of 17
Supplementary Information for
Discovery of quantum phases in the
Shastry-Sutherland compound SrCu
2
(BO
3
)
2
under
extreme conditions of field and pressure
Shi,
et al.
March 31, 2022
1
Supplementary Figures
0
10
20
30
40
0
10
20
30
40
df/dH
0
10
20
30
40
-dM
2
/
d
2
H
0
H
(T)
1/8
Onizuka 2000; H||ab
df/dH
0
10
20
30
40
0.00
0.05
0.10
0.15
b
x
=0
dM
/
dH
(
B
T
-1
per Cu
)
0
H
(T)
1/8
0.4 K
H
|| a
-axis
Onizuka 2000; H||ab
a
Supplementary Fig. 1
:
Identification of the 1/8 plateau at
H
1
.
(a)
dM
/
dH
(left
axis) and
df
/
dH
(right axis) vs
μ
0
H
. (b)
dM
2
/
d
2
H
(left axis) and
df
/
dH
(right axis) vs
μ
0
H
.
dM
/
dH
and
dM
2
/
d
2
H
are calculated based on the magnetization data from Ref.
1
. The same anomaly is clearly seen in both the magnetization measurement
1
and our
TDO measurements.
2
33343536373839
222324252627282930
25303540
2530354045
d
c
b
37.55 T
0
H
(T)
1/4 plateau
1/6 plateau
34.37 T
0 GPa (up to 34.5 T)
0 GPa (up to 45 T)
df/dH
a
2/15 plateau
1.0 GPa
1.1 GPa
sub-1/8
anomaly
1/8 plateau
24.32 T
d
f
/d
H
0
H
(T)
24.34 T
1.7 GPa
1.9 GPa
1/5 plateau (low er)
28.49 T
1/3 plateau
df/dH
H
(T)
1/5 plateau
(low er)
27.31 T
33.27 T
1/5 plateau (upper)
2.3 GPa
1/3 plateau
31.72 T
1/5 plateau
df/dH
H
(T)
Supplementary Fig. 2
:
Identification of the TDO anomalies in
df/dH
vs
μ
0
H
for the pure sample;
T
= 0.3 K.
(
A
) Ambient pressure (
P
= 0); (
B
)
P
= 1.0 GPa
and 1.1 GPa; (
C
)
P
= 1.7 GPa and 1.9 GPa; (
D
)
P
= 2.3 GPa. Local minimum (“dip”)
and local maximum (“bump”) are identified at each anomaly, as indicated by the vertical
short dashed lines. For weak features, dashed lines are drawn as a guide to eye. The
location of the anomaly is identified as the midpoint between the “dips” and “bumps”,
which correspond to a jump in
M
(
H
) (See Fig. Fig. S1 and Ref.
2
). Only data points
that are repeatable and show a systematic trend with pressure are considered. For the
1/3 plateau, the “bump” appear above our maximum field range 45 T for all pressures
except 2.3 GPa. For consistency, we identify the 1/3 plateau using the
H
value at the
“dips” for all pressure, and we note that it would slightly underestimate the onset field
of the 1/3 plateau. This is indeed observed in Fig. 3 and Fig. S1.
3
2530354045
1/3
1/5 (upper)
1/5 (lower)
1/5 (lower)
1/5 (upper)
1/5 (lower)
1.9 GPa (x=0)
1.8 GPa
(x=0.05)
1.7 GPa (x=0)
df/dH
0
H
(T)
Supplementary Fig. 3
:
Identification of the high-
P
1/5 and 1/3 plateaus in
df/dH
vs
μ
0
H
for the
x
=0.05 sample at
P
= 1.8 GPa and
T
= 0.3 K.
4
0 10 20 30 40
5 10 15 20 25 30 35 40 45
10 15 20 25 30 35 40 45
1/3
1/5
upper
1/5
low er
2/15
b
0.3 K
0.44 K
0.57 K
0.74 K
0.95 K
df/dH
0
H
(T)
1.6 K
1.9 GPa
Plaquette -
10x2 SS
10x2 SS -
1/3 SS
c
1/6
sub-1/8
1/3
1/3
1/4
1/8
1/5
5 K
3.4 K
1.8 K
1.5 K
d
f
/d
H
0
H
(T)
10x2 SS -
1/3 SS
0 GPa
0.3 K
1.2 K
0.85 K
d
f
/d
H
0.3 K
2.3 GPa
a
sub-1/8
1/3
1/4
1.6 K
d
f
/d
H
0
H
(T)
0.3 K
pure SCBO
1.1 GPa
1/8
Supplementary Fig. 4
:
Temperature dependence of the high-field anomalies.
(a)
df
/
dH
vs
μ
0
H
at
P
= 1.1 GPa and
T
= 0.3 K and 1.6 K. (b)
df
/
dH
vs
μ
0
H
at
P
=
1.9 GPa for
T
between 0.3 K and 1.6 K. (c)
df
/
dH
vs
μ
0
H
at
P
= 2.3 GPa (left axis) for
various temperatures, in comparison with that at ambient pressure (bottom trace with
red color, right axis) and 0.3 K.
5
4 6 8 10 12 14 16
b
x
= 0
2.4 GPa
1.6 K
0.8 K
0.5 K
df/dH
0.3 K
a
df/dH
0
H
(T)
x
= 0.05
2.4 GPa
1.6 K
0.8 K
0.5 K
0.3 K
Supplementary Fig. 5
:
Doping and temperature dependence of the low-field
anomalies at
P
= 2.4 GPa.
df
/
dH
vs
μ
0
H
for (a)
x
=0 and (b)
x
= 0.05. In both
cases, the low-field anomalies are suppressed at 1.6 K. Notably, the low-field anomalies
are shifted to lower fields in the
x
= 0.05 doped sample.
6
4 6 8 10 12 14 16
df/dH
x
= 0.02
2.1 GPa
1.6 K
0.8 K
0.5 K
0.3 K
a
b
df/dH
0
H
(T)
x
= 0.03
2.1 GPa
1.6 K
0.8 K
0.5 K
0.3 K
Supplementary Fig. 6
:
Doping and temperature dependence of the low-field
anomalies at
P
= 2.1 GPa.
df
/
dH
vs
μ
0
H
for (a)
x
=0.02 and (b)
x
= 0.03. In both
cases, the low-field anomalies are suppressed at 1.6 K. Note that the pressure (2.1 GPa)
is lower than the one (2.4 GPa) used in Supplementary Fig. 5, so caution is needed when
making a comparison.
7
a
b
Supplementary Fig. 7
: (a) iPEPS results for the magnetization curve for
J
/J
= 0
.
68
(
D
= 8) as a function of
h/J
. (b) Energy differences of the competing states with respect
to the ground state as a function of
h/J
.
8
a
b
Supplementary Fig. 8
: (a) iPEPS results for the magnetization curve for
J
/J
= 0
.
7
(
D
= 8) as a function of
h/J
. (b) Energy differences of the competing states with respect
to the ground state as a function of
h/J
.
9
Supplementary Fig. 9
: Spin structure of the 10x2 supersolid state which can be seen
as a descendant of the high-
P
1/5 plateau state with rotated spins. The spin component
in x-direction is reversed between neighboring stripes, leading to a doubling of the unit
cell size with respect to the high-
P
1/5 plateau state. For a more direct comparison with
the spin structure of the high-
P
1/5 plateau state, the color of the spins are chosen here
such that red (black) arrows show spins with a positive (negative) x-component.
10