Progress in understanding thequantum phases of SrCu2(BO3)2
Philippe Corboz, Institute for Theoretical Physics, University of Amsterdam

iPEPS
2D tensor network ansatz

Shastry-Sutherlandmodel

Outline‣ Part I: SCBO under pressure at finite temperature✦ Critical point at finite temperature, analogous to the critical point of water
J. L. Jiménez, S. P. G. Crone, E. Fogh, M. E. Zayed, R. Lortz, E. Pomjakushina,
K. Conder, A. M. Läuchli, L. Weber, S. Wessel, A. Honecker, B. Normand, C. Rüegg,PC, H. M. Rønnow, and F. Mila, Nature 592, 370 (2021).

‣ Part II: SCBO under extreme conditions of field & pressure✦ New type of 1/5 plateau and supersolid phases in the (p, H) phase diagram
Z. Shi, S. Dissanayake, PC, W. Steinhardt, D. Graf, D. M. Silevitch, H. A. Dabkowska,
T. F. Rosenbaum, F. Mila, S. Haravifard, Nat Commun 13, 1 (2022)‣ Part III: SCBO up to the saturation magnetic field✦ Close agreement between iPEPS and experiments & insights into ultrasound velocity
T. Nomura, PC, A. Miyata, S. Zherlitsyn, Y. Ishii, Y. Kohama, Y. Matsuda, A. Ikeda,
C. Zhong, H. Kageyama, F. Mila, arXiv:2209.07652‣ Part IV: Role of interlayer coupling in SCBO
✦ Phase diagram of the SSM with interlayer coupling with new iPEPS approach
P. Vlaar, PC, arxiv:2208.06423; in preparation

The Shastry-Sutherland model and SrCu2(BO3)2
Ĥ = J

0

X

hi,ji

Si · Sj + J

XSi · Sj + hhhi,jiidimerX

SrCu2(BO3)2Siz

Spin-gap system (~35K)

i

Cu2+

carries
S=1/2

Shastry & Sutherland, Physica B+C 108 (1981).

Kageyama et al. PRL 82 (1999)

The Shastry-Sutherland model and SrCu2(BO3)2
Ĥ = J

0

X

hi,ji

Si · Sj + J

XSi · Sj + hhhi,jiidimerX

SrCu2(BO3)2Siz

Spin-gap system (~35K)

i

Cu2+

carries
S=1/2

Dimer phase

Plaquette phase

Néel phase

Kageyama et al. PRL 82 (1999)

previously found in:

J 0 /J
00.675(2) 0.765(15)Corboz and Mila, PRB 87 (2013)

Koga and Kawakami, PRL 84 (2000)
Takushima et al., JPSJ 70 (2001)
Chung et al, PRB 64 (2001)
Läuchli et al, PRB 66 (2002)

The Shastry-Sutherland model and SrCu2(BO3)2
Ĥ = J

0

X

hi,ji

Si · Sj + J

XSi · Sj + hhhi,jiidimerX

SrCu2(BO3)2Siz

Spin-gap system (~35K)

i

Cu2+

carries
S=1/2

Plaquette phase

Dimer phase

Néel phase

Kageyama et al. PRL 82 (1999)

Deconfined QCP?
Lee, You, Sachdev &Vishwanath, PRX 9 (2019)

Intermediate QSL phase?
0.63<latexit sha1_base64="0cIbSyEyqZ9MRXb995eeQWGzjq8=">AAAB63icdVDLSsNAFJ34rPVVdelmsAiuwiQNad0V3bisYB/QhjKZTtqhM0mYmQgl9BfcuFDErT/kzr9x0lZQ0QMXDufcy733hClnSiP0Ya2tb2xubZd2yrt7+weHlaPjjkoySWibJDyRvRAryllM25ppTnuppFiEnHbD6XXhd++pVCyJ7/QspYHA45hFjGBdSMj2a8NKFdmXDd/1fIhshOqO6xTErXs1DzpGKVAFK7SGlffBKCGZoLEmHCvVd1CqgxxLzQin8/IgUzTFZIrHtG9ojAVVQb64dQ7PjTKCUSJNxRou1O8TORZKzURoOgXWE/XbK8S/vH6mo0aQszjNNI3JclGUcagTWDwOR0xSovnMEEwkM7dCMsESE23iKZsQvj6F/5OOazu+jW69avNqFUcJnIIzcAEcUAdNcANaoA0ImIAH8ASeLWE9Wi/W67J1zVrNnIAfsN4+AS+cjbA=</latexit>

00.675(2) 0.765(15)Corboz and Mila, PRB 87 (2013)

J 0 /J

previously
found
in:
Yang,
Sandvik& Wang,PRB 105 (2022)

Magnetization plateaus
SrCu2(BO3)2 in a magnetic field exhibits several magnetization plateaus
1/3 plateau
1/4 plateau

Crystalsof localized triplets
S=0
1/8

Oninzuka, et al.

1/8

1/8 plateau
Onizuka, et al.,JPSJ 69 (2000)

The SSM has almost localized triplet
excitations [Miyahara&Ueda’99, Kageyama et al. ’00]

1/4

1/3

Triplets repel each other(on the mean-field level)
Common assumption:
magnetization plateaus correspond to
crystals of localized triplets!

Onizuka, et al., JPSJ 69 (2000)

ng
=
ic
n
ra

2
Magnetization plateaus below
the 1/4 plateau
0.12

Takigawa, et al., PRL 110 (2013)

1/4

z = 10 mm

Many experimental / theoretical studies

τ/H (a.u.)

Magnetization <Sc>

Kageyama et al, PRL 82 (1999)
0
Onizuka et al, JPSJ 69 (2000)
1/6
0.08
Kageyama et al, PRL 84 (2000)
2/15
Kodama et al, Science 298 (2002)
0.06
z = -10 mm
1/8
Takigawa et al, Physica 27 (2004)
5 10 15 20 25 30 35
Levy et al, EPL 81 (2008)
0.04
H (T)
Sebastian et al, PNAS 105 (2008)
0.02
Isaev et al, PRL 103 (2009)
Jaime et al, PNAS 109 (2012)
0.00
15
20
25
30
35
Takigawa et al, PRL 110 (2013)
H (T)Matsuda et al, PRL 111 (2013)
Miyahara and K. Ueda, PRL 82 (1999)
FIG. 2. (color online) Inset: The magnetic-field dependence Momoi and Totsuka, PRB 61 (2000)
of the torque divided by field at T = 60 mK for the sam- Momoi and Totsuka, PRB 62 (2000)
ple positioned at ±10 mm off the nominal field center. Main Fukumoto and Oguchi, JPSJ 69 (2000)
panel: The thick black line represents the longitudinal mag- Fukumoto, JPSJ 70 (2001)
netization(see the text) with the vertical scale appropriately Miyahara and Ueda, JPCM 15 (2003)
adjusted. The magnetization values at 1/8, 2/15, 1/6, and Miyahara, Becca and Mila, PRB 68 (2003)
1/4 of the saturation are shown by the dashed lines. The
Dorier, Schmidt, and Mila, PRL 101 (2008)
horizontal bars indicate the field range of the plateaus determined by NMR. The open circles show the magnetization Abendschein & Capponi, PRL 101 (2008)
Takigawa et al, JPSJ 79 (2010)
determined from the Cu-NMR shift data from Ref. [8].
Nemec et al, PRB 86 (2012)★ Crystals of triplet-bound states
Matsuda et al., PRL 111 (2013)…
PC,shown
F. Mila,zero, τ /H
inPRL
the 112
inset(2014)of Fig. 2 varies linearly
0.10with H due to the transverse magnetization. To elimi-

Magnetization plateaus at high fields
Matsuda, Abe, Takeyama, Kageyama, PC, Honecker, Manmana, Foltin, Schmidt & Mila, PRL 111 (2013)
0.9
0.8

2x2, D=8, h=1.8, m=0.5
4x4, D=6, h=0.7, m=0.25

0

0.7

0.5

1

1

1.5

1/2
M/MS

0.5

1/2 plateau

1.5

0.6

2

2

0.5

2.5

0.4

ED N=24
3.5
ED N=36
ED N=40
4
Series expansion
4.5
DMRG 8x6
DMRG 10x8
50
DMRG 12x10
iPEPS
Experiment
0

1/4

0.2
0.1
20

40

60
80
Magnetic Field (T)

2.5

3

1/3

0.3

0
0

1/4 plateau

0

J /J = 0.63

0

100

120

• Best fit with experiments for
J’/J = 0.63 using iPEPS, DMRG,
ED, series expansion• Supersolid phases at high fields

3

0.5

1

1.5

2

2.5

3

3.5

4

4.5

1/3 plateau

0

0.5

1

1.5

2

2.5

3

5

1/3 supersolid

6x6, D=8, h=1, m=0.33333

6x6, D=6, h=1.42, m=0.35424

0

140
1

1

2

2

3

3

4

4

5

5

6

6

7

0

1

2

3

4

5

6

7

0 7

1

2

3

4

5

6

7

(NMR) of an intermediate phase with broken spatial
symmetry [22,23], inelastic neutron scattering revealed
an excitation attributed to a PS state [11]. The mode was2
3 2PHYSICALREVIEW
LETTERS 124, 206602 (2020)
only detected at P ¼ 2.15 GPa, andrecently analternative
scenario with no PS phase was proposed [12]. Here weequivalentplaquette
patterns,
order canat T >not0
argue that thePS phaseexistsPS
adjacent
to aappear
previously
alreadyan phaseisolatedlayer.4 K and P ¼ 3–4 GPa.
observedinAF
below
Following
indications
fromnuclear magnetic
resonance
Experiments.—We
have
performed
high-pressure
heat
(NMR)
intermediateonphasecapacity of(C)anmeasurements
SrCu2withðBO3broken
Þ2 singlespatial
cryssymmetry[22,23],ofinelastic
neutronscatteringtals. With support
simulationsof quantum
spin revealed
models,an
attributed
a PS state[11].
The modewas
we excitation
have for the
first timetoextracteda ðP;TÞ phase
diagram,only
detected
at Prange¼ 2.15
andT recently
an alternative
Fig. 2(a),in theof GPa,P andwhere theSS modelscenariono PS
wassamples
proposed[12].
Here andweshould bewith
relevant.
Sixphasedifferent
werestudied,argue
that measured
the PS phase
to a down
previouslyCðTÞ wasfromexists
roomadjacenttemperature
to 1.5notor
observedphase pressures
below 0.63
4 K(usingand Ptwo¼ 3–4
GPa. types
0.4 K at AF
severaldifferent
have performed
heatof Experiments.—We
cryostats and pressure
cells; see high-pressure
the Supplemental
capacity
measurements
on SrCucrys2 ðBOobtained
3 Þ2 singleamong
Material (C)[24]).
Consistent results
werepressuretals.
Withmeasurements.
support of simulationsof quantum
all these
In Figs. 2(b)–2(e)we spinshowmodels,
typicalwehaveforforCðTÞ=T
the firstintime
ðP; TÞ phase
diagram,results
the extracted
different apressure
regions.
In theFig.
2(a), in therange[24]of P
T wheretheP SS
Supplemental
Material
weand
discuss
data for> 4modelGPa,
Drivesystemacrossthephasetransitions!
shouldbe relevant.
Six different
samplesvalid.
were studied, andwhere theSS description
is no longerCðTÞmeasured
fromlow-T
room featurestemperature
down tothere
1.5 orWewas
identify
two mainin CðTÞ=T:
is
0.4
K Waki,several(usingtwo
different
types
etmaximum
al. pressures
J. Phys.that
Soc.
Jpn.
alwaysaatbroad
wewill76,
refer073710
to as the(2007)
hump.
of
cryostatscells;
see emerges
the7,Supplemental
Starting
at P ≈and1.7 pressureGPa,smallerpeak
at lower
T
Haravifard,
et al.aNat.
Commun.
11956(2016)
Material[24]).
results
wereobtainedand prevails
upetConsistenttoal.,
2.4
GPa.
We 13,
will
arguethat thisamong
peak
Zayed,
Nat.
Phys.
962
(2017)
allthesethemeasurements.
In UponFigs. 2(b)–2(e)we show typicalsignalsPS
transition.
furtherincreasing
P,(2018)
the
Sakurai,
et
al.,
J.
Phys.
Soc.
Jpn.
87,
033701
results
for CðTÞ=Tin thedifferent
regions.
In thesmall peakis no longer
detected
at pressure
temperatures
accessible
Guo, et al., PRL 124, 206602 (2020)
Guo,
et
al.,
PRL
124,
206602(2020)
FIG.
2.
(a)
Phase
diagram of SrCu2 ðBO3 Þ2 (crystal structure
Supplemental
Material[24] wehump
discuss
data forP > 4 3GPa,in the experiments.
A broaderappears
betweenandet al,.there
Phys.
Research
2, 012010(2020)the inset) from high pressure CðTÞ measurements. Examples
where
the
SS which
description
is Rev.
longer
4 GPa,Bettler,below
isanopeakat Tvalid.
≈ 2–3.5
K that weCðTÞ=T curves are given in (b)–(e), where the orange arrow… as antwo
We identify
low-T featuresin CðTÞ=T:
there isinterpret
AFmaintransition.
AF orderwas previously
indicate the hump location T h . The green symbols in (a) mark T halwaysthat up
we to
willT referas [11].
the hump.
detecteda broad
only atmaximum
P > 4 GPa
≈ 120to K
This
several samples and the purple curve shows results for the 20-sp

SrCu (BO ) under pressure<latexit sha1_base64="0cIbSyEyqZ9MRXb995eeQWGzjq8=">AAAB63icdVDLSsNAFJ34rPVVdelmsAiuwiQNad0V3bisYB/QhjKZTtqhM0mYmQgl9BfcuFDErT/kzr9x0lZQ0QMXDufcy733hClnSiP0Ya2tb2xubZd2yrt7+weHlaPjjkoySWibJDyRvRAryllM25ppTnuppFiEnHbD6XXhd++pVCyJ7/QspYHA45hFjGBdSMj2a8NKFdmXDd/1fIhshOqO6xTErXs1DzpGKVAFK7SGlffBKCGZoLEmHCvVd1CqgxxLzQin8/IgUzTFZIrHtG9ojAVVQb64dQ7PjTKCUSJNxRou1O8TORZKzURoOgXWE/XbK8S/vH6mo0aQszjNNI3JclGUcagTWDwOR0xSovnMEEwkM7dCMsESE23iKZsQvj6F/5OOazu+jW69avNqFUcJnIIzcAEcUAdNcANaoA0ImIAH8ASeLWE9Wi/W67J1zVrNnIAfsN4+AS+cjbA=</latexit>

Specific heat data (group of H. M. Rønnow)
Jiménez, Crone, et al., Nature 592, 370 (2021)
J

T

JD

critical point
1st order line
Dimer phase

Plaquette phase
0.675(2): 1st order PT

Can we reproduce
this with iPEPS?

https://cnx.org

J 0 /J

Inspiration: fully frustrated Heisenberg bilayer model
Stapmanns, PC, Mila, Honecker, Normand, and Wessel, PRL 121 (2018)

0.4

0.5
0

0.2
0
0

0.5

J⊥/ J|| = 2.315

6Ising critical point4

0.04
first order 0.08
line

1/L

L = 48
L = 36
L = 24
L = 16
L = 12L=88J⊥ / J|| = 2.31

4
0

2

2

DTAF (T = 0)12
1.5

J⊥ / J||DS (T = 0)
3
2.5
3.5

4

Jk J
⇥
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J?
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Dimer triplets

12

(c)

Cmax

/ J||
0.52

8

C

0.6

J× = J||

0.54

T

T / J||

0.8

10

(b)

max

1

Dimer singlets

00

0.5

T / J||

3
ln L

4

1

1.5

Jump in hS · Si on a
dimer at the phasetransition<latexit sha1_base64="dHRZc6UC6K3Ow9M2Oy+1BeTO/R8=">AAACAnicbVDLSsNAFJ3UV62vqCtxM1gEVyURRRcuCm5cVrQPaEKZTCbt0MlMmJkIJRQ3/oobF4q49Svc+TdO0iy09cDlHs65l5l7goRRpR3n26osLa+srlXXaxubW9s79u5eR4lUYtLGggnZC5AijHLS1lQz0kskQXHASDcYX+d+94FIRQW/15OE+DEachpRjLSRBvaBxxAfMgLvoIdDofMuC2Vg152GUwAuErckdVCiNbC/vFDgNCZcY4aU6rtOov0MSU0xI9OalyqSIDxGQ9I3lKOYKD8rTpjCY6OEMBLSFNewUH9vZChWahIHZjJGeqTmvVz8z+unOrr0M8qTVBOOZw9FKYNawDwPGFJJsGYTQxCW1PwV4hGSCGuTWs2E4M6fvEg6pw33vOHcntWbV2UcVXAIjsAJcMEFaIIb0AJtgMEjeAav4M16sl6sd+tjNlqxyp198AfW5w/lp5Zz</latexit>

MPS & PEPS
1D

2D

MPS
Matrix-product state

Bond dimension D

1

2

3

4

5

6

7

PEPS (TPS)
projected entangled-pair state(tensor product state)
Bond dimension D

8Physical indices (lattices sites)
S. R. White, PRL 69, 2863 (1992)
Fannes et al., CMP 144, 443 (1992)Östlund, Rommer, PRL 75, 3537 (1995)
F. Verstraete, J. I. Cirac, cond-mat/0407066
Nishio, Maeshima, Gendiar, Nishino, cond-mat/0401115

Infinite PEPS (iPEPS)
1D

2D

iMPS

iPEPS

infinite matrix-product state

infinite projected entangled-pair state
A

A

A
A

A

A

A

A

A

A

A

A
A

A
A

A
A

A

A

A

A

A

A
A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A
A

Jordan, Orus, Vidal, Verstraete, Cirac, PRL (2008)★ Work directly in the thermodynamic limit:
No finite size and boundary effects!

Infinite PEPS (iPEPS)
1D

2D

iMPS

iPEPS

infinite matrix-product state

infinite projected entangled-pair state
A

B

B
A

B

A

B

A

B

A

A

A
B

B
A

A
B

A

B

B

B

A

A
B

A

A

A

B

B

B

B

A

A

A

A

B

B

B

B

A

B
A

Jordan, Orus, Vidal, Verstraete, Cirac, PRL (2008)★ Work directly in the thermodynamic limit:
No finite size and boundary effects!

iPEPS with arbitrary unit cells
1D

2D

iMPS

iPEPS

infinite matrix-product statewith arbitrary unit cell of tensors
D

A

H
A

B

C

D

A

B

E

D

D
H

B

A
E

F

G

A

H

E

D

C

E

D

G

A

H

C

F

D

G

B

E

C

F

A

H

B

H

A
E

here: 4x2 unit cell
PC, White, Vidal, Troyer, PRB 84 (2011)★ Run simulations with different unit cell sizes andcompare variational energies

iPEPS ground state simulations• Many applications to challenging problems, including frustrated spin,
SU(N), and bosonic systems, t-J / Hubbard models, and more, see e.g.:
P. Corboz, A. M. Läuchli, K. Penc, M. Troyer and F. Mila, PRL 107 (2011)
S. Dusuel, M. Kamfor, R. Orús, K. P. Schmidt, and J. Vidal, PRL 106, 107203 (2011)
H. H. Zhao, C. Xu, Q. N. Chen, Z. C. Wei, M. P. Qin, G. M. Zhang and T. Xiang, PRB 85 (2012)
P. Corboz, M. Lajkó, A. M. Läuchli, K. Penc and F. Mila, PRX 2 (2012)
P. Corboz and F. Mila, PRB 87 (2013); PRL 112 (2014)
Z.-C. Gu, H.-C. Jiang, D. N. Sheng, H. Yao, L. Balents and X.-G. Wen, PRB 88 (2013)
J. Osorio Iregui, P. Corboz and M. Troyer, PRB 90 (2014)
P. Corboz, T. Rice and M. Troyer, PRL 113 (2014)
T. Picot and D. Poilblanc, PRB 91 (2015)
T. Picot, M. Ziegler, R. Orús and D. Poilblanc, PRB 93 (2016)
P. Nataf, M. Lajkó, P. Corboz, A. M. Läuchli, K. Penc and F. Mila, PRB 93 (2016)
H. Liao, Z. Xie, J. Chen, Z. Liu, H. Xie, R. Huang, B. Normand and T. Xiang, PRL 118 (2017)
B.-X. Zheng, et al., Science 358, 1155 (2017)
I. Niesen and P. Corboz, PRB 95 (2017); SciPost Physics 3, 030 (2017); Rev. B 97, 245146 (2018)
R. Haghshenas, W.-W. Lan, S.-S. Gong, and D. N. Sheng, PRB 97 (2018)
J.-Y. Chen, L. Vanderstraeten, S. Capponi, and D. Poilblanc, PRB 98 (2018)
S. S. Jahromi and R. Orús, PRB 98 (2018)
H.-Y. Lee and N. Kawashima, PRB 97 (2018)
H. Yamaguchi, Y. Sasaki, T. Okubo, et al.,PRB 98, 094402 (2018)
R. Haghshenas, S.-S. Gong, and D. N. Sheng, PRB 99, 174423 (2019)
S. S. Chung and P. Corboz, PRB 100 (2019)
B. Ponsioen, S. S. Chung, and P. Corboz, PRB 100 (2019)
C. Boos, S. P. G. Crone, I. A. Niesen, P. Corboz, K. P. Schmidt, and F. Mila, PRB 100 (2019)
Z. Shi, et al, Nature Communications 10, 2439 (2019)
A. Kshetrimayum, C. Balz, B. Lake, and J. Eisert, ArXiv:1904.00028 (2019)
H.-Y. Lee, R. Kaneko, T. Okubo, and N. Kawashima, PRL 123, 087203 (2019).
O. Gauthé, S. Capponi, M. Mambrini, and D. Poilblanc, PRB 101, 205144 (2020).
H.-Y. Lee, R. Kaneko, L. E. Chern, T. Okubo, Y. Yamaji, N. Kawashima, and Y. B. Kim, Nature Communications 11 (2020)
W.-Y. Liu, S.-S. Gong, Y.-B. Li, D. Poilblanc, W.-Q. Chen, and Z.-C. Gu, ArXiv:2009.01821 (2020)
J.-Y. Chen, S. Capponi, A. Wietek, M. Mambrini, N. Schuch, and D. Poilblanc, PRL 125, 017201 (2020)
J. Hasik, D. Poilblanc, and F. Becca, SciPost Physics 10, 012 (2021)… and many more …

Finite temperature simulations with iPEPS‣ Methodological developments (2D):Li et al. PRL 106 (2011); Czarnik et al. PRB 86 (2012); Czarnik & Dziarmaga PRB 90 (2014);
Czarnik & Dziarmaga PRB 92 (2015); Czarnik et al. PRB 94 (2016); Dai et al PRB 95 (2017);
Kshetrimayum, Rizzi, Eisert, Orus, PRL 122 (2019), P. Czarnik, J. Dziarmaga, PC, PRB 99 (2019), …

‣ Wave-function:| i⇡<latexit sha1_base64="3ogdKbCNwjYVKtXUUFZf2LMsJgg=">AAAB/nicbVBNSwMxEM36WevXqnjyEiyCp7JbBT0WvXisYD+gu5RsOm1Ds9mQZMWyFvwrXjwo4tXf4c1/Y9ruQVsfDDzem2FmXiQ508bzvp2l5ZXVtfXCRnFza3tn193bb+gkVRTqNOGJakVEA2cC6oYZDi2pgMQRh2Y0vJ74zXtQmiXizowkhDHpC9ZjlBgrddzDx6CmGQ4UEX0OOCBSquSh45a8sjcFXiR+TkooR63jfgXdhKYxCEM50brte9KEGVGGUQ7jYpBqkIQOSR/algoSgw6z6fljfGKVLu4lypYweKr+nshIrPUojmxnTMxAz3sT8T+vnZreZZgxIVMDgs4W9VKOTYInWeAuU0ANH1lCqGL2VkwHRBFqbGJFG4I///IiaVTK/lm5cnteql7lcRTQETpGp8hHF6iKblAN1RFFGXpGr+jNeXJenHfnY9a65OQzB+gPnM8fE6eVkA==</latexit>‣ Density-operator:⇢ˆ = eA
A

B

Ĥ

<latexit sha1_base64="euJtCkwCgd61qRMjUvsxc09wJi0=">AAACD3icbVA9SwNBEN3z2/gVtbRZDIqN4S4K2giiTcoIRoVcDHObiVncuz1258Rw5B/Y+FdsLBSxtbXz37iJKfx6MPB4b4aZeVGqpCXf//DGxicmp6ZnZgtz8wuLS8XllTOrMyOwLrTS5iICi0omWCdJCi9SgxBHCs+j6+OBf36DxkqdnFIvxWYMV4nsSAHkpFZxM+wC8dB0NecHHC/z7TBCAj6Uq30eQpoafdsqlvyyPwT/S4IRKbERaq3ie9jWIosxIaHA2kbgp9TMwZAUCvuFMLOYgriGK2w4mkCMtpkP/+nzDae0eUcbVwnxofp9IofY2l4cuc4YqGt/ewPxP6+RUWe/mcskzQgT8bWokylOmg/C4W1pUJDqOQLCSHcrF10wIMhFWHAhBL9f/kvOKuVgp1w52S0dHo3imGFrbJ1tsYDtsUNWZTVWZ4LdsQf2xJ69e+/Re/Fev1rHvNHMKvsB7+0TwE6bPQ==</latexit>‣ Symmetric form: e<latexit sha1_base64="EKN00LEZd3nl4H25im+VORJLvRA=">AAACBHicbVA9SwNBEN3zM8avqGWaxSDYGO+ioGXQJmUEo0IuhrnNxCzufbA7J4YjhY1/xcZCEVt/hJ3/xk1MoYkPBh7vzTAzL0iUNOS6X87M7Nz8wmJuKb+8srq2XtjYvDBxqgU2RKxifRWAQSUjbJAkhVeJRggDhZfB7enQv7xDbWQcnVM/wVYIN5HsSgFkpXahiNfZnh8gAfd7QLy2XxlwH5JEx/ftQsktuyPwaeKNSYmNUW8XPv1OLNIQIxIKjGl6bkKtDDRJoXCQ91ODCYhbuMGmpRGEaFrZ6IkB37FKh3djbSsiPlJ/T2QQGtMPA9sZAvXMpDcU//OaKXWPW5mMkpQwEj+LuqniFPNhIrwjNQpSfUtAaGlv5aIHGgTZ3PI2BG/y5WlyUSl7B+XK2WGpejKOI8eKbJvtMo8dsSqrsTprMMEe2BN7Ya/Oo/PsvDnvP60zznhmi/2B8/ENQFiXNQ==</latexit>

Ĥ/2

B

⇡

A

⇡

B

A

B
A

B

B

⇢ˆ( ) ⇡A

<latexit sha1_base64="Piaed43L3faxw/M50oIQDZWDTwk=">AAACBHicbVA9SwNBEN3zM8avqGWaxSDEJtxFQcugjWUE8wG5EOY2m2TJ3u2xOyeGkMLGv2JjoYitP8LOf+MmuUITHww83pthZl4QS2HQdb+dldW19Y3NzFZ2e2d3bz93cFg3KtGM15iSSjcDMFyKiNdQoOTNWHMIA8kbwfB66jfuuTZCRXc4ink7hH4keoIBWqmTy/sDQOrrgaJFP+AIp5RSH+JYq4dOruCW3BnoMvFSUiApqp3cl99VLAl5hEyCMS3PjbE9Bo2CST7J+onhMbAh9HnL0ghCbtrj2RMTemKVLu0pbStCOlN/T4whNGYUBrYzBByYRW8q/ue1EuxdtsciihPkEZsv6iWSoqLTRGhXaM5QjiwBpoW9lbIBaGBoc8vaELzFl5dJvVzyzkrl2/NC5SqNI0Py5JgUiUcuSIXckCqpEUYeyTN5JW/Ok/PivDsf89YVJ505In/gfP4AsmiW2g==</latexit>†⇢ˆ( ) = ⇢ˆ ( )<latexit sha1_base64="xnbBQ00c3kxLCs5n8B6zk5tKXbg=">AAACGHicbVDLSgNBEJyN7/ha9ehlMAh6ibtR0IsgevEYwaiQjaF30kmGzD6Y6RVCyGd48Ve8eFDEa27+jZMYRBMLGoqqbrq7wlRJQ5736eRmZufmFxaX8ssrq2vr7sbmjUkyLbAiEpXouxAMKhljhSQpvEs1QhQqvA07F0P/9gG1kUl8Td0UaxG0YtmUAshKdfcgaAPxQLcTvheESLDP+Sn/Ee+DBrRaqH/Mulvwit4IfJr4Y1JgY5Tr7iBoJCKLMCahwJiq76VU64EmKRT280FmMAXRgRZWLY0hQlPrjR7r812rNHgz0bZi4iP190QPImO6UWg7I6C2mfSG4n9eNaPmSa0n4zQjjMX3omamOCV8mBJvSI2CVNcSEFraW7logwZBNsu8DcGffHma3JSK/mGxdHVUODsfx7HIttkO22M+O2Zn7JKVWYUJ9sie2St7c56cF+fd+fhuzTnjmS32B87gCyF4nfQ=</latexit>by construction

b

2

J
J
Specificheat:
experiments
vs
iPEPS
J
J
D

D

JD

18.2 kbar

Location of max
0.7

J

c
<latexit sha1_base64="um5ylvBO7aMGzdsRqgl+5xvEArQ=">AAAB63icbVBNSwMxEJ2tX7V+VT16CRbRU90VRY9FL9JTBfsB7VKyabYNTbJLkhXK0r/gxYMiXv1D3vw3Zts9aOuDgcd7M8zMC2LOtHHdb6ewsrq2vlHcLG1t7+zulfcPWjpKFKFNEvFIdQKsKWeSNg0znHZiRbEIOG0H47vMbz9RpVkkH80kpr7AQ8lCRrDJpPrpeb1frrhVdwa0TLycVCBHo1/+6g0ikggqDeFY667nxsZPsTKMcDot9RJNY0zGeEi7lkosqPbT2a1TdGKVAQojZUsaNFN/T6RYaD0Rge0U2Iz0opeJ/3ndxIQ3fspknBgqyXxRmHBkIpQ9jgZMUWL4xBJMFLO3IjLCChNj4ynZELzFl5dJ66LqXVXdh8tK7TaPowhHcAxn4ME11OAeGtAEAiN4hld4c4Tz4rw7H/PWgpPPHMIfOJ8/BH6NkA==</latexit>

J 0 /J

60
0 kbar

0.68
0.67
0.66

40
7 kbar

f

0.05
0.045
0.04
0.035
0.03

15 kbar20
16.5 kbar
0<latexit sha1_base64="Jp6AuuPhW1yjxtztYzbTVw7ezJM=">AAAB9HicbVDLSsNAFL3xWeur6tLNYBHqJiS+N0LRjeCmQl/QhjKZTtqhk0mcmRRK6He4caGIWz/GnX/jtM1CWw9cOJxzL/fe48ecKe0439bS8srq2npuI7+5tb2zW9jbr6sokYTWSMQj2fSxopwJWtNMc9qMJcWhz2nDH9xN/MaQSsUiUdWjmHoh7gkWMIK1kbxqh6AbdGY7pcuTh06h6NjOFGiRuBkpQoZKp/DV7kYkCanQhGOlWq4Tay/FUjPC6TjfThSNMRngHm0ZKnBIlZdOjx6jY6N0URBJU0Kjqfp7IsWhUqPQN50h1n01703E/7xWooNrL2UiTjQVZLYoSDjSEZokgLpMUqL5yBBMJDO3ItLHEhNtcsqbENz5lxdJ/dR2L2zn8bxYvs3iyMEhHEEJXLiCMtxDBWpA4Ame4RXerKH1Yr1bH7PWJSubOYA/sD5/AH3Bj/Y=</latexit>

Tc = 3.0(6)K0d

20 kbar

e

0.69

60

0.025 0.05 0.075 0.1

-0.3

g
0.12
22 kbar22.8 kbar

26.5 kbar

-0.5
20

50

with DM interaction
D=10

-0.4

40 D=20
iPEPS,

D = 20Linear fit

40

0.08
30

-0.6
0
4

6

8

10 12 14 16 18

-0.7
0.04

0.08

0.12

20

0.04
-5

-4

-3

-2

10

Temperature [K]

0

0

0.66
0

J /J

Jiménez, Crone, et al., Nature 592, 370 (2021)

0.68
0.7
J 0 /J<latexit sha1_base64="um5ylvBO7aMGzdsRqgl+5xvEArQ=">AAAB63icbVBNSwMxEJ2tX7V+VT16CRbRU90VRY9FL9JTBfsB7VKyabYNTbJLkhXK0r/gxYMiXv1D3vw3Zts9aOuDgcd7M8zMC2LOtHHdb6ewsrq2vlHcLG1t7+zulfcPWjpKFKFNEvFIdQKsKWeSNg0znHZiRbEIOG0H47vMbz9RpVkkH80kpr7AQ8lCRrDJpPrpeb1frrhVdwa0TLycVCBHo1/+6g0ikggqDeFY667nxsZPsTKMcDot9RJNY0zGeEi7lkosqPbT2a1TdGKVAQojZUsaNFN/T6RYaD0Rge0U2Iz0opeJ/3ndxIQ3fspknBgqyXxRmHBkIpQ9jgZMUWL4xBJMFLO3IjLCChNj4ynZELzFl5dJ66LqXVXdh8tK7TaPowhHcAxn4ME11OAeGtAEAiN4hld4c4Tz4rw7H/PWgpPPHMIfOJ8/BH6NkA==</latexit>

0.72

Correlation length
Zooming-in (D=8)
60

iPEPS, D=20
40
20
0

0.64

0.66

0.68

0.70

0.72

0.74

60
40
20
0
0.692

0.693

0.694

0.695

0.696

60

Diverging correlation lengthcompatible with a critical point40
20
00.69600.69620.6964<latexit sha1_base64="um5ylvBO7aMGzdsRqgl+5xvEArQ=">AAAB63icbVBNSwMxEJ2tX7V+VT16CRbRU90VRY9FL9JTBfsB7VKyabYNTbJLkhXK0r/gxYMiXv1D3vw3Zts9aOuDgcd7M8zMC2LOtHHdb6ewsrq2vlHcLG1t7+zulfcPWjpKFKFNEvFIdQKsKWeSNg0znHZiRbEIOG0H47vMbz9RpVkkH80kpr7AQ8lCRrDJpPrpeb1frrhVdwa0TLycVCBHo1/+6g0ikggqDeFY667nxsZPsTKMcDot9RJNY0zGeEi7lkosqPbT2a1TdGKVAQojZUsaNFN/T6RYaD0Rge0U2Iz0opeJ/3ndxIQ3fspknBgqyXxRmHBkIpQ9jgZMUWL4xBJMFLO3IjLCChNj4ynZELzFl5dJ66LqXVXdh8tK7TaPowhHcAxn4ME11OAeGtAEAiN4hld4c4Tz4rw7H/PWgpPPHMIfOJ8/BH6NkA==</latexit>

00.6966J /J

0.69680.6970kbar

Jump inc <S· S> on dimer 0.69

0.7

60

50
0
40
-0.2
30

0.08
20

-0.4
20

0.04
dimer
0

plaquette

d

0
0.66
0.66
60

10
-0.6

1
<latexit sha1_base64="ISrABLte7P6eccfsRnS4LU5I3PU=">AAAB6HicbVDLSgNBEOyNrxhfUY9eBoPgKeyKr2PAi8cEzAOSJcxOepMxs7PLzKwQlnyBFw+KePWTvPk3TpI9aGJBQ1HVTXdXkAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4bua3n1BpHssHM0nQj+hQ8pAzaqzU8Prlilt15yCrxMtJBXLU++Wv3iBmaYTSMEG17npuYvyMKsOZwGmpl2pMKBvTIXYtlTRC7WfzQ6fkzCoDEsbKljRkrv6eyGik9SQKbGdEzUgvezPxP6+bmvDWz7hMUoOSLRaFqSAmJrOvyYArZEZMLKFMcXsrYSOqKDM2m5INwVt+eZW0LqreVdVtXFZq13kcRTiBUzgHD26gBvdQhyYwQHiGV3hzHp0X5935WLQWnHzmGP7A+fwBeFOMrQ==</latexit>

h zi<latexit sha1_base64="spkopbuQI7Vqb5Ky/4Zoi59B2bo=">AAACAXicbVDLSsNAFJ34rPUVdSO4GSyCq5KIr2XBjcsK9gFNLDfTSTp0MgkzE6GGuvFX3LhQxK1/4c6/cZp2oa0HLhzOuZd77wlSzpR2nG9rYXFpeWW1tFZe39jc2rZ3dpsqySShDZLwRLYDUJQzQRuaaU7bqaQQB5y2gsHV2G/dU6lYIm71MKV+DJFgISOgjdS19z0OIuIUe4pFMdw9YE8WQteuOFWnAJ4n7pRU0BT1rv3l9RKSxVRowkGpjuuk2s9BakY4HZW9TNEUyAAi2jFUQEyVnxcfjPCRUXo4TKQpoXGh/p7IIVZqGAemMwbdV7PeWPzP62Q6vPRzJtJMU0Emi8KMY53gcRy4xyQlmg8NASKZuRWTPkgg2oRWNiG4sy/Pk+ZJ1T2rOjenldr5NI4SOkCH6Bi56ALV0DWqowYi6BE9o1f0Zj1ZL9a79TFpXbCmM3voD6zPH0VGlrw=</latexit>

0.68
0.68

0.7
0.7

f

0.05
0.045
0.04
0.035
0.03

0
<latexit sha1_base64="YAR4yiI8h5JeOlOTWaWgo3AtE+c=">AAAB6HicbVDLSgNBEOyNrxhfUY9eBoPgKeyKr2PAi8cEzAOSJcxOepMxs7PLzKwQlnyBFw+KePWTvPk3TpI9aGJBQ1HVTXdXkAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4bua3n1BpHssHM0nQj+hQ8pAzaqzUcPvlilt15yCrxMtJBXLU++Wv3iBmaYTSMEG17npuYvyMKsOZwGmpl2pMKBvTIXYtlTRC7WfzQ6fkzCoDEsbKljRkrv6eyGik9SQKbGdEzUgvezPxP6+bmvDWz7hMUoOSLRaFqSAmJrOvyYArZEZMLKFMcXsrYSOqKDM2m5INwVt+eZW0LqreVdVtXFZq13kcRTiBUzgHD26gBvdQhyYwQHiGV3hzHp0X5935WLQWnHzmGP7A+fwBds+MrA==</latexit><latexit sha1_base64="TBIj0suJt2s6CiNWxt5KiKtbdHQ=">AAAB6XicbVDLSgNBEOyNrxhfUY9eBoPgxbArvo4BLx6jmAckS5idzCZDZmeXmV4hLPkDLx4U8eofefNvnCR70MSChqKqm+6uIJHCoOt+O4WV1bX1jeJmaWt7Z3evvH/QNHGqGW+wWMa6HVDDpVC8gQIlbyea0yiQvBWMbqd+64lrI2L1iOOE+xEdKBEKRtFKD2der1xxq+4MZJl4OalAjnqv/NXtxyyNuEImqTEdz03Qz6hGwSSflLqp4QllIzrgHUsVjbjxs9mlE3JilT4JY21LIZmpvycyGhkzjgLbGVEcmkVvKv7ndVIMb/xMqCRFrth8UZhKgjGZvk36QnOGcmwJZVrYWwkbUk0Z2nBKNgRv8eVl0jyvepdV9/6iUrvK4yjCERzDKXhwDTW4gzo0gEEIz/AKb87IeXHenY95a8HJZw7hD5zPH+GYjOQ=</latexit>

000.025 0.05 0.075 0.1

-0.3

J /J

g
-0.4

40

Clear evidence of a first order linewith a critical point compatible withthe
20 2D Ising universality classD = 20Linear fit-0.5

exact value:= 0.125-0.6
0<latexit sha1_base64="3MhuiJj4wO5mLzXwnbW0nEvbyP0=">AAAB8nicdVBNSwMxEM3Wr1q/qh69BIvgacm23daLUPTisYL9gLaUbJq2odnNkswKpfRnePGgiFd/jTf/jdm2goo+GHi8N8PMvCCWwgAhH05mbX1jcyu7ndvZ3ds/yB8eNY1KNOMNpqTS7YAaLkXEGyBA8nasOQ0DyVvB5Dr1W/dcG6GiO5jGvBfSUSSGglGwUqcbcKCXxPWKfj9fIC6pVIplH6ekVCIlS6q+75Ei9lyyQAGtUO/n37sDxZKQR8AkNabjkRh6M6pBMMnnuW5ieEzZhI54x9KIhtz0ZouT5/jMKgM8VNpWBHihfp+Y0dCYaRjYzpDC2Pz2UvEvr5PA8KI3E1GcAI/YctEwkRgUTv/HA6E5Azm1hDIt7K2YjammDGxKORvC16f4f9Isul7FJbflQu1qFUcWnaBTdI48VEU1dIPqqIEYUugBPaFnB5xH58V5XbZmnNXMMfoB5+0T15aQUg==</latexit>

-0.7
0.04

1

0.72
0.72

0

4 16 18

0.67
0.66

40

5 kbar

0.68
2D classical Ising model in a field

iPEPS D=10 (with DM)

0.12

e

0.08

0.12

-5

-4

-3

-2

Open challenges
JDJ

too low T to obtainreliable data with
iPEPS (currently)‣ Inelastic neutron scattering:full plaquette phase (FPP), notempty plaquette phase (EPP)
Zayed, et al., Nat. Phys. 13, 962 (2017)

full plaquettephase (FPP)J1

Competing plaquette phases ~ex

0J
1 PC, Schmidt & Mila, PRB 100 (2019)
Boos, Crone, Niesen,

J2

J20 EPP and FPP phase‣ Distorted Shastry-Sutherland model: competition between

EPP(a) J10 = J20

1

0.76

dimer

0.046
-0.211-0.3760.149
0.035-0.267-0.350vs

FPP(b) J1 = J2
iPEPS
SE

0.75

0.6

EPP

0.98
0.96

0.72

0.8

(c) J20 /J2 = 0.68

1

0.74

0.85

0.7
0.55

(c)

0.68

iPEPS
SE

0.64

0.65

0.7
0
J /J2

0.75

EPP

0.7

FPP/Haldane

0.8

0.84

‣ Small deformation leads to FPP phase!

FPP/
Haldane

0.94
0.92

FPP/
Haldane

0.66

0.62
0.62

J1 /J2J1 /J20.9

Néel

J20 /J

0.95

EPP

0.78

(b)

dimer

0.9

iPEPS
SE

0.88
0.66

0.7
J10 /J

0.74

0.78

1

1.02 1.04 1.06 1.08 1.1 1.12
J10 /J20

But precise model still unclear…

Finite T iPEPS study of the m=1/2 plateau in SCBO
P. Czarnik, M. M. Rams, PC, and J. Dziarmaga, PRB 103, 075113 (2021)

High-field magnetization data

Systematic scaling analysis

0.9
0.8

Matsuda et al.
PRL 111 (2013)

0.7

M/M

S

0.6

1/2

0.5
0.4
0.3
0.2
0.1
0
0

ED N=24
ED N=36
ED N=40
Series expansion
DMRG 8x6
DMRG 10x8
DMRG 12x10
iPEPS
Experiment

1/3
1/4

T=2.1K
20

40

60
80
Magnetic Field (T)

100

120

Critical exponents compatible
with 2D Ising universality class140

Tc = 0.043(2)J ⇡ 3.5(2)K<latexit sha1_base64="3YzgmIsrLNzcH/XEHqsh5low99Q=">AAACCXicbVDLSsNAFJ3UV62vqEs3g0Wom5D0gW6EohvRTYW+oA1hMp20QyeZMDMRS+nWjb/ixoUibv0Dd/6N0zYLbT1w4XDOvdx7jx8zKpVtfxuZldW19Y3sZm5re2d3z9w/aEqeCEwamDMu2j6ShNGINBRVjLRjQVDoM9Lyh1dTv3VPhKQ8qqtRTNwQ9SMaUIyUljwT1j0ML6Bt2eVSoXgKb2AXxbHgD7BkVbRw65l5bc4Al4mTkjxIUfPMr26P4yQkkcIMSdlx7Fi5YyQUxYxMct1EkhjhIeqTjqYRCol0x7NPJvBEKz0YcKErUnCm/p4Yo1DKUejrzhCpgVz0puJ/XidRwbk7plGcKBLh+aIgYVBxOI0F9qggWLGRJggLqm+FeIAEwkqHl9MhOIsvL5Nm0XIqln1Xzlcv0ziy4AgcgwJwwBmogmtQAw2AwSN4Bq/gzXgyXox342PemjHSmUPwB8bnD9lclf4=</latexit>

Part II: SCBO under extreme conditions of field & pressure
Shi, Dissanayake, PC, William Steinhardt, Graf, Silevitch, Dabkowska, Rosenbaum, Mila, Haravifard, Nat Commun 13, 1 (2022)‣ Experiments:
tunnel diode oscillator (TDO) technique<latexit sha1_base64="UJoWOl3WFxr1aBdJXnQ198tPXn0=">AAACEnicbVDLSsNAFJ3UV62vqEs3g0XQhSUpRV1JwU03QgX7gCYtk8mkHTpJhpmJUEK+wY2/4saFIm5dufNvnLZZaOuBgcM593LnHI8zKpVlfRuFldW19Y3iZmlre2d3z9w/aMs4EZi0cMxi0fWQJIxGpKWoYqTLBUGhx0jHG99M/c4DEZLG0b2acOKGaBjRgGKktDQwz5xAIJz6QZb6jQw6XMRcxfAc5nq/ejt1+tVsYJatijUDXCZ2TsogR3Ngfjl+jJOQRAozJGXPtrhyUyQUxYxkJSeRhCM8RkPS0zRCIZFuOouUwROt+DCIhX6RgjP190aKQiknoacnQ6RGctGbiv95vUQFV25KI54oEuH5oSBhUIee9gN9KghWbKIJwoLqv0I8QroLpVss6RLsxcjLpF2t2BeV2l2tXL/O6yiCI3AMToENLkEdNEATtAAGj+AZvII348l4Md6Nj/lowch3DsEfGJ8/5+admg==</latexit>

df
/
dH

‣ Non-zero df/dH

d2 M
dH 2

slope change in M

‣ Identify anomalies → phase boundaries‣ Compare with iPEPS phase diagram

SrCu2(BO3)2 under extreme conditions of field & pressure
Shi, Dissanayake, PC, William Steinhardt, Graf, Silevitch, Dabkowska, Rosenbaum, Mila, Haravifard, Nat Commun 13, 1 (2022)

J1

~ex
Nature of the 1/5 plateau

J10
J2

J20
<latexit sha1_base64="0q7rDYvpOhku1ybhmw0ZNC/jq6Y=">AAACC3icbVC7TsMwFHXKq5RXgJHFokJioUoqBIwVLIxF6ktqo8hxndaq88C+Qaqi7Cz8CgsDCLHyA2z8DW6aAVqOZOv4nHt1fY8XC67Asr6N0srq2vpGebOytb2zu2fuH3RUlEjK2jQSkex5RDHBQ9YGDoL1YslI4AnW9SY3M7/7wKTiUdiCacycgIxC7nNKQEuueTzwJaFpy7Vxy7XwWX7rV5YO1L2EtJ5lrlm1alYOvEzsglRRgaZrfg2GEU0CFgIVRKm+bcXgpEQCp4JllUGiWEzohIxYX9OQBEw5ab5Lhk+0MsR+JPUJAefq746UBEpNA09XBgTGatGbif95/QT8KyflYZwAC+l8kJ8IDBGeBYOHXDIKYqoJoZLrv2I6Jjoc0PFVdAj24srLpFOv2Rc1++682rgu4iijI3SMTpGNLlED3aImaiOKHtEzekVvxpPxYrwbH/PSklH0HKI/MD5/AIEamXg=</latexit>(b)
T1<latexit sha1_base64="MNTm8jAApZWpLgyz6ls/tBJqoAI=">AAAB6nicdVBNS8NAEJ3Ur1o/WvXoZbEInkJiS7W3ohePFfsFbSib7aZdutmE3Y1QQn+CFw+KePUXefPfuGkrqOiDgcd7M8zM82POlHacDyu3tr6xuZXfLuzs7u0XSweHHRUlktA2iXgkez5WlDNB25ppTnuxpDj0Oe360+vM795TqVgkWnoWUy/EY8ECRrA20l1r6A5LZceuZ6ihJanUDXGqrlOpINd2FijDCs1h6X0wikgSUqEJx0r1XSfWXoqlZoTTeWGQKBpjMsVj2jdU4JAqL12cOkenRhmhIJKmhEYL9ftEikOlZqFvOkOsJ+q3l4l/ef1EB5deykScaCrIclGQcKQjlP2NRkxSovnMEEwkM7ciMsESE23SKZgQvj5F/5POue3WbPe2Wm5creLIwzGcwBm4cAENuIEmtIHAGB7gCZ4tbj1aL9brsjVnrWaO4Aest093do3w</latexit>

T1<latexit sha1_base64="MNTm8jAApZWpLgyz6ls/tBJqoAI=">AAAB6nicdVBNS8NAEJ3Ur1o/WvXoZbEInkJiS7W3ohePFfsFbSib7aZdutmE3Y1QQn+CFw+KePUXefPfuGkrqOiDgcd7M8zM82POlHacDyu3tr6xuZXfLuzs7u0XSweHHRUlktA2iXgkez5WlDNB25ppTnuxpDj0Oe360+vM795TqVgkWnoWUy/EY8ECRrA20l1r6A5LZceuZ6ihJanUDXGqrlOpINd2FijDCs1h6X0wikgSUqEJx0r1XSfWXoqlZoTTeWGQKBpjMsVj2jdU4JAqL12cOkenRhmhIJKmhEYL9ftEikOlZqFvOkOsJ+q3l4l/ef1EB5deykScaCrIclGQcKQjlP2NRkxSovnMEEwkM7ciMsESE23SKZgQvj5F/5POue3WbPe2Wm5creLIwzGcwBm4cAENuIEmtIHAGB7gCZ4tbj1aL9brsjVnrWaO4Aest093do3w</latexit>

T1 T0 T0 T1
(c) p
2
0.149
0.035
-0.267-0.350

TFPP
1<latexit sha1_base64="MNTm8jAApZWpLgyz6ls/tBJqoAI=">AAAB6nicdVBNS8NAEJ3Ur1o/WvXoZbEInkJiS7W3ohePFfsFbSib7aZdutmE3Y1QQn+CFw+KePUXefPfuGkrqOiDgcd7M8zM82POlHacDyu3tr6xuZXfLuzs7u0XSweHHRUlktA2iXgkez5WlDNB25ppTnuxpDj0Oe360+vM795TqVgkWnoWUy/EY8ECRrA20l1r6A5LZceuZ6ihJanUDXGqrlOpINd2FijDCs1h6X0wikgSUqEJx0r1XSfWXoqlZoTTeWGQKBpjMsVj2jdU4JAqL12cOkenRhmhIJKmhEYL9ftEikOlZqFvOkOsJ+q3l4l/ef1EB5deykScaCrIclGQcKQjlP2NRkxSovnMEEwkM7ciMsESE23SKZgQvj5F/5POue3WbPe2Wm5creLIwzGcwBm4cAENuIEmtIHAGB7gCZ4tbj1aL9brsjVnrWaO4Aest093do3w</latexit>‣ Vertical stripes separated by dimer singlets (along red dashed lines)‣ Strong triplets in the center of each stripe, with a weaker pair of triplets in between✦ Neither a crystal of triplets, nor a crystal of bound states!
‣ Full-plaquette formation: reminiscent of full plaquette phase (FPP)✦ FPP: triplets on dimers within plaquette and singlets on dimers outside of plaquette,
where the triplets form effective S=1 Haldane chains‣ Effective description: S=1 diamond chain with m=2/3‣ Also found in a thin SSM tube made of 2 orthogonal dimer chains
Manmana, Picon, Schmidt, and Mila, EPL 94, 67004 (2011)

Nature of the 10x2 supersolid‣ Descendant of the 1/5 plateau state‣ Alternating rotation of the spins of successive stripes clockwise orcounterclockwise by 90 degrees‣ Finite component in the field direction, also on the boundary between stripes

Full plaquette physics appearing at finite magnetic field!

Part III: SrCu2(BO3)2 up to the saturation field
Nomura, PC, Miyata, Zherlitsyn, Ishii, Kohama, Matsuda, Ikeda, Zhong, Kageyama, Mila, arXiv:2209.07652

II.

(a)

0.2 c x10
11

Δv/v0

0.0

c66

Hc9

-0.2

Hc8

-0.4
-0.6

(b)

EXPERIMENTAL METHOD

The pulsed magnetic fields up to 150 T were generated
Comparison
iPEPS
resultssystem (STC)‣ with help of thewith
verticalsingle-turn-coil

Hc7

1.0
0.5

Hc8
Hc9

0.0
-0.5

Hc7
Hc6

-1.0

(c)
2+

M (µB/Cu )in the ISSP, University of Tokyo [35]. We used a liquid4decrease
theof‣ StrongHe bathcryostat to ofkeep
the soundsample atvelocity
3–4.2 K. We
note that the sample temperature can change due to thethec66 acoustic
in the 1/2 plateau?
magnetocalorice↵ect during the pulsed field (⇠ 6 µs)[17, 30]. High-quality single crystals of SCBO were grownby a traveling solvent floating zone method [36]. We used
the one (2⇥1⇥1 mm3 ) for the ultrasound and another one(2 ⇥ 1 ⇥ 0.3 mm3 ) for the magnetostriction measurement.
Magnetic fields were always applied along the c axis in
our experiments.
We performed the ultrasound measurements by using
the continuous-wave excitation technique [37]. Ultra-

0.4

ΔLc (a. u.)

the earlier studies have pointed out that the spin-latticecoupling plays an important role for the high-field properultrahigh fields up to 150T!
‣ Experiments:
ties of SCBO [15–17, 28–31]. In particular, the c66 mode(with ±"xy strain) shows extremely large anomalies. In& magnetostriction‣ Ultrasoundthis study, we velocity
discuss the magneto-structuralproperties
of SCBO at ultrahigh magnetic fields supported by thetheoretical calculations
based on above
the infinite projected
new anomalies‣ Identified
entangled pair state (iPEPS) tensor-network algorithm
the[32–34].
1/2 plateau
116T,of127T,
and 139T
The elastic atproperties
novel supersolid
phasesabove 100 T are discussed in terms of the spin-latticecoupling.
field: 139T
‣ Saturation

2

0.6

M/Ms 1/8 1/4

1/3

2/5

1/2

1/1

0.4
0.2
0.0
0

Y. H. Matsuda et al.
50

100

150

µ0H (T)

FIG. 2.

The ultrahigh-field data (H//c) obtained iniPEPS phase diagram
period-14 SSP

2.8

period-8 SSP

saturation
2.7

npetitio
m
o
c
g
on

es
of statstr

2.6

period-14 SSP

h/J

2.5
type a2.4
2.3
type b

1/3 SSP

2.2

1/2 plateau
2.1

period-10 SSP

type a2
0.6

1/2 plateau
0.61

0.62

0.63

0.64

0.65

0.66

0.67

0.68

J'/J
period-10 SSP
1/3 SSP type a

1/3 SSP type b

period-8 SSP

Results for J’/J=0.63?
period-10 SSP

iPEPS:

1

?

105.5
105.7

121.6

0.9

137

1/3 supersolid

1/2
M/M s

128.2

sat

0.8
period-14
SSP

0.7
0.6

type b

1/3 SSP a

type a

1/3 SSP b

0.5
2

1/2 plateau

105
10

-3

110

115

120

125

130

135

140

h [T]

2.048
2.362
2.489
2.659
2.053
Anomalies Hc6, Hc8, and Hc9 compatible with iPEPS results
1.5

p-10 SSP
p-14 SSP
7/8 plateau
saturation

2
Reduction of ultrasound velocity
in 1/2 plateau?
(a)

0.4

Strain of c66 mode:

0.2 c x10
11

Δv/v0

0.0

c66

Hc9

-0.2

‣ Strong reduction in 1/2 plateau

Hc8

-0.4
-0.6

(b)

1/8 1/4

1/3

1/2

1/1

Hc7

‣ Tiny reduction in 1/3 plateau

1.0

ΔLc (a. u.)

Both E’ and E’’ of the magnetic energy contribute to the elastic constant
‣ 0.5
Hc8

Hc9

‣ E’ has large magnitude in 1/2, but vanishing for 1/3 plateau
0.0

-0.5

H

-1.0

2+

M (µB/Cu )

(c)

0.6

M/Ms 1/8 1/4

1/3

c7
Checkerboard
structure
Hc6 (triplet) andwith positive
negative (singlet) bonds
contributions to E’
2/5 →1/2
1/1addup → large magnitude

0.4
0.2

odd periodicity → cancellation

2
Reduction of ultrasound velocity
in 1/2 plateau?
(a)

0.4

Strain of c66 mode:

0.2 c x10
11

Δv/v0

0.0

c66

Hc9

-0.2

‣ Strong reduction in 1/2 plateau

Hc8

-0.4
1/8 1/4

-0.6

(b)

1/3

1/2

Hc7

1/1

‣ Tiny reduction in 1/3 plateau

1.0

ΔLc (a. u.)

Both E’ and E’’ of the magnetic energy contribute to the elastic constant
‣ 0.5
Hc8

Fit:
‣ 0.0

<latexit sha1_base64="IPAQ57mNx7kCn9jsUjOdGkc+xOI=">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</latexit>

0
E
+
1

00
E
⇠
2

c/c,

-0.5

Hc7
Hc6

-1.0

2+

M (µB/Cu )

(c)

0.6
0.4
0.2

c = ⇢v 2Hc9

M/Ms 1/8 1/4

1/3

2/5

1/2

1/1

Estimates in good
agreement withexperimentPart IV: SSM with interlayer coupling‣ Extent of the plaquette phase is smaller in experiments than in theory

CAL REVIEW LETTERS 124, 206602 (2020)

an appear at T > 0

Guo, et al(2020)

magnetic resonanceth broken spatial
cattering revealed1]. The mode wasently an alternativeed [12]. Here weo a previously not¼ 3–4 GPa.
high-pressure heatBO3 Þ2 single crys≈ 0.7
ntum spin models,TÞ phase diagram,
J 0 /J
ere the SS model
Dimer 0.675
Plaquette
AF
0.765
were studied, andure down to 1.5 oro different types‣ Pressure model: Shi, et al. (2022)
the Supplemental
e obtained among
0
0
p
=
1.8GP
a
$
J
/J = 0.675 J = 81.5K
p
$
J
/J
=
0.63
c
0
e) we show typicalure regions. In theJ’(p), J(p): linear functions. J’(p) changes by 5% between p0 and pc (ESR)
ata for P > 4 GPa,valid.
Sakurai, et al., J. Phys. Soc. Jpn. 87, 033701 (2018)n CðTÞ=T: there is
fer to as the hump.
<latexit sha1_base64="ExjaJBKnxzJUcQ12DsQnGRbQTGg=">AAAB63icbVBNSwMxEJ31s9avqkcvwSJ6qrtS1GPRi/RUwX5Au5Rsmm1Dk+ySZIWy9C948aCIV/+QN/+N2XYP2vpg4PHeDDPzgpgzbVz321lZXVvf2CxsFbd3dvf2SweHLR0litAmiXikOgHWlDNJm4YZTjuxolgEnLaD8V3mt5+o0iySj2YSU1/goWQhI9hkUv3sot4vld2KOwNaJl5OypCj0S999QYRSQSVhnCsdddzY+OnWBlGOJ0We4mmMSZjPKRdSyUWVPvp7NYpOrXKAIWRsiUNmqm/J1IstJ6IwHYKbEZ60cvE/7xuYsIbP2UyTgyVZL4oTDgyEcoeRwOmKDF8YgkmitlbERlhhYmx8RRtCN7iy8ukdVnxrirVh2q5dpvHUYBjOIFz8OAaanAPDWgCgRE8wyu8OcJ5cd6dj3nripPPHMEfOJ8/BhqNlQ==</latexit><latexit sha1_base64="N2BemtVxsC8pEZlCKDIZ5rpz5vQ=">AAACEHicbVC7TgJBFJ3FF+Jr1dJmIjFYrbsGgQZDYqGhwkQeCRAyO8zChNlHZu5qCOETbPwVGwuNsbW0828cHoWCJ7nJyTn35t573EhwBbb9bSRWVtfWN5Kbqa3tnd09c/+gpsJYUlaloQhlwyWKCR6wKnAQrBFJRnxXsLo7uJr49XsmFQ+DOxhGrO2TXsA9TgloqWNmog7FRexYhesKwS3BPJC81wciZfiAy5mzctG2cvmLjpm2LXsKvEycOUmjOSod86vVDWnsswCoIEo1HTuC9ohI4FSwcaoVKxYROiA91tQ0ID5T7dH0oTE+0UoXe6HUFQCeqr8nRsRXaui7utMn0FeL3kT8z2vG4BXaIx5EMbCAzhZ5scAQ4kk6uMsloyCGmhAqub4V0z6RhILOMKVDcBZfXia1c8vJWdnbbLp0OY8jiY7QMTpFDsqjErpBFVRFFD2iZ/SK3own48V4Nz5mrQljPnOI/sD4/AE2BZor</latexit><latexit sha1_base64="J9BxJgHzodTe6Y7wVtvQ58BajJk=">AAACBnicbVDLSgNBEJyNrxhfqx5FGAyip7irIXpRAl4kpwjmAUkIs5PZZMjszjLTq4QlJy/+ihcPinj1G7z5N04eB00saCiquunu8iLBNTjOt5VaWFxaXkmvZtbWNza37O2dqpaxoqxCpZCq7hHNBA9ZBTgIVo8UI4EnWM3rX4/82j1TmsvwDgYRawWkG3KfUwJGatv7UdvBTcF8ULzbA6KUfMClo5PSpZMrnLXtrJNzxsDzxJ2SLJqi3La/mh1J44CFQAXRuuE6EbQSooBTwYaZZqxZRGifdFnD0JAETLeS8RtDfGiUDvalMhUCHqu/JxISaD0IPNMZEOjpWW8k/uc1YvAvWgkPoxhYSCeL/FhgkHiUCe5wxSiIgSGEKm5uxbRHFKFgksuYENzZl+dJ9TTnFnL523y2eDWNI4320AE6Ri46R0V0g8qogih6RM/oFb1ZT9aL9W59TFpT1nRmF/2B9fkDq52XTw==</latexit><latexit sha1_base64="YZRWTXZRQX6MYwIE1orUxTWiJLY=">AAAB7nicbVBNS8NAEJ3Ur1q/oh69LBbBU0ikai9KwYvopYL9gDaUzXbTLt1swu5GKKE/wosHRbz6e7z5b9y2OWjrg4HHezPMzAsSzpR23W+rsLK6tr5R3Cxtbe/s7tn7B00Vp5LQBol5LNsBVpQzQRuaaU7biaQ4CjhtBaObqd96olKxWDzqcUL9CA8ECxnB2kitu6uq55zf9+yy67gzoGXi5aQMOeo9+6vbj0kaUaEJx0p1PDfRfoalZoTTSambKppgMsID2jFU4IgqP5udO0EnRumjMJamhEYz9fdEhiOlxlFgOiOsh2rRm4r/eZ1Uh1U/YyJJNRVkvihMOdIxmv6O+kxSovnYEEwkM7ciMsQSE20SKpkQvMWXl0nzzPEunMpDpVy7zuMowhEcwyl4cAk1uIU6NIDACJ7hFd6sxHqx3q2PeWvBymcO4Q+szx+IH45j</latexit>

SSM with interlayer coupling‣ Series expansion results

Koga, J. Phys. Soc. Japan 69 (2000)

J 00<latexit sha1_base64="Y9re3dRnmgSVr/QvwWy+Nv+zfOc=">AAAB6nicbVDLSgNBEOz1GeMr6tHLYJB4CrsS1GPQi3iKaB6QLGF20psMmZ1dZmaFEPIJXjwo4tUv8ubfOEn2oIkFDUVVN91dQSK4Nq777aysrq1vbOa28ts7u3v7hYPDho5TxbDOYhGrVkA1Ci6xbrgR2EoU0igQ2AyGN1O/+YRK81g+mlGCfkT7koecUWOlh7tSqVsoumV3BrJMvIwUIUOtW/jq9GKWRigNE1Trtucmxh9TZTgTOMl3Uo0JZUPax7alkkao/fHs1Ak5tUqPhLGyJQ2Zqb8nxjTSehQFtjOiZqAXvan4n9dOTXjlj7lMUoOSzReFqSAmJtO/SY8rZEaMLKFMcXsrYQOqKDM2nbwNwVt8eZk0zsveRblyXylWr7M4cnAMJ3AGHlxCFW6hBnVg0IdneIU3RzgvzrvzMW9dcbKZI/gD5/MHZDuNOQ==</latexit>

AF phase becomes favored over plaquette phase with increasing J’’‣ Predicted values for J’’ (no consensus yet)
✦ J’’/J = 0.09 … 0.21 from fits to susceptibility [Miyahara & Ueda (2000), Knetter et al (2000)]✦ J’’/J ≈ 0.03 from ab-initio calculations[Radtke et al., PNAS 112 (2015)]

iPEPS for layered systems
Vlaar, PC, arxiv:2208.06423
(a)

Dxy

Dz

(b)

Ansatz:(c)‣ 3D tensor network ansatz (coupled iPEPS)‣ Dxy > Dz for weak interlayer coupling
21 → product state of iPEPSs‣ Dz =DxyPatrick Vlaar

Dz2

(a)

(b)

(c)

(d)

(e)

(f)

Contraction:‣ Dz = 1 : contract individual layers (2D)
‣ Dz > 1 : perform effective decouplingaway from center → 2D contraction‣ Interlayer correlations beyond meanfield level are included by the Dz > 1bonds in the center‣ Layered corner transfer matrix(LCTM) method

Benchmarks for 3D anisotropic Heisenberg model
Vlaar, PC, arxiv:2208.06423<latexit sha1_base64="+rTf9VncW3YNPot5nIri3bkUm78=">AAAB9XicbVDLSsNAFL3xWeur6tLNYBFcxUSKuhGKbqSrCvYBbQyT6aQdOnkwM1Fj6H+4caGIW//FnX/jtM1CWw9cOJxzL/fe48WcSWVZ38bC4tLyymphrbi+sbm1XdrZbcooEYQ2SMQj0fawpJyFtKGY4rQdC4oDj9OWN7wa+617KiSLwluVxtQJcD9kPiNYaemu5j4d19zsMR1dWKbtlsqWaU2A5omdkzLkqLulr24vIklAQ0U4lrJjW7FyMiwUI5yOit1E0hiTIe7TjqYhDqh0ssnVI3SolR7yI6ErVGii/p7IcCBlGni6M8BqIGe9sfif10mUf+5kLIwTRUMyXeQnHKkIjSNAPSYoUTzVBBPB9K2IDLDAROmgijoEe/bledI8Me1Ts3JTKVcv8zgKsA8HcAQ2nEEVrqEODSAg4Ble4c14MF6Md+Nj2rpg5DN78AfG5w9PXZG/</latexit>

Jz /Jxy = 0.1

E
improvement from Dz =1 to DF z = 2‣ Substantial‣ Values close to the extrapolated QMC result‣ In agreement with more expensive full 3D contractions
Vlaar & PC, PRB 103, 205137 (2021)

D

Phase diagram: SSM with interlayer coupling(work in progress)

Dz = 2<latexit sha1_base64="PzS97/1STsevXuBvdQrK336Bj54=">AAAB7HicbVBNSwMxEJ3Ur1q/qh69BIvgqeyWol6Eoh48VnBroV1KNs22odnskmSFuvQ3ePGgiFd/kDf/jWm7B219MPB4b4aZeUEiuDaO840KK6tr6xvFzdLW9s7uXnn/oKXjVFHm0VjEqh0QzQSXzDPcCNZOFCNRINhDMLqe+g+PTGkey3szTpgfkYHkIafEWMm76T1d1nrlilN1ZsDLxM1JBXI0e+Wvbj+macSkoYJo3XGdxPgZUYZTwSalbqpZQuiIDFjHUkkipv1sduwEn1ilj8NY2ZIGz9TfExmJtB5Hge2MiBnqRW8q/ud1UhNe+BmXSWqYpPNFYSqwifH0c9znilEjxpYQqri9FdMhUYQam0/JhuAuvrxMWrWqe1at39Urjas8jiIcwTGcggvn0IBbaIIHFDg8wyu8IYle0Dv6mLcWUD5zCH+APn8AIp6OQQ==</latexit>

AF
Dimer

Plaquette≈ 0.03

Guo et al (2020)

Estimate for the strength of interlayer coupling: J’’/J ≈ 0.03
LCTM: Promising approach also for other layered systems!

Conclusion✓ SrCu2(BO3)2 under pressure / in a magnetic field exhibits very rich physics!
✓ Finite temperature Ising critical point, analogous to critical point of water✓ New type of 1/5 plateau and supersolid phases at high pressure & field✓ Results up to saturation in good agreement with experiments &understanding of the reduction of ultrasound velocity in the 1/2 plateau✓ Reduction of plaquette phase due to interlayer coupling‣ Progress with iPEPS: versatile tool for ground state and finite temperature
calculations + extensions to layered systems

Thank you for your attention!
Acknowledgements:

F. Mila, S. Crone, P. Vlaar, P. Czarnik, J. Dziarmaga, L. Weber, A. Wietek, S. Wessel, B. Normand,
A. Honecker, A. Läuchli, J. Larrea Jimenez, E. Fogh, R. Lortz, M. E. Zayed, K. Conder,
E. Pomjakushina, Ch. Rüegg, H. M. Rønnow, T. Nomura, A. Miyata, S. Zherlitsyn,Y. Ishii,
Y. Kohama,Y. Matsuda, A. Ikeda, C. Zhong, H. Kageyama, Z. Shi, S. Dissanayake, W. Steinhardt,
D. Graf, D. M. Silevitch, H. A. Dabkowska, T. F. Rosenbaum, S. Haravifard