
K
,
we
explore
the
temperature evolution of the (
0
,
0
,
6
) order
in Sm
2
Ir
2
O
7
at
P
=
6.
26
GPa
, just below
P
c
.
The
m
osaic profile
is
measured
up
to 40
K
for one azimuthal condition
,
with
the energy resonance
profile measured
at selected
T
(Fig.
3
a
)
.
T
he integrated
diffraction
intensity continuously
approach
es
a constant beyond
27
K
(Fig
.
3
b
)
,
demonstrating
a
second
-
order
thermal AIAO
phase
transition at 6.
26
GPa.
High
-
resolution l
ongitudinal
scans of
the
(0,
0,
6) order
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
coherence length of at least
1
45
0
Å.
At 27 K,
the line
shape broadens
to a diffusive
shape
, indicating
a short
ened
spin
correlation length
of
a
pproximately
4
5
0
Å
at the
magnetic
transition
.
Furthermore
,
our high
-
resolution study of
the
2
q
value of
the
(0,
0,
6)
diffraction
reveals
that
t
he lattice constant
a
shrinks with increasing
T
from
4
to
27 K
due to
a
decrea
sing <
M
>
(Fig.
3
b
inset)
.
Th
is
anomal
ous
a
(
T
) is a reflection of
the overall magnetostriction, which was also observed in
the
antiferromagnets Nd
2
Ir
2
O
7
and
NiS
2
at ambient pressure [
2
5
,
2
6
].
As
a
(
T
) evolve
s
similarly to
the diffraction intensity
in Fig.
3
b
,
there is a
con
sistent
relationship of
D
a
(
T
)
~
I
(006)
(
T
)
~<
M
>
2
.
T
he magnetostriction
D
a
/
a
~5
10
-
4
in Sm
2
Ir
2
O
7
(
T
N
=
26.7K
)
is
comparable
in size
to
that
in
Nd
2
Ir
2
O
7
(
T
N
=33K
)
[
2
5
]
.
At
P
=
6.7
GPa,
above
P
c
,
there is
no
observed
magnetic resonance (Fig.
2
d
)
,
and a similar temperature
study of the mosaic profile generates no temperature dependence up to
2
0
K
.
We fit b
oth
the
thermal and pressure evolution
(Figs.
3
b
,
4
a
)
of the
resonant
magnetic
diff
raction intensity
to
critical
power law form
s:
I
(006)
~(
T
c
-
T
)
2
b
and
I
(006)
~
(
P
c
-
P
)
2
g
.
We find
T
c
=26.8
±
0.3
K
and
b
=
0.
41
±
0.05
at
P
=
6.
26
GPa
,
and
P
c
=
6.3
0
±
0.05
GPa
and
g
=
0.15
±
0.03 at
T
=
4
.0
K
.
The
exponent
b
is
between
the
mean
-
field
expectation
of 0.5
and three
-
dimensio
n
al
Heisenberg spin fluctuation
s
of 0.37
,
but
the
order parameter
evolves
more rapidly
under
pressure
with
a small
g
.
For
the
P
-
T
phase diagram
of
AIAO order
in Sm
2
Ir
2
O
7
,
w
e
scale
T
N
(
P
)
by
the magnetic
diffraction intensity
I
(006)
in Fig.
4a
.
This mapping of
m
agnetic intensity to the phase boundary
is justified by the consideration
that within
this
small pressure range
(
D
a
~
0.1
0
Å
or
D
a
/
a
~1
.0
%
)
,
the order parameter strength, defined as the staggered moment <
M
>,
should
connect to
the
energy scale of
T
N
as
I
(006)
(
P
)
~<
M
>
2
~
T
N
(
P
).
This relation
has been
demonstrated
previously
in
several antiferromagnet
s
und
er press
ure [
4
,
2
4
].
The projected phase boundary
T
N
(
P
) is
consistent with the three observed phase points (Fig.
4c
)
,
identified
through
mag
n
e
tization
M
(
T
)
at ambie
nt
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 moment
s
in Sm
2
Ir
2
O
7
have
an
estimated
size
of 0.1
μ
B
/
Sm
3+
, and order at
T
~10K
[
27
]
. Both are much smaller
than
the
Ir
moment size of 0.3
μ
B
/
Ir
4+
and
the
ordering
temperature
T
N
~110 K
[
27
]
.
By
comparison to
several
pyrochlore
iridate
s with
large
A
-
site
moments
(2.6
-
9
μ
B
per Nd
3+
, Er
3+
,
or
Tb
3+
)
[
28
,
29
],
t
he
magnetic
coupling strength between
Sm
3+
4
f
moments
is
likely
much below 0.1 meV
[
28
]
,
and
Sm
3+
o
rdering would rely
on
the
assistance
of
the
Ir
4+
molecular field,
making
it
parasitic to
the
Ir
AIAO
order
.
W
e
thus
consider
Sm
3+
ions
as
disorder
ed
at
P
c
=6.30 GPa and
T
=
4
K.
A
lthough
both p
ressure and chemical
variation of
R
(from Eu, Sm, Nd, to Pr)
in
R
2
Ir
2
O
7
[
19
]
are effective in
suppress
ing
T
N/MIT
to zero whil
e the lattice persists
in the cubic
Fd
-
3m
4
symmetry,
there
exist
microscopic
difference
s
between these two
tuning
mechanisms
.
For
the
two structural parameters
a
and
x
of the pyrochlore lattice
[
9
]
,
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.
4d
)
.
T
he parameter
x
indicates
compressive trigonal distortion of
the octahedron
IrO
6
.
For
Eu
2
Ir
2
O
7
and
Pr
2
Ir
2
O
7
a
t ambient pressure
,
x
reduces
from 0.339 to 0.3
30
with chemical variation
(Fig.
4d
)
.
However,
x
is
largely constant
if not increasing
in Sm
2
Ir
2
O
7
under pressure
(Fig. 1)
. In
other 5
d
AIAO
pyrochlores,
x
inc
r
eases
from 0.330 to 0.335 in
Eu
2
Ir
2
O
7
over 17 GPa
at 295
K
[
30
]
,
and from
0.319 to
about
0.325 in
Cd
2
Os
2
O
7
over
40 GPa at 4 K
[
2
4
]
.
AIAO order exists i
n pyrochlore
lattice
s
due to both
the electron
correlation
,
U
/
t
,
and
local Ising spin anisotropy
.
From symmetry considerat
i
o
n
s
,
the
p
yrochlore structure
naturally
host
s
the
Dz
yaloshinskii
-
Moriya interaction
,
and the
direct
DM interaction
lead
s
to
the AIAO
order
[
31
].
As t
he DM strength is
proportional
to the
spin
-
orbit coupling
l
, AIAO order
has
been
discovered
in many
5
d
pyrochlores
.
The
trigonal distortion of local crystal field
s
is
of
similar
strength
to the spin
-
orbit coupling
[
14
,
32
]
,
and
varies
the Ir
-
O
-
Ir bond
angle
to
affect
all ranks of the
super
-
exchange
interaction
.
The increasing
x
sharpen
s
the Ir
-
O
-
Ir bond angle
from
~
13
1
.
3
o
at
x
=0.3
30
in Pr
2
Ir
2
O
7
to
~12
6.
7
o
at
x
=0.3
39
in Eu
2
Ir
2
O
7
.
By contrast
, a
shrinking
a
(
P
)
directly increase
s
the 5
d
bandwidth or equivalently the hopping strength
t
. A
3% volume
reduction at 6.7 GPa would inject ~
6
00
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 exten
t
of Ir 5
d
orbitals under pressure reduce
s
U
/
t
. Conversely,
the
lattice
expansion in
the
chemical series
R
2
Ir
2
O
7
reduce
s
the hopping strength
t
towards
the
paramagnetic metal
.
W
e
thus
expect
t
to be
predominantly
affected
by
the
lattice constant
,
while
the
spin
anisotropy is controlled by
x
.
The contrasting
effects
of
p
ressure and chemical
tuning
now
become clear
and are
captured
in Fig.
4e
.
P
ressure 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 spin state,
mainly
through
a
weakened
Dzyaloshinskii
-
Moriya
interaction
from
a
more
obtuse
Ir
-
O
-
Ir angle
[
7
,
10
]
.
There are likely
two
separate
path
way
s
across
the
quantum
phase boundary
in
U
/
t
-
l
/
t
parameter
space
(Fig.
4e
)
[
19
]
,
accompanied by
different types of spin fluctuations at
the
respective
critical points
.
A
metal
-
insulator trans
ition runs concurrently
with
t
he
AIAO
magnetic
order in
R
2
Ir
2
O
7
(
R
= Eu, Sm, and Nd)
at ambient pressure
, and
was measured
in
pelleted
polycrystalline
Sm
2
Ir
2
O
7
up
to 2.2 GPa
[
21
]
.
The
pressure suppressi
on of
the insulating phase
is
consistent
with
our measured AIAO magnetic phase boundary
(Fig.
4
c
)
for the limited region they overlap
.
S
everal theoretical
simulations
[
6
-
7, 9
]
have
suggest
ed
the existence of
a
Weyl semi
metallic
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
span
s
a
finite
width of
D
U
~
0.2
eV.
As
the
pressure
-
driven
AIAO quantum phase transition
happens
with
in
~
0.
3
eV
change in
t
from the ambient condition
,
a
span
of
D
U
~
0.2
eV
would
likely
cover
the whole pressure range
of
AIAO order
evolution
in Sm
2
Ir
2
O
7
.
Nevertheless, g
iven
the
small critical exponent
g
and
the strongly con
vex 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)
.
Sm
2
Ir
2
O
7
represents
one of
the
cleanest
system
s to explore the AIAO type of
antiferromagnetic quantum criticality
with
Ir
moments
maintain
ing
local Ising
anisotropy
5
under pressure
and
being much
larger
in
size
than
the
Sm
moments
. With no breaking of
inversion
symmetry through the quantum critical point, it
provide
s
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
,
most theoretical simulations
agree
in
general
that
a metallic paramagnetic phase
exists
beyond the AIAO order
.
Whether o
r
not there
exists
a
magnetic
Weyl semimetal phase
,
a
strong electronic
evolution
likely
exist
s
close
to
the
AIAO quantum critical point
,
providing
the quantum critical region
strong
-
coupling
characteristics
with intertwined spin and
charge
fluctuations.
Acknowledgments
Y.F.
acknowledges the support from Okinawa Institute of Science and Technology
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 UK 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 US
Department of Energy Basic Energy Sciences under Contract No.
NEAC02
-
06CH11357.
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7
Figur
e captions
:
Fig. 1.
(
a
) Lattice constant
a
(
P
)
at 4.0
±
0.3 K
is
f
i
t
to a two
-
parameter Birch equation,
with
a
bulk modulus
B
0
=
215.6
±
4.8
GPa
, and its derivative
B
’=
3.9
±
0.5.
(inset) Representative
longitudinal
(
q
-
2
q
)
scans of
the
(2, 2, 0)
and
(1, 1, 1)
order
s
at five pressures
, showing
single
peaks with minimal traces of a
stressed lattice condition
.
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
-
c
) Diffraction intensities of
the
(2, 2, 0)
and
(1, 1,
1)
orders
,
norm
alized by those
of
the
(4, 4, 0)
and
(2, 2, 2)
orders
,
respectively
. Different symbols (circle,
square, and diamond)
in Figs.
1
b
,
1
c
,
4
a
,
and
4
a
represent each of three
individual
samples
studied under pressure. (
d
) The ATS resonance
at
four
pressure
s and
4
K.
The spectral shape
has no azimuthal dependence, from measurements at (2, 4, 0) and (0,
-
2, 4) orders with
y
~0
o
and ~35
o
, respectively, up to 21 GPa (not shown).
Fig. 2. Raw
x
-
ray magnetic diffraction
profiles
of both mosaic and energy
resonance
at both
(
a
-
d
) below
,
and (
e
-
f
) above the critical pressure
P
c
=
6.3
0
GPa
from one sample
.
The
common
spectral weight (marked by
solid
red
circles
)
between
energy spectra
at
all
azimuthal angles,
defines the
multiple
-
scattering
-
free
resonance profile.
All counting rates are normalized to a
100
-
mA synchrotron ring
-
current
in Figs. 2 and 3
.
Fig. 3.
Temperature
evolution
of the r
esonant x
-
ray
m
agnetic diffra
ction
at 6.26 GPa, measured
at
the
(0,
0,
6) order in the
p
-
s
channel
.
(
a
) Mosaic scans and energy profiles.
A
bove
27
K
, t
he
magnetic
resonance fully disappears
and the mosaic profile no
longer
varies
.
Given the similar
shapes of three mosaic profiles at azimuthal
y
=137
o
~140
o
in
Fig. 2
c
, the
residual
mosaic
form
is likely due to dislocations and voids, instead of multiple scattering
, and
could be attributed to
a small
s
component in the
p
-
polarized synchrotron light, sharing the same origin
as
the
minimal (0, 0, 6) diffraction intensity in the
p
-
p
’ channel (Fig. 4
b
).
(
b
) Integrated mosaic
intensity
in Fig.
3
a
as a function of temperature. The data is fit to a power law p
lus a constant.
(inset) Lattice expansion at low temperature
demonstrates a noticeable
magnetostriction
effect
.
(
c
)
Longitudinal scans of
the
(0
,
0
,
6) order
at three different temperatures, fit to
resolution
-
limited Pseudo
-
Voigt
line shapes
at 4
K and 23
K, and
a
diffusive Lorentzian
shape plus
a
linear background at 27 K.
Fig. 4.
(
a
)
I
ntegrated
magnetic
diffraction intensity
of
the
(0
,
0
,
6) order
in the
p
-
s
channel
,
normalized by
that
of
the
(0
,
0
,
4) order in the
p
-
p
’
channel
.
A fit to a power law plus a constant
(solid line)
reveals
the AIAO quantum phase transition
at
P
c
=6.30 GPa at
T
=
4.0
±
0.
3 K
.
(
b
)
Integrated intensities of
the
(0
,
-
2
,
4) and (0
,
0
,
6) orders
in the
p
-
p
’ channel
, normalized by
that
of
the
(0
,
0
,
4) order,
indicat
e
the
Fd
-
3m
space group
persist
s
to 21 GPa.
(
c
) The projected
P
-
T
phase diagram of
Sm
2
Ir
2
O
7
(gray area)
is
scaled
from the power
-
law fit of the magnetic
intensity in
(a)
,
with
magnetic phase boundary points
measured on our sample
s
(orange
symbols)
,
and
metal
-
insulator transition
points
(green circles)
from
the literature
[
2
3
].
(
d
)
Correlation between pyrochlore lattice parameters
x
and
a
in
R
2
Ir
2
O
7
series for elements
R
=
Gd (down triangle), Eu (diamonds), Sm (squares), Nd (circles), and Pr (up triangles)
,
from
seven independent research groups
(Supplementary Table I, Ref.
[
2
1,
2
5
,
30,
3
4
-
3
7
]
)
.
(
x
,
a
)
values of the same group but of different
element
R
are
connected by l
inear segments,
highlighting the correlation
between
x
and
a
(grey dashed line)
despite systematic variations
between different
crystal
grower
s
. (
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
solid circles
)
and
the
R
series (
solid
blue squares)
with
the generally agreed
T
N
[
19
], the average lattice
constant for each
R
element,
and
the grey
x
-
a
line in (
d
).
T
N
of dopi
ng series (Sm, Nd, Pr)
2
Ir
2
O
7