1.
Magnon crystallization in kagomé
antiferromagnets
Ryutaro Okuma
University of Oxford (PD in R. Coldea’s group)
R. Okuma et al. Nat. Commun. 10, 1229 (2019).
R. Okuma et al. PRB 102, 104429 (2020).

2.
Outline
1. A series of magnetization plateaus in CdK up to 160 T
2. 1/3 magnetization plateau in herbertsmithite over 150 T
MegaGauss Lab in
ISSP Univ. Tokyo
Prof. Shojiro
Takeyama
Dr. Daisuke
Nakamura
“Low”
fi
eld experiment
Prof. M. Tokunaga, Prof. K. Kindo,
Dr. A. Miyake, & Dr. A. Matsuo
Calculation of magnetization process
Dr. T. Okubo & Prof. N. Kawashima
Prof. Zenji
Hiroi

3.
Hexagonal magnon in a kagomé antiferromagnet
Magnon is localized
inside a hexagon[1]
H = J
∑
Si ⋅ Sj − H
∑
Sz
i (H > Hs = 3J)
Kagomé lattice under high
fi
elds
Excitation from a polarized state
Sz = 2
Flat band
[1]. J. Schulenberg et al. PRL
(2002).
Molecular entity in frustrated lattices
Star of David[4]
Trimeron[3]
Spin loop[2]
ns severely under-constrained, no adjustment is possible with-
fecting spins on the outer perimeter. A general lowest-energy
configuration on the pyrochlore lattice should therefore have
like soft modes that extend throughout the lattice. However, if
x hexagon spins are antiparallel with each other (Fig. 4c), then
aggered magnetization vector for a single hexagon, which for
y we shall call the spin-loop director, is decoupled from the 12
spins, and hence its reorientation embodies a local zero energy
for the pyrochlore lattice12
. It is possible to assign simul-
usly all spins on the spinel lattice to hexagons, thus producing
where N is total number of spins) weakly interacting degrees of
om (Fig. 1c). Accordingly, states that account for 1/6 of the
py of the magnet are accessible through local fluctuations from
gurations where all spins are bunched into directors. Indeed,
easured entropy of 0.15 R ln 4 (where R is the gas constant) just
a low-temperature spin-Peierls-like phase transition4,13
is
to the predicted entropy of (1/6) R ln 4 for uncorrelated
ors in a spin-3/2 magnet. In contrast, there are no local soft
s for a general spin configuration in the low-energy manifold.
distinction between a general state from the low-energy
old and states from the director protectorate is probably the
for the stability of the latter.
r finding that fluctuations in ZnCr2O4 involve spin clusters
ot individual spins provides a natural explanation for a range
operties of geometrically frustrated magnets. It explains why
mperature-dependent susceptibility of frustrated magnets is
ately described by exact diagonalization of judiciously chosen
clusters14–16
. Moreover, the so-called ‘undecouplable’ muon
esonance response17
is recognized as being a consequence of
ns sensing slow, large-amplitude fluctuations of select spins in a
director. A director protectorate also provides a natural
nation for the coexistence of low characteristic energy scales
rigid short-range order, as evidenced by specific-heat18
and
-elastic neutron data19
.
e director protectorate hints at an organizing principle for
ated systems: if macroscopic condensation is not possible,
cting degrees of freedom combine to form rigid composite
es or clusters with weak mutual interactions. Exploring the
generality and basis for such a principle should be an interesting
focus for theoretical work and experiments on a wider class of
systems. Composite degrees of freedom are common in strongly
interacting many-body systems. Quarks form hadrons, hadrons
form nuclei, nuclei plus electrons form atoms, and atoms form
molecules, which in turn are the basis for complex biological
functionality. Planets, stars, galaxies and galaxy clusters are
examples of clustering on a grander length scale. However, to our
knowledge, the emergence of a confined spin cluster degree of
freedom has not previously been documented in a uniform gapless
magnet. The discovery is important because magnets offer an
opportunity not afforded by the above-mentioned systems to
monitor emergent structure in complex interacting systems with
microscopic probes such as neutron scattering and NMR. The
collapse of a geometrically frustrated magnet into a director
protectorate could, for example, be a useful template for exploring
aspects of protein folding2,20
. A
Methods
Three crystals of ZnCr2O4 (total mass 200 mg) were co-mounted for the inelastic neutron
scattering measurements. The measurements were performed using the cold neutron
triple-axis spectrometer SPINS at the National Institute of Standards and Technology
Center for Neutron Research. A vertically focusing pyrolytic graphite (002)
monochromator, PG(002), extracted a monochromatic beam with energy Ei ¼ 6.1 meV
from a 58
Ni-coated cold neutron guide. Scattered neutrons were analysed with seven or
eleven 2.1 cm £ 15 cm PG(002) analyser blades that reflected neutrons with Ef ¼ 5.1 meV
onto a 3
He proportional counter. Cooled Be filtered the scattered beam.
Received 22 February; accepted 4 July 2002; doi:10.1038/nature00964.
1. Debenedetti, P. G. & Stillinger, F. H. Supercooled liquids and the glass transition. Nature 410, 259–267
(2001).
2. Wolynes, P. G. & Eaton, W. A. The physics of protein folding. Phys. World 12, 39–44 (1999).
3. Bramwell, S. T. & Gingras, M. J. P. Spin ice state in frustrated magnetic pyrochlore materials. Science
294, 1495–1501 (2001).
4. Ramirez, A. P. in Handbook on Magnetic Materials (ed. Busch, K. J. H.) Vol. 13, 423–520 (Elsevier
Science, Amsterdam, 2001).
5. Moessner, R. & Chalker, J. T. Properties of a classical spin liquid: the Heisenberg pyrochlore
antiferromagnet. Phys. Rev. Lett. 80, 2929–2932 (1998).
6. Canals, B. & Lacroix, C. Pyrochlore antiferromagnet: a three-dimensional quantum spin liquid. Phys.
Rev. Lett. 80, 2933–2936 (1998).
7. Stormer, H. L., Tsui, D. C. & Gossard, A. C. The fractional quantum Hall effect. Rev. Mod. Phys. 71,
S298–S305 (1999).
8. Laughlin, R. B. & Pines, D. The theory of everything. Proc. Natl Acad. Sci. USA 97, 28–31 (2000).
9. Lovesey, S. W. Theory of Thermal Neutron Scattering from Condensed Matter (Clarendon, Oxford, 1984).
10. Birgeneau, R. J. et al. Instantaneous spin correlations in La2CuO4. Phys. Rev. B 59, 13788–13794 (1999).
11. Lee, S.-H. et al. Spin-glass and non-spin-glass features of a geometrically frustrated magnet. Europhys.
Lett. 35, 127–132 (1996).
12. Moessner, R. & Chalker, J. T. Low-temperature properties of classical, geometrically frustrated
antiferromagnets. Phys. Rev. B 58, 12049–12062 (1998).
13. Lee, S.-H., Broholm, C., Kim, T. H., Ratcliff, W. & Cheong, S-W. Local spin resonance and spin-
Peierls-like phase transition in a geometrically frustrated antiferromagnet. Phys. Rev. Lett. 84,
3718–3721 (2000).
14. Moessner, R. & Berlinsky, A. J. Magnetic susceptibility of diluted pyrochlore and SrCr9-9xGa3þ9xO19.
Phys. Rev. Lett. 83, 3293–3296 (1999).
15. Garcia-Adeva, A. J. & Huber, D. L. Quantum tetrahedral mean-field theory of the pyrochlore lattice.
Can. J. Phys. 79, 1359–1364 (2001).
16. Mekata, M. & Yamada, Y. Magnetic-ordering process in SrCr9(Ga-In)3O19. Can. J. Phys. 79,
1421–1426 (2001).
17. Uemura, Y. J. et al. Spin fluctuations in frustrated kagomé lattice system SrCr8Ga4O19 studied by
muon spin relaxation. Phys. Rev. Lett. 73, 3306–3309 (1994).
18. Ramirez, A. P., Espinosa, G. P. & Cooper, A. S. Elementary excitations in a diluted antiferromagnetic
kagomé lattice. Phys. Rev. B 45, 2505–2508 (1992).
19. Broholm, C., Aeppli, G., Espinosa, G. P. & Cooper, A. S. Antiferromagnetic fluctuations and short
range order in a kagomé lattice. Phys. Rev. Lett. 65, 3173–3716 (1990).
20. Pande, V. S., Grosberg, A. Yu. & Tanaka, T. Heteropolymer freezing and design: Towards physical
models of protein folding. Rev. Mod. Phys. 72, 259–314 (2000).
ters to nature
ZnCr2O4
DBB 5 2.967 Å on average), but asymmetric connectivity leads to off-
centre distortions and long B–B bonds in 2 of 16 cases.
The linear three-site (trimeron) distortions (Fig. 4d) significantly
perturb charge order in the Cc magnetite structure and couple the
magnetism to the complex overall distortion. Only two B ions are
available per minority-spin electron, so trimeron units are constrained
to share sites. End-sharing connections with angles of 60u, 120u and
180u are possible, but 180u linkages are apparently avoided to maximize
charge transfer because two or three different t2g orbitals are used by an
Fe31
ion participating in two or three trimerons. This is analogous to
the formation of two or three short cis-bonds in some d0
transition
metal oxides and oxynitrides26
, and creates off-centre Fe31
displace-
ments that contribute strongly to the acentric lattice modes and electric
polarization. High connectivity seems favourable because four of the
eight Fe31
sites participate in the maximum of three trimerons. The
orderings of linear three-site, one-electron units along the infinite B
chains correspond to charge density waves with wavevector magnitude
qc 5 1/N for a repeat sequence of one trimeron for every N B sites. The
structural average is qc 5 1/6 but only chains with qc 5 0, 1/8 and 1/4
are present, in a 1:6:5 ratio. The unique qc 5 0 chain contains only Fe21
ions (Fig. 4a, b) but their three-site distortions are always within other
chains.
The linear three-site distortions we attribute to trimerons also
account for the non-Verwey distribution of radial amplitudes (QRad)
and BVS values for the B-site FeO6 octahedra (Fig. 3a). Each Fe21
ion
donates minority-spin t2g electron density to two B-site acceptors
within a trimeron (Fig. 4c), so the eight Fe21
-like sites have a narrow
QRad distribution and BVS < 2.4. However, Fe31
ions are acceptors in
varying numbers of trimerons. Two Fe31
sites do notparticipate in any
trimerons and so have the two lowest QRad values and the highest
B-site BVS values, ,3.0 (Fig. 3a). The other six Fe31
sites are acceptors
for one to three trimerons and thus have a spread of higher Q values
C
O
B
C
O
B
C
O
B
a b
c d
A
A
A
Figure 4 | Charge, orbital and trimeron order in the low-temperature
magnetite structure. a, Distribution of Fe21
and Fe31
states (blue and yellow
spheres, respectively) in the first-approximation Verwey-type model, shown in
the !2a3 !2a3 2a Cc supercell (a is the high-temperature cubic-cell
parameter). Nearest-neighbour Fe21
–Fe21
connections (blue lines) describe
irregular chains, parallel to the a axis, derived from a more symmetric, centric
LETTER RESEARCH
[2] S-H. Lee et al. Nature (2002).
[3] M. S. Senn et al. Nature (2012).
[4] L. Ma et al. Nat. Commun. (2016).
Fe3O4 TaS2
Sz = 3
S = 1/2

4.
Magnon crystallization in the spin-1/2 kagomé antiferromagnet
1/9
1/3
5/9
7/9
H
M
M/Msat =1/3 5/9
Kagome solid plateaus
Long range orders are
Magnetic unit ce
Close packing of hexagonal magnon
~crystalline phase
Sz = 2
Sz = 1
Sz = 0
Calculated M-H curve[5]
3J
1
[5] S. Nishimoto et al. Nat.
Commun. (2012).
Ultra-high magnetic
fi
eld is necessary to observe this phenomena!

5.
Experimentally synthesized S = 1/2 kagomé magnets
Distortion Antisite mixing TCW (K) Hs (T) TN (K)
Volborthite[6] Heavy No -110 246 1.1
Herbertsmithite[7] No Large -300 582 <0.05
Cd-Kapellasite[8] No No -60 134 4
Volborthite Herbertsmithite
dx2−y2
d3z2−r2
[6] Z. Hiroi et al., JPSJ (2001). [7] S. Helton et al., PRL (2007). [8] E. A. Nytko et al., Inorg. Chem.
(2009).
Cl-
Cu2+
O2-

6.
Ideal kagomé lattice in Cd-Kapellasite
• dx2-y2 type of S = 1/2 Cu2+
• Negative Vector Chirality order
• DM interaction ~ 0.1J
a
b
c
Cd Cu
OH
[9] R. Okuma et al. PRB (2017). [10] M. Elhajal et al. PRB (2002).
NVC
q = 0 order
<Si> = 0.3µB
Net moment
// a
J+DM model
I
(a.u.)
7.0
6.0
5.0
4.0
d (A)
I1.6K - I10K fit
0.01
2
4
0.1
2
4
1
2
M
(10
-3
µ
B
Cu
-1
)
20
15
10
5
0
T (K)
B = 10 Oe
B // a
B // c
TN = 4 K

7.
light
M
electric current
B
sample
beam splitter
single turn coil
Is
Ip
High
fi
eld measurement by faraday rotation up to 200 T
1mm
Optical method is immune to E-B noise compared
with induction method[11]
→Best for M - H measurement over 100 T
[11] E. Kojima et al. PRB (2008).
Used single crystal
M ∝ θF = cos−1
(Is − Ip)/(Is + Ip)
200
150
100
50
0
B
(T)
8
6
4
2
0
time (µs)
2.0
1.5
1.0
0.5
0.0
I
p
,
I
s
(V)
B
Is
Ip
T = 5 K

8.
Whole magnetization process of CdK at T = 5~8 K
120
100
80
60
40
20
0
θ
F
(deg.)
150
100
50
0
B (T)
1.5
1.0
0.5
0.0
M
(µ
B
/
Cu)
4.2 K
5 K
6 K
8 K
Many anomalies appeared
Are they related to plateaux?
[12] R. Okuma et al. Nat. Commun. (2019).
1.0
0.8
0.6
0.4
0.2
0.0
M
/
M
S
150
100
50
0
B (T)
4
3
2
1
0
dM
/
dB
(10
-2
µ
B
/
Cu)
1/3
T = 5 K
B // c, down sweep
5/9
7/9

9.
7 plateau-like anomalies appeared at T = 5 K
µ
µ
Hp
dM/dH
M
H
Mp
Hp
J + DM model predicts no
plateau
Deviation starts from where
hexagonal magnons can appear
[12] R. Okuma et al. Nat. Commun. (2019).

10.
How to understand high-
fi
eld phases
Background
High
fi
eld plateau phases in nearest neighbor model is
a closest packing of hexagonal magnons
Ad hoc hypothesis
1. Even if extra interactions exist, magnetic structures are based on
tiling of hexagonal magnons
2. Magnetic structures hold three 3 or 6 fold symmetry
Sz = 2
Sz = 1
Sz = 0
Qmag = 9, 12, 21, 36, …can have 3 or 6 fold symmetry

11.
Superlattice of hexagonal magnons
Sz = 2
Sz = 1
Sz = 0

12.
Unit cell of possible plateau phases in CdK
m1
m2
m3
m4
m5
m6
m7
M / Ms
Qmag =
observed
not observed
1
0
8/12
6/12
10/12
12
19/21
17/21
15/21
21
34/36
32/36
30/36
36
3/9
7/9
5/9
9
1.3
1.2
1.1
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
M
(µ
B
Cu
-1
)
150
100
50
0
B (T)
4
3
2
1
0
dM
/
dB
(10
-2
µ
B
Cu
-1
T
-1
)
B4
B2
B3
B1
B6
B5
M1
B7
BS
M2
M3
M4
M5
M6
M7
CdK: B // c, T ~ 5 K
[12] R. Okuma et al. Nat. Commun. (2019).

13.
Formation of extremely large magnon crystals
m1 = 3/9
m = 7/9
m = 5/9
Qmag = 9
m5 = 10/12
m2 = 6/12
m3 = 8/12
Qmag = 12
m6 = 19/21
m = 15/21
m4 = 17/21
Qmag = 21
m7 = 34/36
m = 30/36
m = 32/36
Qmag = 36
Sz = 2
Sz = 1
Sz = 0

14.
Origin of superstructure: diagonal interaction
a b
c
ΔE= 2E1
ΔE = 2E1 – J2/9
d
ΔE = E1
ΔE = 2E1 – Jd/18
J2 Jd
Cd Cu
OH
Antiferromagnetic Jd stabilize Qmag = 12
Ferromagnetic J2 destabilize Qmag = 9 J2
Jd

15.
Celebrated kagomé antiferromagnet Herbertsmithite
[13] D. E. Freedman et al. JACS (2010). [14] M. Fu et al. Science (2016).
[15] P. Khuntia et al. Nat Phys. (2020).
Zn1-xCu4-x(OH)6Cl2 (x>0.15)
Cu2+
Zn2+/Cu2+
a
b
c
viously measured powder-averaged 35
Cl and 17
O
NMR line shapes (22, 23), we can resolve the five
Iz = m to m + 1 transitions (Iz, z component of the
nuclear spin angular momentum; magnetic quan-
tum number m = –5/2, –3/2, –1/2, +1/2, +3/2),
which are separated by a nuclear quadrupole
3 6 2
ence in the transverse relaxation that affects the
apparent signal intensities (fig. S1), we estimated
the population of the three sites as 13 ± 4%, 28 ±
5%, and59± 8%, inagreementwith earlier2
DNMR
observations of three corresponding sites in a
deuterated single crystal of ZnCu3(OD)6Cl2 (24).
51.5 52.0 52.5 53.0 53.5
fo
NN
NNN
Frequency (MHz)
Main
295K (9T||c)
0 50 100 150 200 250 300
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2
K
(c)
NN
(%)
17
K
(%)
Temperature (K)
Main (B||c)
Bext
= 9T
NN (B||c)
Main1 (B||a*)
NN1 (B||a*)
NN2 (B||a*)
-0.5
0.0
0.5
1.0
2
D NN (B||c)
NN
fo
Main
30K
4.2K
50K
170K
10K
48 50 52 54 56
Frequency (MHz)
Spin
echo
intensity
(arb.
units)
120K
Spin
echo
intensity
(arb.
units)
Fig. 2. Influence of the Cu2+
defects and NMR Knight shift in 9 T. (A) 17
O NMR line shape at 295 K in
Bext = 9 T || c, fitted with three sets of five peaks arising from the main (nQ
(c)
~ 360 kHz), NN (nQ
(c)
~ 310 kHz),
and NNN (nQ
(c)
~ 350 kHz) sites.The vertical dashed line represents the zero-shift frequency fo = (gn/2p)Bext
17 (c) 17 (a*) 17 (c)
Fig. 1. Kagome lattice and the structure of
ZnCu3(OH)6Cl2. (A) A kagome lattice formed by
corner-sharedtriangles.Theantiferromagneticinterac-
tions between NN sites frustrate the spin system
(black arrows indicate two spins forming a singlet
pair; the red double-headed arrow with the question
mark represents a frustrated spin). (B) Top view of
the Cu3O3 kagome layer in ZnCu3(OH)6Cl2. The
local geometries of the main1 and main2 17
O sites
(marked as 1 and 2, respectively) are not equivalent
under the presence of B || a*. (C) Zn2+
sites (purple)
on
January
18,
2021
http://science.sciencemag.org/
Downloaded
from
ARTICLES NATURE
1 10
0.1
1
1/T
1
(ms
–1
)
T (K)
~T
0.84(3)
~T
ν = 39.054 MHz and B = 6.60 T (KM
= 2.45%)
~T × e
–0.09/T
1 10 100
–0.2
0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
Shift,
K
M
(%)
T (K)
2.6 T
4.3 T
6.7 T
Ref. 41
B||a*
0
1
2
3
Gapped series (ref. 42)
Gapless series
χ
calc,kagome
(e.m.u.
mol
−1
f.u.
−1
×
10
3
)
6.4 6.7 7.0
39.054 MHz
Field (T)
T = 5 K
4.1 4.3 4.5
25.013 MHz
2.5 2.6 2.7
15.338 MHz
6.2 6.4 6.6 6.8 7.0
Field (T) @ 5 K
Normalized
intensity
–0.10 –0.05 0 0.05
(B/Bpeak – 1) / w
1.7–10 K
0 5 10 15 20
0
0.2
0.4
0.6
0.8
1.0
Shift,
K
M
(%)
T (K)
2.6 T
4.3 T
6.7 T
No contrast,
this work
B||a*
0
0.5
1.0
1.5
2.0
χ
calc,kagome
(e.m.u.
(mol
f.u.
−1
)×
10
3
)
a
b
c
d
Gapless
series
Fig. 3 | Gapless ground state from shift and T1 measurements. a, Low-temperature spectra obtained by sweeping the field B at a constant fr
ν, corresponding to the reference field (Bref ¼ 2πν
17γ
I
) with 17
γ!=!2π!×!5.7718!Mrad!T−1
. The shift of the main line, KM, is obtained from the peak pos
the spectrum at a field Bpeak
Bref
Bpeak
¼ 1 þ KM
! "
. Left: the spectra are rescaled by a factor w proportional to their width and shifted horizontally t
Gapless or gapped
quantum spin liquid[15]
. Field dependence of the 1.7 K macroscopic magnetization (black squares) of
ithite normalized to the saturated magnetization of 1 mole of Cu2+ S=1/2 spins.
etization of the defective inter-layer Cu2+ spins (red open squares) is estimated by
g the linear contribution of the kagome planes (blue line) from the total magnetization.
kagome plane susceptibility at 1.7 K is shown as the blue circle with large downward
Defect spin dominates
bulk property[13]
17O NMR spectra[14]

16.
Spin-gap like behavior and plateau in Faraday rotation
Spin gap? (Δ~30 T)
No upturn due to free spin
Plateau-like behavior
cf. Hs = 400-600T
[16] T. H. Han et. al. arXiv (2014).
[17] R. Okuma et al. PRB (2020).
35
30
25
20
15
10
5
0
θ
F
(˚)
150
100
50
0
B (T)
M
(a.u.)
B // [101]
T ~ 6 K
T ~ 10 K
Bp
Bulk M
[19]

17.
Light seems to see only the kagomé plane
Cl
O
Cu
Cu/Zn
O
kagomé defect
PHYSICAL REVIEW B 96, 241114(R) (2017)
−3
−2
−1
0
1
Γ M K Γ
(a)
E
−
E
F
(eV)
(c)
x2
− y
2
z2
xz (yz)
xy
1.18 eV 1.39 eV 1.42 eV 1.75 eV
(b)
DOS
x2
− y2
z2
xz (yz)
xy
FIG. 5. Calculated GGA (a) band structure and (b) Cu 3d orbital
resolved density of states (DOS) for ZnCu3(OH)6Cl2 at U = 0.
(c) Energy level diagram for on-site Cu 3d orbitals. The energy
differences of the three levels 1.18, 1.39, and 1.42 eV correspond
to the experimental peaks in Fig. 3(a).
“allowing” the dipole-forbidden d-d transition [20–22,47,51].
In our case we can explain the main features in terms of mixing
a1g
x2-y2,z2
eg
θtotal = αkag*Mkag + αdef*Mdef
Subtraction of
paramagnetic moment
reproduces FR data
0.12
0.10
0.08
0.06
0.04
0.02
0.00
M
/
M
S
60
40
20
0
B (T)
Bulk M
Subtracted
FR (10 K)
FR (6 K)
[18] A. Pustogow et al. PRB (2017).
At hν = 2.33 eV,
αdef ~ 0
RAPID COMMUNICATIONS
PHYSICAL REVIEW B 96, 241114(R) (2017)

18.
Robust 1/3 plateau in herbertsmithite
Magnon crystal is robust
against disorder!
[19] H. Nakano et al. JPSJ (2018)
0
1/9
1/3
M
/
M
S
250
200
150
100
50
0
B (T)
Theory
[12]
Theory
[19]
FR (6K)
FR (10K)
J = 250 K
Origin of large spin gap is unclear.

19.
Summary
• High
fi
eld phases of kagomé antiferromagnet is
characterized by crystallization of hexagonal magnons
• Various plateau-like anomalies were observed in CdK
• Crystallization of hexagonal magnons can account for
magnetization plateaus in CdK
• 1/3 plateau observed in herbertsmithite also suggests
presence of magnon crystal