PHYSICAL REVIEW B 96,024445 (2017)
J1-J2square lattice antiferromagnetism in the orbitally quenched insulator MoOPO4
L. Yang,1,2M. Jeong,1,*P. Babk e v i ch, 1V. M . K a tu ku r i , 3B. Náfrádi,2N. E. Shaik,1A. Magrez,4H. Berger,4J. Schefer,5
E. Ressouche,6M. Kriener,7I. Živkovi ´
c,1O. V. Yazyev,3L. Forró,2and H. M. Rønnow1,7,†
1Laboratory for Quantum Magnetism, Institute of Physics, Ecole Polytechnique Féderale de Lausanne, CH-1015 Lausanne, Switzerland
2Laboratory of Physics of Complex Matter, Institute of Physics, Ecole Polytechnique Féderale de Lausanne, CH-1015 Lausanne, Switzerland
3Chair of Computational Condensed Matter Physics, Institute of Physics, Ecole Polytechnique Féderale de Lausanne,
CH-1015 Lausanne, Switzerland
4Crystal Growth Facility, Institute of Physics, Ecole Polytechnique Féderale de Lausanne, CH-1015 Lausanne, Switzerland
5Laboratory for Neutron Scattering and Imaging, Paul Scherrer Institute, CH-5232 Villigen, Switzerland
6Université Grenoble Alpes, CEA, INAC, MEM, F-38000 Grenoble, France
7RIKEN Center for Emergent Matter Science, Wako, Saitama 351-0198, Japan
(Received 17 May 2017; published 28 July 2017)
We report magnetic and thermodynamic properties of a 4d1(Mo5+) magnetic insulator MoOPO4single crystal,
which realizes a J1-J2Heisenberg spin-1/2 model on a stacked square lattice. The speciﬁc-heat measurements
show a magnetic transition at 16 K which is also conﬁrmed by magnetic susceptibility, ESR, and neutron
diffraction measurements. Magnetic entropy deduced from the speciﬁc heat corresponds to a two-level degree of
freedom per Mo5+ion, and the effective moment from the susceptibility corresponds to the spin-only value. Using
ab initio quantum chemistry calculations, we demonstrate that the Mo5+ion hosts a purely spin-1/2 magnetic
moment, indicating negligible effects of spin-orbit interaction. The quenched orbital moments originate from the
large displacement of Mo ions inside the MoO6octahedra along the apical direction. The ground state is shown
by neutron diffraction to support a collinear Néel-type magnetic order, and a spin-ﬂop transition is observed
around an applied magnetic ﬁeld of 3 .5 T. The magnetic phase diagram is reproduced by a mean-ﬁeld calculation
assuming a small easy-axis anisotropy in the exchange interactions. Our results suggest 4 dmolybdates as an
alternative playground to search for model quantum magnets.
DOI: 10.1103 /PhysRevB.96.024 4 4 5
The 4 dtransition-metal oxides naturally bridge the two
different regimes of the strongly correlated 3dcompounds
and the 5 dcompounds with strong spin-orbit coupling (SOC)
. To what extent the 4 dcompounds represent either regime
or display original properties is largely an open question of
current interest . Most notably, for instance, it is intriguing
that seemingly similar Ca2RuO4and Sr2RuO4display totally
different behaviors: the former is a Mott insulator [3–6],
while the latter is a metal and becomes superconducting at
low temperature [5–8]. Despite great interest, however, purely
4dquantum (spin-1/2) magnets are rather rare [9–12]asthe
electronic structure is often complicated by the presence of
other types of 3 dor 4 fmagnetic orbitals [13 ].
Among the few known 4d1magnets [9,11,12]themolyb-
denum phosphate MoOPO4is reported [14 ]. The MoO6
octahedra with Mo5+ions are corner shared to form a chain
along the crystallographic caxis of the tetragonal structure
[Fig. 1(a)], and these chains are further coupled to each
other via corner sharing PO4tetrahedra [Fig. 1(b)][14 ,15 ].
Previous susceptibility data on a powder sample of MoOPO4
shows a Curie-Weiss behavior with antiferromagnetic !CW =
−14 .5Kandamagnetictransitionat18K. The 31P
NMR on a powder evidences a substantial exchange through
the PO4tetrahedra, and a sharp powder ESR line infers a
rather isotropic gfactor . However, so far there have not
been any studies on the magnetic structure in the ordered
state or magnetic properties of a single crystal. Moreover,
any discussion on the possible interplay between the crystal
electric ﬁeld and SOC is absent.
Here we report the magnetic and thermodynamic properties
of a MoOPO4single crystal using speciﬁc heat, susceptibility,
magnetization, ESR, and neutron diffraction experiments. We
also elucidate the electronic states and magnetic aspects in
light of SOC and crystal-ﬁeld effects, with the help of ab initio
II. EXPERIMENTAL DETAILS
High-quality single crystals of MoOPO4were grown
following the procedure described in Ref. [14 ]. H2MoO4
was mixed with concentrated phosphoric acid and heated
up to 1000 ◦C for reaction in an open platinum crucible.
After being cooled to room temperature, the resulting dark-
blue solid was dissolved in a large amount of hot water.
The yellow transparent crystals were obtained in a platelike
shape [Fig. 1(c)]. Large crystals have a typical dimension
of 3 ×2×0.4mm
3with the caxis normal to the plate.
The crystal belongs to the space group P4/n,withlattice
parameters of a=b=6.204 4 ˚
by single-crystal x-ray diffraction, in agreement with Ref. [14 ].
Speciﬁc heat was measured using a physical properties
measurement system (PPMS, Quantum Design, Inc.), and
magnetization was measured using a magnetic properties
measurement system (MPMS, Quantum Design, Inc.). ESR
measurements were performed using a Bruker X-band spec-
trometer with a TE102 resonant cavity around 9.4 GHz.
24 69-995 0/2017/96(2)/024 4 4 5 (8) 024 4 4 5 -1 ©2017 American Physical Society
L. YANG et al. PHYSICAL REVIEW B 96,024445 (2017)
FIG. 1. Crystal structure of MoOPO4projected onto (a) the ac
planes, showing a chainlike arrangement of MoO6octahedra (yellow),
and (b) the ab planes, showing the coupling between the chains via
PO4tetrahedra (blue). Dashed lines represent the unit cells. Possible
in-plane (J1and J2) and out-of-plane (Jc)exchangecouplingsare
also shown. (c) Photograph of a representative single crystal.
Neutron diffraction experiments were performed on TRICS
and D23 beamlines at the Paul Scherrer Institute and Institut
Laue-Langevin, respectively. An incident neutron wavelength
of 2.3 109 ˚
A was employed.
A. Speciﬁc heat
Figure 2(a) shows the speciﬁc heat Cpmeasured from 2
to 15 0 K in zero ﬁeld and in a magnetic ﬁeld of 14 T. Cp
above 25 K for both ﬁelds is essentially the same, increasing
monotonically with increasing temperature. In zero ﬁeld a
pronounced peak is found at 16.1 K, while the peak is
shifted to a slightly lower temperature of 15.4 K at 14 T.
These peaks correspond to a transition into a magnetically
long range ordered phase, as evidenced by other experimental
measurements discussed in later sections.
In order to extract the magnetic part of the speciﬁc heat Cmag
and to deduce the corresponding entropy Smag,wesimulatethe
lattice contribution from the high-temperature data by taking
into account the Debye and Einstein contributions. We ﬁt the
Cpdata above 3 0 K by a lattice-only model, Cp=CD+
!iCE,i, where CDand CE,i represent the Debye and Einstein
terms, respectively. The Debye term is expressed as
and the Einstein term is expressed as
where Rdenotes the gas constant, !Dand !Eare the Debye
and Einstein temperatures, and nDand nEare the numbers of
FIG. 2. (a) Speciﬁc heat Cpas a function of temperature in zero
ﬁeld (circles) and at 14 T (squares). The solid line represents the best
ﬁt of the simulated lattice contribution using the Debye (dash-dotted
line) and Einstein (dotted line) terms. The inset provides an enlarged
view of the low-temperature region. (b) Left axis: magnetic part of
the speciﬁc heat Cmag divided by temperature (circles). Right axis:
the solid line is the entropy calculated from Cmag.
the corresponding modes, respectively; the sum nD+nEis the
total number of atoms per formula unit. For our purpose, we
consider that a phenomenological ﬁt using nD,nE,!D,and
!Eas free parameters is sufﬁcient. The best ﬁt for the zero ﬁeld
was obtained when using one Debye and two Einstein terms,
which yields the characteristic temperatures !D=1177 K,
!E,1=372K, and !E,2=15 4 K and the numbers nD=4,
nE,1=2, and nE,2=1. The solid line in Fig. 2(a) is the
best-ﬁt result for the total lattice contribution, while the
dash-dotted and dotted lines are the corresponding Debye and
Einstein contributions, respectively. While the parameters in
the phenomenological phonon ﬁt may not be directly physical,
they provide a parametrization of the lattice contribution to the
speciﬁc heat, which can be substracted to estimate the magnetic
Figure 2(b) shows the resulting Cmag/T in zero ﬁeld (cir-
cles, left axis) obtained by subtracting the lattice contribution
from the measured Cp. The solid line in Fig. 2(b) plots
Smag(T)obtainedbyintegratingCmag/T over temperature
(right axis). Smag(T)isfoundtoreachandstayatRln 2 at
high temperatures, indicating two-level degrees of freedom.
The thin colored band in Fig. 2(b) represents the entropy range
obtained when ﬁtting the Cpdata by varying the lower bound
of temperature between 25 and 3 5 K to conﬁrm the negligible
dependence of the result on the chosen ﬁt range. A similar
analysis for the 14 T data (not shown) indicates negligible
024 4 4 5 -2
J1-J2SQUARE LATTICE . . . PHYSICAL REVIEW B 96,024445 (2017)
FIG. 3. (a) The dc magnetic susceptibility χ(T) in a ﬁeld of H=0.1 T applied parallel (circles) and perpendicular (squares) to the c
axis. The dashed line represents the Curie-Weiss ﬁt for H∥c, and the solid line shows the high-temperature series expansion using the Padé
approximant (see the text). The inset shows an enlarged view of the low-temperature region. (b) Isothermal magnetization M(H)forH∥c
(solid symbols) and H⊥c(open symbols) at several different temperatures. The inset plots the ﬁeld derivative dM/dH versus Hfor H∥c.
(c) Magnetic phase diagram from the susceptibility (squares), speciﬁc heat (upward triangles), magnetization (circles), and neutron diffraction
(downward triangles) data. Lines are guides to the eye. The colored background represents the result from the mean-ﬁeld calculations (see
B. Susceptibility and magnetization
Figure 3(a) shows the dc magnetic susceptibility χ=
M/H, where Mis magnetization, in a ﬁeld of H=0.1T
applied parallel and perpendicular to the caxis. For both cases,
However, for H∥c,χ(T) exhibits a sharp drop toward zero as
temperature is decreased across 17 K, while the one for H⊥c
remains only weakly temperature dependent. This is indicative
of an antiferromagnetic transition where the ordered moments
at low temperatures are collinear to each other and parallel to
The nearly isotropic, high-temperature part of χ(T)could
be well ﬁt by the Curie-Weiss formula, χ(T)=C/(T−
!CW )+χ0,where!CW is the Curie-Weiss temperature and
χ0is a temperature-independent diamagnetic and background
term that may arise from the plastic sample holder or the small
amount of grease used. The best and stable ﬁt is obtained
in the 5 0–3 00 K range, which yields the effective moment
µeff =1.67(1)µBper Mo5+ion, !CW =−6(1) K, and χ0=
2.2(1) ×10−4emu/mol for H∥cand µeff =1.69(1)µB,
!CW =−4(1)K,andχ0=4.6(1) ×10−4emu/mol for H⊥
c.ThebestﬁtforH∥cis shown as a dashed line in
Fig. 3(a). The negative !CW indicates that antiferromagnetic
interactions are dominant. The effective moments indicate a
spin-only value consistent with the speciﬁc-heat results.
The isothermal magnetization M(H)forH∥cand H⊥c
at several temperatures is shown in Fig. 3(b).At5 K,M(H)
increases slowly with the ﬁeld H∥cup to 3 T but then sharply
increases in a narrow ﬁeld range of 3 –4 T until it eventually
converges to the high-temperature M(H)dataobtainedat16or
20 K. This stepwise increase of M(H)becomessmearedoutas
temperature is increased. On the other hand, no such stepwise
behavior was observed at any temperatures for H⊥c.These
are typical signatures of a spin-ﬂop transition which occurs
when the ﬁeld is applied along an easy axis, along which
the ordered moments align: the spins on the two sublattices
rotate to attain components perpendicular to the applied ﬁeld
direction as a result of competition between antiferromagnetic
coupling, magnetic anisotropy, and the Zeeman energy.
The magnetic phase diagram is thus mapped out by combin-
ing the above bulk magnetic and speciﬁc-heat results, as shown
in Fig. 3(c).Theantiferromagnetictransitiontemperaturesin
different ﬁelds are obtained from the peaks in χ(T)andCp(T),
and the spin-ﬂop transition ﬁelds at different temperatures are
obtained from the peak positions in the dM/dH versus Hplot
[inset of Fig. 3(b)].
C. Electron spin resonance
In order to gain microscopic insight into the magnetic
properties, we have performed ESR measurements as a
function of ﬁeld orientation and temperature. Figure 4(a)
plots the obtained room-temperature gfactor as the ﬁeld
direction is rotated by φin the ab and ac planes. The g
factor in the ac plane shows a φvariation as large as 2%
with characteristic cos2φangular dependence. On the other
hand, the gfactor in the ab plane remains essentially constant,
as expected from the tetragonal symmetry, within the error of
0.08%, which might have arisen from a slight misorientation
of the crystal. We obtain the gfactor along the principal
axes as ga=1.926(2) and gc=1.889(2). The average value
g=(2ga+gc)/3=1.913 (2) agrees with the one previously
obtained by powder ESR . These gvalues correspond to
the effective moments of 1.64 µBand 1.66µBfor spin-1/2for
effective moment values obtained from the Curie-Weiss ﬁt in
the previous section.
For a system with tetragonal symmetry with short distances
between the transition-metal and ligand ions, one would expect
c. However, we ﬁnd an opposite structure for the g
024 4 4 5 -3
L. YANG et al. PHYSICAL REVIEW B 96,024445 (2017)
FIG. 4. (a) Angular dependence of the gfactor at room temperature from the ESR measurements, where solid symbols are for the ﬁeld
orientation varied on the ac plane and open symbols are for the ﬁeld orientation varied on the ab plane. (b) Resonance ﬁeld B0(solid circles,
right axis) and linewidth $B0(open circles, left axis) of the ESR spectrum as a function of temperature. The inset plots normalized spin
susceptibility χs(T)/χs(3 00 K) as a function of temperature. (c) Temperature evolution of the spectrum for B∥cacross the transition (open
circles). The solid line is a sum of two contributions from intrinsic (dotted line) and defect (dashed line) susceptibilities. At 5 K, the data for
B⊥c(solid circles) are overlaid.
factor in MoOPO4, even though the orbital energy diagram for
the Mo5+ion is expected to be similar to that of tetragonally
compressed octahedron with a stabilized dxy orbital (see Fig. 7
below). As explained in Sec. III E,themultiorbitalcharacterof
the ground state in MoOPO4results in the observed gvalues.
Figure 4(b) shows the temperature dependence of the
resonance ﬁeld B0and the linewidth $B0of the ESR spectrum.
B0slowly decreases as temperature is lowered from 300 down
to 24 K, which may be attributed to a lattice contraction.
As temperature is further lowered below 24 K, B0starts
increasing sharply, which indicates that a magnetic transition is
approached. Similarly, $B0slowly decreases as temperature is
lowered down to 25 K but then starts broadening signiﬁcantly
as temperature is further lowered down to 15 K due to critical
spin ﬂuctuations. The inset of Fig. 4(b)plots the temperature
dependence of the local spin susceptibility, which is obtained
from the spectral area at each temperature normalized by
the one at 3 00 K, χs(T)/χs(3 00 K). The data could be ﬁt
to the Curie-Weiss formula with !CW =−8.9 K, which is in
reasonable agreement with the bulk susceptibility result shown
in Fig. 3(a).
Across the transition, the ESR line changes in shape and
intensity as shown in Fig. 4(c).Thelinesustainsaperfect
Lorentzian shape down to 16 K. On the other hand, the
line below 16 K close to the transition ﬁts better to a
sum of two Lorentzians: one corresponds to the intrinsic
sample susceptibility, while the other may correspond to
some defects. Indeed, the ESR signal at the paramagnetic
resonance ﬁeld position below 15 K corresponds to about
0.1% concentration of paramagnetic impurities. The response
below 15 K represents the summation of the possible defect
contribution and the intrinsic susceptibility. The tiny intrinsic
response below the transition temperature may represent
clusters of spins that continue to ﬂuctuate within the ESR time
window, which essentially disappears at lower temperatures
below 14 K. At 5 K, a broad hump of weak signal is observed
around 0.27 T for B∥cwhich is absent for B⊥c.Thissignal
may correspond to an antiferromagnetic resonance.
D. Neutron diffraction
To determine the microscopic magnetic structure, we
have performed neutron diffraction measurements. Magnetic
intensity appears at the position of the k=(100) wave vector at
change in scattering is found close to (001) between 5 and 25 K,
as shown in Fig. 5(b). A small shoulder of the (001) reﬂection
is likely to originate from a closely oriented secondary grain.
FIG. 5 . Neutron diffraction measurements of rotation scans
through (a) (100) and (b) (001) reﬂections recorded at 5 and 25 K.
(c) Temperature evolution of the Bragg peak integrated intensity. The
solid line is a power-law ﬁt I(T)∝(TN−T)2βwith the parameters
β=0.23 and TN=16.17 ±0.06 K. (d) The (010) peak counts as a
function of magnetic ﬁeld parallel to the caxis.
024 4 4 5 -4
J1-J2SQUARE LATTICE . . . PHYSICAL REVIEW B 96,024445 (2017)
FIG. 6. Schematic phase diagram of the spin-1/2J1-J2Heisen-
berg model on a square lattice with corresponding compounds
. Different ground states are expected depending on the J2/J1
ratio as represented in the diagram, where CAF, NAF, and FM
refer to columnar antiferromagnetic, Néel antiferromagnetic, and
ferromagnetic ground states, respectively. The present compound,
related by a spatial inversion being antiparallel. Due to the
dipolar nature of the magnetic interaction, only magnetization
perpendicular to the scattering wave vector gives a nonzero
structure factor. As no change is observed for the (001)
reﬂection upon cooling below TN,wecanconcludethatthe
moments are parallel to the caxis. To verify that this is
consistent with the symmetry of the lattice and rule out any
other magnetic structures, we utilize BASIREPS andoutline
the results here.
The magnetic representation is decomposed into six one-
dimensional irreducible representations &νwhose resulting
basis functions are shown in Table I. Examining the results
of the irreducible magnetic representations, we ﬁnd that only
&2is consistent with our observations. These results are in
contrast to the closely related AMoO(PO4)Cl (A=KandRb)
materials. Unlike the tilted arrangement of MoO6octahedra
FIG. 7. Single-particle energy-level diagram of dstates in an
octahedral arrangement of the oxygen ligands (in red) and Mo ion (in
yellow) in MoOPO4.
TABLE I. Basis functions of irreducible representation &νfor
k=(100) separated into real (Re) and imaginary (Im) components
and resolved along the crystallographic axes. The two equivalent Mo1
and Mo2ions are related by an inversion through the origin.
1 Re (0,0,1) (0,0,1)
2 Re (0,0,1) (0,0,¯
3 Re (1,0,0) (1,0,0)
3 Im (0,¯
4 Re (1,0,0) (¯
4 Im (0,¯
5 Re (1,0,0) (1,0,0)
5 Im (0,1,0) (0,1,0)
6 Re (1,0,0) (¯
6 Im (0,1,0) (0,¯
and PO4tetrahedra in MoOPO4,AMoO(PO4)Cl possesses a
higher symmetry where the octahedra and tetrahedra are ar-
ranged untilted in the abplane . Powder neutron diffraction
measurements on AMoO(PO4)Cl reveal an antiferromagnetic
structure where Mo moments are instead conﬁned to the ab
Figure 5(c) shows the temperature dependence of the
(100) Bragg peak integrated intensity. By ﬁtting a power-law
dependence to the intensity, we ﬁnd TN=16.17 ±0.06 K,
which is consistent with the magnetization and speciﬁc-
heat measurements. The order parameter exponent is found
to be β=0.23 , corresponding to the two-dimensional XY
universality class. However, dedicated measurements with
better resolution and separating critical scattering would be
needed before any conclusions could be drawn from this. In
Fig. 5(d) we show the magnetic Bragg peak intensity as a
function of applied ﬁeld along the caxis recorded at 2 K. Above
to saturate above 5 T. The change in the Bragg peak intensity
is consistent with a spin-ﬂop transition that is observed in
the magnetization measurements shown in Fig. 3(b). This
corresponds to a tilt of the moments by approximately 3 5 ◦
away from the caxis for the ﬁelds above 5 T.
E. Model calculations
In order to gain insight into the magnetic interactions,
we ﬁt the experimental susceptibility shown in Fig. 3(a)
using a high-temperature series expansion  assuming a
J1-J2spin-1/2 Heisenberg model on a square lattice. The
best ﬁt [solid line in Fig. 3(a)] returns J1=11.4(0.4)K
and J2=−5.2(1.0) K, corresponding to J2/J1=−0.46.This
ratio supports a collinear Néel order for the ground state (see
Fig. 6) in agreement with the neutron diffraction result. Using
the mean-ﬁeld expression for the Curie-Weiss temperature,
where ziis the number of neighbors for the corresponding
couplings (4 for both J1and J2in the present case), the high-
temperature expansion ﬁt yields !CW =−6.2K,whichagrees
with the value obtained from the simple Curie-Weiss ﬁt. Next,
024 4 4 5 -5
L. YANG et al. PHYSICAL REVIEW B 96,024445 (2017)
we simulate the phase diagram using a mean-ﬁeld calculation.
The results are presented by the colored background in
Fig. 3(c).Aslightexchangeanisotropy,$=0.02, has been
introduced in the Hamiltonian,
where ⟨i,j⟩and ⟨i,k⟩refer to the nearest and the next-nearest
neighbors in the abplane, to account for the spin-ﬂop transition
in a spin-1/2systemwheresingle-ionanisotropyisnot
expected to be present. We note that the mean-ﬁeld calculation
reproduces the temperature dependence of the spin-ﬂop ﬁeld.
From the mean-ﬁeld expression for the Néel temperature,
we obtain TN=16.6(1.4)K, whichis in excellent agreement
with the actual value from the experiments. In the above
analyses, we do not include Jcexplicitly: although an ar-
bitrarily small Jcis necessary in the actual system to give
rise to the (three-dimensional) long-range magnetic ordering,
including this parameter in the mean-ﬁeld calculation produces
an insigniﬁcant change in the phase diagram. In addition,
Jcconnects only two neighbors instead of the four of the
other couplings in the ab plane, and thus its effect should be
Our methods of analysis do not necessarily select the best
model, but rather test the validity and consistency of a proposed
one. For instance, ferromagnetic Jccomparable in strength
to antiferromagnetic J1, with negligible J2, may similarly
reproduce our experimental data. However, the ground-state
wave function from our ab initio quantum-chemistry calcula-
tions (see the next section) indicates zero contribution from
the out-of-plane orbitals and thus no direct virtual hopping
channels for Jcto be appreciable, in contrast to the other
coupling on the ab plane.
With strong ferromagnetic second-nearest-neighbor inter-
actions, MoOPO4populates a region of the J1-J2phase
diagram which has so far seen rather few investigations (see
Fig. 6). In the context of (π,0) zone-boundary anomalies on
the square lattice, linear spin-wave theory would for MoOPO4
predict a dispersion with signiﬁcantly higher energy at (π,0)
than at (π/2,π/2), opposite the case of weak antiferromagnetic
J2in Cu(pz)2(ClO4)2. Compared to the 3 9% reduction in
ordered moment due to quantum ﬂuctuations for the nearest-
neighbor Heisenberg model, the estimate for J2/J1=−0.46
is only a 24 % reduction of the ordered moment. Adding
the weak anisotropy for MoOPO4yields a 21% reduction in
ordered moment. Hence quantum ﬂuctuations are likely much
weaker in MoOPO4than in, e.g., Cu(DCOO)2·4D
CFTD for short [23 –26]orSr
2CuTeO6, and it would be
interesting in future investigations to examine whether this
leads to a similar suppression of the quantum dispersion and
continuum around (π,0).
TABLE II. Relative energies of d-level states of the Mo5+ion
obtained from CASSCF/NEVPT2 calculations. The corresponding
wave functions without (coefﬁcients) and with (weights) SOC at
the CASSCF level are also provided, where the up and down arrows
signify the Szvalues of +1
2,respectively. At the NEVPT2
level, the wave function would also contain contributions from the
inactive and virtual orbitals. For simplicity only the weights of the
SOC wave function are provided as the coefﬁcients are complex.
2gStates Relative Wave function (CASSCF)
without SOC energies (eV) coefﬁcients
|φ1⟩1.79 0.98 |yz⟩+0.21 |zx⟩
|φ2⟩1.79 0.21 |yz⟩−0.98 |zx⟩
|φ3⟩3.68 0.32|xy ⟩+0.95 |x2−y2⟩
|φ4⟩4.42 1.00 |z2⟩
2gStates Relative Wave function (CASSCF)
with SOC energy (eV) normalized weights (%)
|ψ0⟩086.0|φ0,↑⟩ +14 .0|φ0,↓⟩
|ψ1⟩1.75 5 0.0|φ1,↑⟩ +50.0|φ2,↓⟩
|ψ2⟩1.82 4 6.0|φ1,↑⟩ +46.0|φ2,↑⟩
|ψ3⟩3.70 88.0|φ3,↑⟩ +12.0|φ3,↓⟩
F. Ab initio calculations
An interesting feature in MoOPO4is that the axial position
of the Mo4+ion inside the MoO6octahedron is heavily shifted
such that the short and long distances to the apical oxygens
are 1.65 2 and 2.64 1 ˚
A, respectively. As a consequence, the
octahedral symmetry around the Mo ion is reduced, resulting
in the removal of octahedral orbital degeneracies and an
orbitally mixed electronic ground state. To elucidate the
electronic levels of the Mo5+ion in low-symmetry crystal
ﬁelds in MoOPO4, we performed ab initio quantum-chemistry
calculations using the cluster-in-embedding formalism .
6octahedron along with
surrounding nearest-neighbor (NN) PO4tetrahedra within the
plane and the out-of-plane MoO6octahedra embedded in an
array of point charges that reproduces the effect of the solid
environment  was considered for many-body calculations.
The NN polyhedra were included within the cluster region
to better describe the charge density within the active MoO6
region. Such calculations have provided excellent insights into
the interplay of crystal ﬁeld and SOC effects for several 4d
and 5 dtransition-metal compounds [30–33].
A perfect octahedral arrangement of the oxygen ligands
around the transition-metal ion splits the dlevels into high-
energy egand low-energy t2gmanifolds (see Fig. 7). In
t2gand eglevels of the Mo5+ion, resulting in an orbital singlet
ground state. In Table II the ground-state wave function and the
d-dexcitations of the Mo5+ion are summarized. These were
obtained from many-body multiconﬁgurational self-consistent
ﬁeld (MCSCF) andN-electron valence-state perturbation
theory (NEVPT2) calculationsfortheatomsintheactive
cluster region. All-electron Douglas-Kroll-Hess (DKH) basis
sets of triple-zeta quality wereusedtorepresentthe
024 4 4 5 -6
J1-J2SQUARE LATTICE . . . PHYSICAL REVIEW B 96,024445 (2017)
TABLE III. Computed gfactors of MoOPO4at the NEVPT2
level of theory. The ground-state multiconﬁguration wave function as
shown in Table II produces the correct structure for the gfactors.
active orbital space
Mo and oxygen ions in the central MoO6octahedron, and
for the Mo and P ions in the NN polyhedra we employed
effective core potentials [37,38]withvalencetriple-zeta
and a single basis function, respectively. The oxygen ions
corresponding to the NN MoO6and PO4polyhedra were
expanded in two sand one patomic natural orbital type 
functions. All the calculations were performed using the ORCA
quantum-chemistry package .
In the complete active space formalism of the MCSCF
(CASSCF) calculation, a self-consistent wave function was
constructed with an active space of one electron in ﬁve Mo
dorbitals. On top of the CASSCF wave function, NEVPT2
was applied to capture the dynamic electronic correlation.
Table II shows that the ground state is predominantly of dxy
character but has signiﬁcant contributions from the dx2−y2
orbital. The ﬁrst orbital excitations are nearly degenerate at
1.79 eV and are composed of dyz-anddzx -like orbitals. This
scenario is in contrast to the situation in other t2g-active class
of compounds with regular transition-metal oxygen octahedra
where the t2gmanifold remains degenerate with an effective
orbital angular momentum ˜
l=1. In the latter scenario the
spin-orbit interaction admixes all the t2gstates to give rise to
atotalangularmomentumJeff ground state [42,43]. Due to
the large noncubic crystal-ﬁeld splittings in the t2gmanifold in
the Mo5+ground state ψ0(see the with SOC results in Table II).
However, the orbital angular momentum is unquenched in
dzx and dyz, and hence the SOC results in the splitting of
the high-energy states ψ1and ψ2. Our calculations result in
excitation energies of 3.68 and 4 .42 eV into the egstates.
To understand the unusual structure of gfactors deduced
from the ESR experiments, we computed them from the
ab initio wave function as implemented in ORCA . In
Table III,thegfactors obtained from CASSCF calculations
with two different active orbital spaces, only t2gand t2g+eg,
are presented. With only t2gorbitals in the active space, we
cas expected for tetragonal symmetry with the
dxy -like orbital occupied in the ground state. By enlarging the
active space, the wave function now contains conﬁgurations
involving the egorbitals as well, and this is crucial to produce
the experimentally observed gfactors with ga>g
We have show n with a va r iety of exp e rimental a n d
computational techniques that MoOPO4realizes a spin-1/2
magnetic system of 4 d1electrons, with the quenched orbital
moment due to the large displacement of the Mo ions inside
the MoO6octahedra. The magnetic ground state supports a
Néel-type collinear staggered order on the square lattice with
the moments pointing normal to the plane, while the moments
align ferromagnetically along the stacking axis. The compound
likely realizes a spin-1/2HeisenbergmodelonaJ1-J2square
lattice, with an unfrustrated conﬁguration of antiferromagnetic
J1and ferromagnetic J2, while a small interlayer coupling
Jcwould lead to the observed magnetic ordering transition.
The spin-ﬂop transition suggests a small easy-axis anisotropy
in the dominant antiferromagnetic exchange, and the mean-
ﬁeld calculation reproduces the experimental magnetic phase
diagram. The small anisotropy in the gfactor observed in ESR,
which is reproduced by the quantum-chemistry calculations,
indicates that the ground state involves the higher-energy eg
orbitals in addition to the t2gorbitals. Our results suggest that
4dmolybdates provide an alternative playground to search for
model quantum magnets other than 3 dcompounds.
We tha n k R. Scopell i ti and O. Zaha r ko for thei r h elp with
x-ray and neutron diffraction, respectively. We also thank
V. Fav re a n d P. H u a n g f o r th e i r h e l p wi t h t h e s p ec i ﬁ c - h ea t
analysis. V.M.K. is grateful to H. Stoll for discussions on
effective core potentials. This work was supported by the
Swiss National Science Foundation, the MPBH network,
and European Research Council grants CONQUEST and
TopoMat (No. 3 065 04 ). M.J. is grateful for support from the
European Commission through the Marie Skłodowska-Curie
Action COFUND (EPFL Fellows). M.K. is supported by a
Grant-in-Aid for Scientiﬁc Research (C) (JSPS, KAKENHI
No. 15 K05 14 0). The ab initio calculations were performed
at the Swiss National Supercomputing Centre (CSCS) under
project s675 .
 D. Khomskii, Transition Metal Compounds (Cambridge Uni-
versity Press, Cambridge, 2014 ).
 W. Witczak-Krempa, G. Chen, Y. B. Kim, and L. Balents, Annu.
Rev. Condens. Matter Phys. 5,57(2014 ).
[3 ] S. Nakatsuji, S.-i. Ikeda, and Y. Maeno, J. Phys. Soc. Jpn. 66,
[4 ] G. Cao, S. McCall, M. Shepard, J. E. Crow, and R. P. Guertin,
Phys. Rev. B 56,R2916(R) (1997).
[5 ] S. Nakatsuji and Y. Maeno, Phys. Rev. Lett. 84,2666 (2000).
 C. G. Fatuzzo, M. Dantz, S. Fatale, P. Olalde-Velasco, N. E.
Shaik, B. D. Piazza, S. Toth, J. Pelliciari, R. Fittipaldi, A.
Vecchione, N. Kikugawa, J. S. Brooks, H. M. Rønnow, M.
Grioni, Ch. Rüegg, T. Schmitt, and J. Chang, Phys. Rev. B
91 ,15 5 104 (2015 ).
 Y. Maeno, H. Hashimoto, K. Yoshida, S. Nishizaki, T. Fujita, J.
Bednorz, and F. Lichtenberg, Nature (London) 372 ,532(1994 ).
 K. Ishida, H. Mukuda, Y. Kitaoka, K. Asayama, Z. Mao, Y.
Mori, and Y. Maeno, Nature (London) 396,65 8 (1998).
 M. A. de Vries, A. C. Mclaughlin, and J.-W. G. Bos, Phys. Rev.
Lett. 104,177202 (2010).
 T. Aharen, J. E. Greedan, C. A. Bridges, A. A. Aczel, J.
Rodriguez, G. MacDougall, G. M. Luke, T. Imai, V. K.
024 4 4 5 -7
L. YANG et al. PHYSICAL REVIEW B 96,024445 (2017)
Michaelis, S. Kroeker, H. Zhou, C. R. Wiebe, and L. M. D.
Cranswick, Phys. Rev. B 81 ,224 4 09 (2010).
 L. Clark, G. J. Nilsen, E. Kermarrec, G. Ehlers, K. S. Knight, A.
Harrison, J. P. Attﬁeld, and B. D. Gaulin, Phys. Rev. Lett. 113,
117201 (2014 ).
 H. Ishikawa, N. Nakamura, M. Yoshida, M. Takigawa, P.
Babkevich, N. Qureshi, H. M. Rønnow, T. Yajima, and Z. Hiroi,
Phys. Rev. B 95,064 4 08 (2017).
[13 ] J. S. Gardner, M. J. P. Gingras, and J. E. Greedan, Rev. Mod.
Phys. 82 ,53 (2010).
[14 ] P. Kierkegaard and J. M. Longo, Acta Chem. Scand. 24,427
[15 ] E. Canadell, J. Provost, A. Guesdon, M. Borel, and A. Leclaire,
Chem. Mater. 9,68 (1997).
 L. Lezama, K. Suh, G. Villeneuve, and T. Rojo, Solid State
 L. Lezama, G. Villeneuve, M. Marcos, J. Pizarro, P. Ha-
genmuller, and T. Rojo, Solid State Commun. 70,899
 A. Abragam and B. Bleaney, Electron Paramagnetic Resonance
of Transition Ions (Clarendon, Oxford, 1970).
 J. Rodríguez-Carvajal, Phys. B (Amsterdam, Neth.) 192,55
 M. Borel, A. Leclaire, J. Chardon, J. Provost, and B. Raveau,
J. Solid State Chem. 137,214 (1998).
 H.-J. Schmidt, A. Lohmann, and J. Richter, Phys. Rev. B 84,
104 4 4 3 (2011).
 N. Tsyrulin, F. Xiao, A. Schneidewind, P. Link, H. M. Rønnow,
J. Gavilano, C. P. Landee, M. M. Turnbull, and M. Kenzelmann,
Phys. Rev. B 81 ,13 4 4 09 (2010).
[23 ] H. M. Rønnow, D. F. McMorrow, and A. Harrison, Phys. Rev.
Lett. 82 ,3152 (1999).
[24 ] H. M. Rønnow, D. F. McMorrow, R. Coldea, A. Harrison, I. D.
Youngson, T. G. Perring, G. Aeppli, O. Syljuåsen, K. Lefmann,
and C. Rischel, Phys. Rev. Lett. 87,03 7202 (2001).
[25 ] N. B. Christensen, H. M. Rønnow, D. F. McMorrow, A. Harrison,
T. Perring, M. Enderle, R. Coldea, L. Regnault, and G. Aeppli,
Proc. Natl. Acad. Sci. USA 104,15 264 (2007).
 B. Dalla Piazza, M. Mourigal, N. B. Christensen, G. Nilsen, P.
Tregenna-Piggott, T. Perring, M. Enderle, D. F. McMorrow, D.
Ivanov, and H. M. Rønnow, Nat. Phys. 11,62 (2015 ).
 P. Babkevich, V. M. Katukuri, B. Fåk, S. Rols, T. Fennell, D.
c, H. Tanaka, T. Pardini, R. R. P. Singh, A. Mitrushchenkov,
O. V. Yazyev, and H. M. Rønnow, Phys. Rev. Lett. 117,23 7203
 L. Hozoi and P. Fulde, in Computational Methods for Large
Systems, edited by J. R. Reimers (Wiley, Hoboken, NJ, 2011),
pp. 201–224 .
 M. Klintenberg, S. Derenzo, and M. Weber, Comput. Phys.
Commun. 131,120 (2000).
[3 0] V. M. Katukuri, K. Roszeitis, V. Yushankhai, A.
Mitrushchenkov, H. Stoll, M. van Veenendaal, P. Fulde,
J. van den Brink, and L. Hozoi, Inorg. Chem. 53,4833 (2014 ).
[3 1] V. M. Katukuri, H. Stoll, J. van den Brink, and L. Hozoi, Phys.
Rev. B 85,2204 02(R) (2012).
[3 2] N. A. Bogdanov, V. M. Katukuri, H. Stoll, J. van den Brink, and
L. Hozoi, Phys. Rev. B 85,23 5 14 7 (2012).
[3 3 ] X. Liu, V. M. Katukuri, L. Hozoi, W.-G. Yin, M. P. M. Dean,
M. H. Upton, J. Kim, D. Casa, A. Said, T. Gog, T. F. Qi, G. Cao,
A. M. Tsvelik, J. van den Brink, and J. P. Hill, Phys. Rev. Lett.
,15 74 01 (2012).
[3 4 ] T. Helgaker, P. Jørgensen, and J. Olsen, Molecular Electronic-
Structure Theory (Wiley, Chichester, UK, 2000).
[3 5 ] C. Angeli, R. Cimiraglia, and J.-P. Malrieu, Chem. Phys. Lett.
[3 6] D. A. Pantazis, X.-Y. Chen, C. R. Landis, and F. Neese, J. Chem.
Theory Comput. 4,908 (2008).
[3 7] K. A. Peterson, D. Figgen, M. Dolg, and H. Stoll, J. Chem. Phys.
126,124 101 (2007).
[3 8] G. Igel-Mann, H. Stoll, and H. Preuss, Mol. Phys. 65,13 21
[3 9] K. Pierloot, B. Dumez, P.-O. Widmark, and B. Roos, Theor.
Chim. Acta 90,87 (1995 ).
[4 0] F. Neese, WIREs Comput. Mol. Sci. 2,73 (2012).
[4 1] The spin-orbit coupling results in an admixture of the t2gstates
with different Sz; see Ref. .
[4 2] B. J. Kim, H. Ohsumi, T. Komesu, S. Sakai, T. Morita, H. Takagi,
and T. Arima, Science 32 3,13 29 (2009).
[4 3 ] B. J. Kim, H. Jin, S. J. Moon, J.-Y. Kim, B.-G. Park, C. S. Leem,
J. Yu, T. W. Noh, C. Kim, S.-J. Oh, J.-H. Park, V. Durairaj, G.
Cao, and E. Rotenberg, Phys. Rev. Lett. 101,0764 02 (2008).
[4 4 ] F. Neese, Mol. Phys. 105,25 07 (2007).
024 4 4 5 -8