Science 13Mar2020

the magnetic peaks at (–1/3, 2/3, 1) (fig. S4B)
and (1/3, 4/3, 1) versus the strength of the mag-
netic field along thebaxis at 4 K. Overall, the
intensity decreases with increasing field and
disappears atH> 3.2 T, with sudden changes at
the metamagnetic transitions depicted in Fig. 1,
C and D, suggesting the shrinking of the mag-
neticunitcellinfield.Toobtainfurtherinfor-
mation, we refined the magnetic structures at
the three majorM(H) plateaus from the neu-
tron scattering. The magnetic field breaks the
threefold rotational symmetry and turns the
ground state magnetic space group P-6′m2′
into Am′m2′, with the nine Ho moments in

the√3×√3 unit cell forming six nonequivalent

groups (Fig. 2G).

Figure 2, G to I, shows the magnetic struc-

tures at the three major plateaus obtained from

the neutron data taken at 1.8 K andH=1.5,

2.5, and 4 T along thebaxis (also see Table 1

for the refinement factors). One first notices

that all of them can be obtained from the ground

state by reversing certain Ho spins, with negli-

giblerotationfromtheirlocalIsingaxis( 31 ).

This is strong evidence for the Ising-like ani-

sotropy of the Ho moments, with the local easy

axes defined by a perpendicular mirror plane

through each atom. The Ising-like anisotropy

is further confirmed by our crystalline electric

field (CEF) calculationsbelow. Moreover, in all

threestructures,thespinsarealwaysreversed

in such a way that the one-in-two-out or two-in-

one-out ice rule is satisfied but the total mag-

netic moment alongbincreases with increasing

field. AtH= 4 T, the magnetic unit cell becomes

identical to the structural unit cell ( 14 , 15 , 18 )

and has the largest possible net moment allowed

bytheicerule.Thisisfurthercorroboratedby

the identical magnetization jump of 1.7mB/Ho

at the three metamagnetic transitions at 1.8 K

(fig. S2B). Assuming the magnetic structures

in Fig. 2, G to I, this jump can be translated to

Zhaoet al.,Science 367 , 1218–1223 (2020) 13 March 2020 3of6

4681012

0

20

40

60

(1, 0, 0)

Integrated Intensity

).u.a(ytisnetnI detargetnI

T(K)

T(K)

(1/3, 1/3, 0)

4 6 8 10 12

2

4

H

1

2

5

6

3

4

1

3

5

A

D E

G

1

2

2

HI

1

2

3

B

01234

0

10

20

30

HoAgGe; H//b; T=4K

(-1/3, 2/3, 1)

(1/3, 4/3, 1)

).u.a(ytisnetnI detargetn

I

μ 0 H(T)

1.5T(4K) 2.5T(1.8K) 4T(1.8K)

10K

4K

T

H//b

3

3

1

1

2

2

F

C

Fig. 2. Magnetic structures of HoAgGe versus temperature and field with
H//b.(A) Integrated intensity of the magnetic peak (1/3, 1/3, 0) (fig. S4A) from
13 K down to 3.8 K according to the neutron diffraction, with the integrated
intensity of nuclear site (1, 0, 0) as an inset. (B) Refined magnetic structures of
HoAgGe at 10 K. The magnetic unit cell is indicated by the green rhombus, with
the three inequivalent Ho sites Ho1, Ho2, and Ho3 labeled by 1, 2, and 3,
respectively, for simplicity. (C) Counterclockwise hexagons of spins in the partially
ordered structure of HoAgGe at 10 K, with 1/3 spins not participating in the
long-range order. (D) Integrated intensity of magnetic peak (–1/3, 2/3, 1)

(fig. S4B) and (1/3, 4/3, 1) versus field at 4 K. (E) Refined magnetic structure of

HoAgGe at 4 K. (F) Clockwise and counterclockwise hexagons of spins in the

magnetic structure of HoAgGe at 4 K, which is exactly the expected√3×√3 ground

state of kagome spin ice. (G) Refined magnetic structure of HoAgGe atH= 1.5 T

andT= 4 K. The refinement was done in the 3 ×√3 light-green rectangle. The

six inequivalent Ho sites are labeled by numbers 1 to 6 for simplicity. (H) Refined

magnetic structure of HoAgGe atH= 2.5 T andT= 1.8 K. (I) Refined magnetic

structure of HoAgGe atH= 4 T andT= 1.8 K, with the two inequivalent Ho sites

labeled by 1 and 2. The field direction is marked by the red arrow for (G) to (I).

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