Science 13Mar2020

planes ( 31 ). Another piece of experimental evi-
dence is the critical exponentb= 0.321(3) of
the (1/3, 1/3, 0) peak intensity nearT 2 (fig. S4A),
which indicates the 3D Ising nature of the
magnetic order ( 42 ).

Discussion

Although the Monte Carlo simulations of the
classical spin model described above are in
partial agreement with our experiments, they
do not explain the experimental value of the
magnetic entropySm=10.38J·mol−^1 K−^1 ≈
1.248RatT 2 , which is very different from the
0.231Rgiven by the model. Qualitatively, the
discrepancy should be a result of the thermal
population of multiple low-lying CEF levels
of Ho3+; however, this leads to the question of
why the classical Ising Hamiltonian is appli-
cable. In pyrochlore systems such as Dy 2 Ti 2 O 7 ,
the classical Ising behavior is a consequence of
the dominance of CEF splitting over exchange
and dipolar interactions. This leads to an ef-
fective pseudospin-1/2 Hamiltonian with only
the pseudospin components along the local easy
axes present ( 43 ). These pseudospin components
are thus good quantum numbers, justifying the
use of the classical Ising Hamiltonian.
In HoAgGe, metallicity simultaneously sup-
presses the CEF splitting of Ho3+ions and en-
hances the exchange coupling between them,
making the two energy scales comparable at
least for the low-lying CEF levels. Thus, the
large (J=8)Ho3+moments in HoAgGe at
moderately low temperatures, when multiple
CEF levels are occupied, are closer to semi-
classical spins with strongsingle-ion anisotropy.
Such a semiclassical model can still be mapped
to an Ising model at the expense of introducing
further-neighbor exchange interactions ( 44 ),
which serves as an explanation for the apparent
validity of the classical Ising Hamiltonian for
HoAgGe. A complete understanding of the
entropy data awaits a full quantum mechanical
description of the system. It is worth noting
that the deviation from an ideal spin-1/2 system
can also lead to stronger quantum fluctuations, as
in the cases of Tb 2 Ti 2 O 7 ( 45 )andTb 2 Sn 2 O 7 ( 46 ).
The metallic nature of HoAgGe not only
makes it a high-temperature (compared with
pyrochlore spin ices) kagome ice, but may also
lead to exotic phenomena such as the interac-
tion between electric currents and the magnetic
monopoles or the toroidal moments, the rela-
tionship between the noncollinear ordering
and the anomalous Hall effect ( 47 – 50 ), and

metallic magnetoelectric effects caused by

broken inversion symmetry ( 51 ). Our results

suggest that ZrNiAl-type intermetallic com-

pounds are a prototypical family of kagome

spin systems that may host other exotic phases

beyond the classical spin liquid ( 52 , 53 )and

deserve further investigation.

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ACKNOWLEDGEMENTS

We thank Y. Tokiwa, S. Nakatsuji, O. Tchernyshyov, J. Xu,

J. Song, V. Peçanha-Antonio, and M. Meven for helpful

discussions; A. Jesche for providing access to the MPMS;

and T. Delazzer for help with the CEF calculations. K.A.R.

acknowledges J. Rau and M. Gingras for CEF discussions. We

also thank K. Kiefer and colleagues from Helmholz Zentrum

Berlin for providing access and assistance using the VM4

magnet at MLZ. The technical assistance of W. Luberstetter

in setting up the Oxford Instruments magnet on POLI is

appreciated.Funding:The work in Augsburg was supported

by the German Science Foundation through SPP1666 (project

no. 220179758) and TRR80 (project no. 107745057). The

instrument POLI at Heinz Maier-Leibnitz Zentrum (MLZ), Garching,

Germany, was operated by RWTH Aachen University in cooperation

with FZ Jülich (Jülich Aachen Research Alliance JARA). Inelastic

neutron experiments were conducted at the time-of-flight

spectrometer NEAT operated by Helmholtz Zentrum Berlin. This

work utilized the RMACC Summit supercomputer, which is

supported by the National Science Foundation (awards

ACI-1532235 and ACI-1532236), the University of Colorado–Boulder,

and Colorado State University. The Summit supercomputer is

a joint effort of the University of Colorado–Boulder and Colorado

State University. The work in Prague was supported by the Czech

Science Foundation through project no. 18-10504S.Author

contributions:K.Z. and P.G. proposed the experiments; K.Z.

synthesized single crystals and measured magnetic properties

and specific heat; H.D. and V.H. conducted single-crystal elastic

neutron scattering; V.P., H.D., and K.Z. refined magnetic structures;

G.G. and M.R. conducted inelastic neutron scattering; H.C.

provided theoretical analysis and MC simulation; K.A.R. performed

CEF calculations; K.Z., H.C., and P.G. wrote the manuscript with

input from all authors.Competing interests:The authors declare

no competing interests.Data and materials availability:The

data presented in this paper can be found on Zenodo ( 54 ).

SUPPLEMENTARY MATERIALS

science.sciencemag.org/content/367/6483/1218/suppl/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S22

Tables S1 to S4

References ( 55 – 65 )

26 November 2018; accepted 14 February 2020

10.1126/science.aaw1666

Zhaoet al.,Science 367 , 1218–1223 (2020) 13 March 2020 6of6

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