Charge and Orbital Orderings, and Frustration in Quasi-one-dimensional Ferrimagnetic Insulator $β$-V$_2$O(PO$_4$)
Seo-Jin Kim K. -W. Lee




mckwan@korea.ac.kr

 ^1Department of Applied Physics, Graduate School, Korea University, Sejong 30019, Korea

 ^2Division of Display and Semiconductor Physics, Korea University, Sejong 30019, Korea




Using ab initio calculations based on the correlated band theory,
we have investigated 

the quasi-one-dimensional chain system β-V_2O(PO_4), 
showing both charge and spin orderings. 

Even in the uncorrelated region,
the pure transition from the tetragonal to the monoclinic structure
leads to a sizable charge difference between the two types of V ions,
regardless of magnetic orders.
In the ferrimagnetic phase, 
inclusion of the on-site Coulomb repulsion U leads to a full orbital-polarization 
of V1 (t_2g^3↑, S=3/2) and V2 (a_1g^1↓e_g^'1↓, S=1) 
above U^c_eff≈3.5 eV, leading to local spin moments of 2.30 and –1.54 μ_B, respectively,
with small orbital moments of several hundredth μ_B.
So, the net moment is nearly 1 μ_B per formula unit, which is about 2–3 times larger than the experimental value.

Our results show significant variations, strongly depending on the strength of U_eff, 
in energy differences between various magnetic states as well as a small magnetic anisotropy.
These results suggest that the substantial difference between the calculated and experimental moments 
is attributed to quantum fluctuation of the pyrochlore-like weakly linked V_4 tetrahedral structure.
Our findings are expected to provide a good platform to investigate the interplay 
among the charge-, spin-, and lattice-degrees of freedom, and geometrical frustration.

Charge and Orbital Orderings, and Frustration in Quasi-one-dimensional Ferrimagnetic Insulator 
 β-V_2O(PO_4) 

    Kwan-Woo Lee^1,2
    December 30, 2023
===============================================================================================================






§ INTRODUCTION

In condensed matters, the interplay among the charge-, spin-, orbital-, and lattice-degrees of freedom
leads to abundant interesting phenomena. 
Over the past several decades, low-dimensional spin systems have been intensively investigated
due to their exotic phenomena such as spin-Peierls (or dimers),<cit.>
spinon confinement,<cit.> and even high T_c superconductivity.<cit.>



In particular, quasi-one-dimensional (1D) chains with (weakly) frustrated structures 
have been attracted significant interest 
owing to various characteristic quantum magnetic states driven by the competition between the low dimensionality 
and frustration.<cit.>
Most studies have been carried out on 1D  chains with weakly linked triangular lattices,
e.g., Ca_3Co_2O_6,<cit.> β-TeVO_4,<cit.> 
and Sr_2Rh_4O_12-family.<cit.>
Another example is the trans-1,4-cyclohexanedicarboxylate of alternating Fe^2+ and Fe^3+ 
with a 1D chain and spin ladder structure.<cit.>

Many vanadium oxides exhibit metal-insulator transition (MIT), 
charge ordering (CO), and even orbital ordering, but their mechanisms have been debated
over the past several decades.
For example, the binary and perovskite-type ternary vanadium oxides 
such as VO_2 (with V^4+ d^1) and V_2O_3 (with V^3+ d^2)<cit.> 
have been intensively investigated
owing to their peculiar MITs and possible applications to information technology and, 
electronic and photonic devices.<cit.>
In the quasi-1D paramagnetic VO_2, MIT is induced by the dimeralization of V ions.<cit.>
On the other hand, for the antiferromagnetic monoclinic V_2O_3
it is not clear whether MIT accompanies orbital-ordering.<cit.>
In addition, V_4O_7 consisting of two independent cation chains
exhibits an unconventional MIT involved in dimeralized spin-singlet V^4+ ions and magnetically ordered V^3+.<cit.>

In this paper, we provide a promising candidate 
of a distinct low-dimensional and frustrated system showing
a quasi-1D ferrimagnetic Heisenberg spin chain structure
with a weak pyrochlore-like tetrahedral link of vanadium ions in the β-  vanadium phosphate.
In the formal charge concept of (PO_4)^3- and O^2-, 
V_2O(PO_4) is a mixed valent system with two V^2.5+ ions in average.  
As will be addressed below, 
this system shows charge and spin orderings intertwined with the lattice degree of freedom.
Therefore, below the spin ordering temperature, there are chains of alternating 
V^2+ (d^3, S=3/2) and V^3+ (d^2, S=1) ions.
The chain structure exists more rarely than in a V^4+ system.

  was initially reported to be tetragonal at room temperature.<cit.>
Based on the assumption of a ferromagnetic (FM) state in the tetragonal structure,
through theoretical investigations,
Jin et al. proposed that this system is a magnetic Weyl semimetal.<cit.>
However, recently,  Pachoud et al.<cit.> revealed
a tetragonal-to-monoclinic structure transition at T_s≈600 K,
leading to two distinct V sites.
At T_s, a transition of negative to positive thermal expansion simultaneously occurs.
Through a resistivity measurement demonstrating the semiconducting behaviors in both phases 
and analysis of the temperature-dependent bond valence sums,
they concluded that CO occurred in the 1D V-V chain at T_s(=T_CO).
This suggested that the very rare negative thermal expansion in the high-temperature (tetragonal) phase
was driven by loss of CO.

In the low-temperature (monoclinic) phase, a ferrimagnetic (FI) spin ordering is observed 
at the Curie temperature of T_C=164 K.
Isothermal magnetization measurements indicated the FI order between nearest neighbor (NN) V ions,
which was confirmed by a neutron powder diffraction measurement.
Xing et al.<cit.> confirmed most of these results with a single crystalline sample,
but observed a lower T_C=128 K than that of the polycrystalline sample.
The magnetic susceptibility above T_C provides a high Curie-Weiss temperature of Θ_CW=-900 K
and effective moment of 3.7 μ_B per V ion.
The effective moment is closer to the value of the V^2+ ion than to that of the V^2.5+ ion,
implying a complex magnetic interaction.
The total moment, incompletely saturated even at 6 T, is 0.27 – 0.31 μ_B/V 
with a remnant moment of 0.22 – 0.25 μ_B,
comparable to the value of 0.6 μ_B/V obtained by Pachoud et al.<cit.> 
The magnitudes of the V local moments are approximately 1.2 – 1.5 μ_B for V^3+ and 1.4 – 1.8 μ_B for V^2+,
which are substantially reduced from the nominal values of S=1 and 3/2, respectively.
From the resistivity data,
the energy gap of 0.48 eV could be estimated.<cit.> 


As mentioned above, this vanadium phosphate shows various peculiar properties.
In addition to the negative thermal expansion in the high T phase, which will not be discussed here,
this system exhibits several interesting properties in the low T phase: (i) 1D mixed spin chain 
with a promising frustrated structure, (ii) CO FI insulator, and
(iii) magnetic moments substantially reduced from the nominal values. 

Through ab initio calculations, including correlation effects,
we have investigated the electronic and magnetic structures of the low T monoclinic phase,
which has not been previously reported in detail.








§ CRYSTAL STRUCTURE AND CALCULATION METHODS

Our calculations were carried out for the monoclinic V_2O(PO_4) containing two formula units, 
displayed in Fig. <ref>(a),
with the experimental lattice parameters of a=7.5552 Å, b=7.5979 Å, c=7.2110 Å,
and the angle β=121.2^∘.<cit.>
In the structure (space group: No. 15, C2/c), V1 and V2 atoms occupy at 4c (1/4,1/4,0) and 
4b (0,1/2,0) sites, respectively.
O1 and O2 lie at 4f (x,y,z) sites. O3 and P atoms reside at 4e (0,y,1/4) sites.
O1 and O2 are shared by VO_6 octahedra and PO_4 tetrahedra, 
whereas O3 atoms are vertically connected only to the VO_6 octahedra.
We used the internal parameters experimentally obtained at 10 K.<cit.>
As summarized in Table <ref>,
the V1-O_6 octahedron has an approximately 3% (in average) longer V-O bond length  
than that in the V2-O_6 octahedron.
The octahedron of V2 is more irregular than that of V1,
indicating a substantial trigonal distortion in the V2-O_6 octahedra.
These lead to the charge and orbital orderings, as will be shown below.

Basically, the crystal consists of substantially distorted VO_6 octahedra 
and nearly regular PO_4 tetrahedra, as shown in Fig. <ref>(a).
The PO_4 tetrahedra, at layers of b≈1/4 and 3/4, are corner- and edge-shared 
with four oxygen ions of the octahedra.
At the b=1/2 layer, the V-V chains are stretched in the direction perpendicular to the V-V chains
at the b=0 layer.
Here, the chain is on the ac-plane.
The face-sharing VO_6 octahedra form a quasi-1D chain 
with a notably short intrachain V-V distance of 2.68 Å.
The interchain V-V ionic distances are 5.35 Å (in-plane) and 3.60–3.63 Å (interlayer). 
Remarkably, due to the relatively short interlayer interchain V-V distance,
the network of V ions leads to a pyrochlore-like tetrahedral link,
as displayed in Fig. <ref>(b).




The calculations were based on the generalized gradient approximation (GGA)
with the Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional,<cit.> 
implemented in the accurate all-electron full-potential wien2k code.<cit.>
The correlation effects are treated by the GGA+U approach.<cit.>
The strength of the effective on-site Coulomb repulsion U_eff=U-J is varied 
in the range of 2 – 5 eV,<cit.>
where J is the Hund's rule coupling.
We also obtained results with  a separate U and J scheme, but both results were very close to each other.
So, only the results obtained from the U_eff scheme are presented.
In wien2k, the basis size was determined by R_mtK_max=7
and augmented-plane-wave sphere radii (1.96 a.u. for V, 1.28 a.u. for O, and 1.61 a.u. for P).
The Brillouin zone was sampled with a k-mesh of 21× 17× 21.









§ UNDERLYING ELECTRONIC STRUCTURE

Before considering the complications of spin order and correlation effects,
we consider the nonmagnetic state within the GGA level to understand 
the basic underlying features of the electronic structure.

Figure <ref> shows the nonmagnetic total and atom-resolved densities of states (DOSs),
indicating a (V_2)^5+ configuration, as expected from the formal charge.
The V d orbitals in the range of –2 to 3.2 eV 
are separated with a large hybridization gap of 2.2 eV from the O p orbitals, 
which are spread in the range of –8 to –4.2 eV.

The VO_6 octahedral structure leads 
to a crystal field splitting of triplet t_2g and doublet e_g manifolds.
The crystal field splitting measured from the center of each manifold 
is about 2.5 eV.
In this system, the triplet t_2g manifold splits into the doublet e_g^' (or e_g^π) 
and singlet a_1g orbitals due to the sizable trigonal distortion in the octahedra.
Setting the z-axis along the V-V chain, 
the a_1g orbital has a d_z^2-like shape,
thus leading to a strong V-V direct interaction along the chain.
The e_g^' orbitals can be represented by<cit.> 

    e_g,1^'   =   1/√(3)(d_zx-√(2)d_x^2-y^2),
    
    e_g,2^'   =   1/√(3)(d_yz+√(2)d_xy).

A small monoclinic distortion, as that in this system, leads to a small splitting of the e_g^' orbitals.





In the V d-orbital region, the blowup band structure with the a_1g fatband 
is given in the left panel of Fig. <ref>.
Interestingly, the partially filled e_g^' orbital is relatively well localized with a width of 0.7 eV,
leading to a sharp peak at 0.1 eV in the DOS.
The bonding and antibonding a_1g orbitals below and above the e_g^' orbital
have  an approximately twice larger bandwidth than that of the e_g^' orbital 
due to the strong ddσ interaction.
This is illustrated in the orbital-projected DOSs of the singlet a_1g, doublet e_g^', 
and doublet e_g manifolds, in the right panel of Fig. <ref>.
 
As given in the Inset of Fig. <ref>, the atom-projected DOSs of V1 and V2 already show
some distinction, leading to a sizable charge difference of 0.08e between them 
obtained from the Bader charge decomposition.
This value is almost independent of the magnetic states studied here.
This is consistent with the experimental observations,<cit.> 
in which the pure tetragonal-to-monoclinic structure transition leads to the CO.






§ FERRIMAGNETIC STATE

We have considered a few possible spin-ordered states including two different FI orders.
The FI1 state has the antiferrimagnetic intrachain and ferromagnetic interchain configuration,
whereas the FI2 state involves both intra- and inter-chain antiferrimagnetic interactions.
So in the FI2 state, the spins of the NN same-type V ions in the interlayer chain 
along the b̂ direction, which is vertical to the ferrimagnetic V1-V2 chain, are also antialigned.
The magnitudes of the V local moments in FI1 and FI2 are almost identical,
but the net moment of FI2 is precisely compensated.  
In the GGA, as shown in Fig. <ref>(a),
FI1 is energetically favored over the nonmagnetic (by 1.5 eV/f.u.),  FM (by 0.1 eV/f.u),
and FI2 (by 12 meV/f.u.) states, consistent with the experimental suggestions.<cit.>
Such a large magnetization energy is induced by the large spin moments of the V ions of 1.5–2.3 μ_B.
For comparison, the energy gain due to a simple Stoner instability IM^2/4 leads to a similar value,
where the Stoner parameter I≲1eV can be roughly estimated 
by the exchange splitting Δ_ex≈ IM≈2.3 eV (see below). 
Remarkably, the energetic preference among the spin-ordered states substantially depends on strength of U_eff,
suggesting a delicate magnetic character.
This issue will be further considered in the Discussion section.
In this section, we will focus on the FI1 state, which has been suggested from the experiments.<cit.>







 §.§ Uncorrelated regime: GGA level

In the FI1 state, the V local moments are 1.94 (V1) and –1.71 (V2) μ_B, 
leading to a net moment of 0.64 μ_B/f.u.
The net moment is close to the value obtained in the polycrystalline sample,<cit.>
but about twice that of the single crystalline sample.<cit.>
Considering the interstitial moment, these values are consistent with that of V^2.5+,
which is reflected in the DOS, presented in Fig. <ref>(a).

Figure <ref>(a) shows the orbital-projected DOSs of the V1 and V2 ions in the FI1 state.
E_F is just above the peaks at –50 meV (–100 meV) for the up (down) channel.
The exchange splitting of the t_2g manifold is about 2.3 eV in both V ions,
while the hybridization gap between the O p orbital and this manifold is 2.5 eV,
yielding the well-isolated t_2g manifolds with a bandwidth of 2.5 eV around E_F.
The partially filled t_2g manifolds in the up and down channels

are the V1 and V2 character, respectively.
In each majority t_2g manifold, 
the centers of the completely filled a_1g orbitals are approximately 1 eV lower than
those of the e_g^' orbitals.
Besides, the V2 e_g^' orbital in the down channel has 
a center higher by 50 meV and significantly longer unfilled tails than those of V1, 
which is in the up channel and is almost filled.
This feature leads to a sizable charge difference of 0.08e between the V1 and V2 ions.
Additionally, upon the application of U to the V ions, 
the almost filled V1 e_g^' manifold is simply pushed down,
and a Mott transition occurs in the effectively half-filled e_g^' manifold of V2 in the down channel (see below).  
 







 §.§ Correlated regime: fully charge-ordered state

The effects of correlations were considered with the GGA+U approach 
to obtain the observed insulating phase.
Figure <ref>(b) shows a plot of U_eff vs. energy gap E_g.
A gap opens at U_eff^c≈3.5 (4.0) eV for the FI1 (FM) state,
similar to that of VO_2.<cit.>
Interestingly, with the increase in U_eff in the FM state, 
a transition from metal to half-semimetal with linearly crossing bands near E_F (not shown here)
occurs at U_eff≈1 eV, and then the energy gap is completely open at U_eff≈4 eV.
Therefore, in the range of U_eff≈1-4 eV, a magnetic Weyl semimetallic phase appears,
as suggested for the presumed tetragonal FM .<cit.> 
On the other hand, in the FI2 state, U_eff≈1.5 eV is sufficient to open a gap,
similar to the value of the antiferromagnetic V_2O_3.<cit.>





Further, we focus on the FI1 insulating state at U_eff=4 eV, where a gap is clearly visible.
The band structure and total DOS in the region of the V d-orbitals are given in Fig. <ref>.
The corresponding orbital-projected DOSs are shown in Fig. <ref>(b).
In the down channel, the V2 a_1g and one of the V2 e_g^' orbitals around –2 eV are completely filled,
whereas the V1 t_2g manifold is fully filled in the up channel.
The upper Hubbard band, i.e., the other unfilled e_g^' orbital, below 1 eV,
has the e_g,2^' character, as shown in the Inset of Fig. <ref>.
Thus, the application of U leads to an orbital ordering of a_1g^1↓e_g^'1↓ 
in V2 (3d^2), in contrast to the e_g^'2 configuration in the antiferromagnetic V_2O_3.
This leads to a spin-asymmetric gap of 0.25 eV between the spin-up V1 t_2g and -down V2 e_g^' bands.

In the state, the local moments are 2.30 μ_B (V1) and –1.54 μ_B (V2),
consistent with S=3/2 and 1, respectively.
Considering the interstitial moment and small contributions from the other ions, 
the total spin moment is 1 μ_B/f.u.,
which is about three times larger than the experimentally observed value in the single crystalline sample.<cit.>
This issue is further discussed below.




The CO can be analyzed with a few different approaches.<cit.>
Our calculated Bader charges Q_B are Q_B(V1)=+1.58, Q_B(V2)=+1.94, Q_B(PO_4)=–2.23,
and Q_B(O3)=–1.30, 
leading to a large charge difference of 0.36e between the V ions.
Compared with the formal values of +2, +3, –3, and –2, respectively, in the CO state,
this charge difference seems to be significantly smaller. 
However, this value is significantly larger than the value of 0.2e 
observed in the charge disproportionated Na_1/2Co_2
with Co^3+ and Co^4+ ions.<cit.>
The summation of the occupation matrix gives 2.64 (2.38) for V1 (V2),
indicating the charge difference of 0.26e between the two V ions,
consistent with the Bader charges.
Another approach is to analyze the radial charge densities 4π r^2ρ(r), 
as suggested by Quan, Pardo, and Pickett,<cit.> 
which have a clear difference in their tails in a CO state.
Figure <ref>(a) shows the densities of the two V ions, obtained from wien2k.
Below approximately 0.5 a.u., the charge densities insignificantly differ,
while a difference is observed above this value.
This distinction is more clearly observed in the spin-resolved charge densities in Fig. <ref>(b),
particularly in the majority densities.






Figure <ref> shows the charge density isosurface in GGA+U at U_eff=4 eV.
The V1 ion has the shape of the equally occupied t_2g orbital, 
i.e., t_2g^3↑.
On the other hand, the shape of the V2 ion indicates  a combination of the a_1g and e_g,1^' orbitals.
So, the lower Hubbard band at –2 eV in the spin down is the e_g,1^' orbital.
Consequently, the V2 ions show the a_1g^1↓e_g,1^'1↓ orbital order.

However, this order leads to only a small orbital moment, 
since the occupation matrix of V2 is nearly symmetric.
Our GGA+U+spin-orbit coupling (SOC) calculations show that the orbital moment is about 0.04 μ_B in each V ion,
opposite to each spin moment.
This indicates that this order cannot be attributed to the significantly reduced net moment 
in the experiments.<cit.>
It is worth noting that the O3 ion has a small moment, but exhibits a clear p_π character. 
This implies a considerable pdπ interaction,
which would lead to the next nearest neighbor (NNN) superexchange pattern 
with an angle of 130^∘ along the b̂ direction. 





§ DISCUSSION ON QUANTUM FLUCTUATION

As mentioned above,
the experimentally observed moments in the system are approximately 1/3 of our calculated values.
These discrepancies are significant,
although the theoretically calculated moment should not be expected to be identical 
to the experimentally observed value, considering the theoretical specifications
(and some spin in the interstitial region) in the theoretical approach. 
Below, we discuss a possible mechanism of such a large reduction.




 §.§ Experimental indications

There are several experimental indications of complex unusual magnetic properties
of this system.<cit.>
The magnetization exhibits a small magnetic anisotropy,
and the magnetic entropy is considerably smaller than those of V^2+ and V^3+.
The observed Curie-Weiss and ordering temperatures lead to 
a sizable frustration parameter of |θ_CW|/T_C≈ 7.03,
comparable with the typical values of frustrated systems.<cit.>
As mentioned above, the V ions form a pyrochlore-like link consisting of
the V_4 tetrahedra with two sides of 2.67 Å  and four sides of ∼3.62 Å.
The long sides are longer by 1/3 than the short sides, but the length is relatively short,
compared with those of the existing candidates of 1D chain and frustrated structure.
For example, the NNN distance is 4.40 – 4.92 Å  
in β-TeVO_4 of the NN 3.64 Å,<cit.>.
In Ca_3Co_2O_6 and Sr_2Rh_4O_12 
the NNN distances are twice longer than the NN distances of ∼2.6 Å.<cit.>
Additionally, this system shows a small coercivity of the order of mT,<cit.> 
which implies an easily variable magnetization direction by an external magnetic field or magnetic impurity.
It may be expected that the complex  behavior originates from strong quantum fluctuations or frustration.




 §.§ Evaluation of the magnetic ground state

We considered three magnetic states, FM, FI1, and FI2.
Figure <ref>(a) shows the evaluation of differences in energy between these three states,
as varying the strength of U_eff.
Remarkably, with the increase in U_eff,
the difference in energy between FM and FI1 is reduced and becomes negative around U_eff≈2.6 eV,
indicating that FM is favored over FI1 in this regime.
Above this value, the energy difference increases again, leading to an energetically favored FI1 state.
Comparing FI1 with FI2, the FI2 state is energetically favored over the FI1 state 
in the range of U_eff=1.5 to 4.7 eV.
Even at U_eff=5 eV, the energy difference between FI1 and FI2 is just a few meV.
This indicates that the NNN superexchange interaction is also crucial in this system. 
Since this behavior is observed in the reasonable range of U_eff for this vanadium oxide,
our results suggest that frustration would be a reliable scenario in this system.


Additionally, as expected from the tiny orbital moment,
this system shows a little magnetic anisotropy, obtained from GGA+SOC calculations.
We considered two quantized directions parallel and perpendicular to the chain.
The anisotropy energy is less than a few tenths meV/f.u., 
which indicates that the application of a small magnetic field
can vary the spin direction.
This also supports the proposed scenario.






 §.§ Fixed spin moment studies

In order to affirm the delicate magnetic behavior, fixed spin moment (FSM) calculations were performed,<cit.>
as varying the strength of U_eff.
In these calculations, two states, FI1 and FM, were considered by choosing a proper initial condition.
Figure <ref> shows the energy difference Δ E(M) plotted against the total moment M, 
with respect to the energy at M=0 for each U_eff, 
where the net moment is completely canceled by the antialigned V1 and V2.
These results are consistent with our self-consistent results, given in Fig. <ref>(a).

Interestingly, two minima appear, regardless of the value of U_eff.
One of them is in the range of 0.5 – 1 μ_B for the FI1 state, 
while the other is in the range of 4.5 – 5 μ_B for the FM state.
In the GGA (U_eff=0),
a large range of magnetic field Δ B=Δ E/Δ M 
is required for the magnetic phase transition from FI1 to FM.
However, as increasing U_eff,
the energy difference Δ E between the two states decreases 
and becomes close to zero around U_eff=2.5 eV.
So, only a small magnetic field is required for the phase transition around this value of U_eff.
Above this value, Δ E monotonously increases reaching 37 meV at U_eff=4 eV, 
where both states are insulating.

Thus, these results also suggest that this system is close to a quantum fluctuation.

Note that the Δ E(M) curves are discontinuous and  
both FI1 and FM states appear coincidentally in the range of M=2.5 – 3 μ_B in the GGA+U cases,
leading to hysteresis. 
This has been observed in the density functional theory plus U approach, which often leads to multisolutions
in a critical region.<cit.>











§ SUMMARY

Through ab initio calculations including correlation effects, 
we have investigated the insulating β-  with a quasi-1D ferrimagnetic Heisenberg spin chain structure,
exhibiting the charge and spin orderings
at T_CO≈600 K and T_SO≈125 K, respectively.
Remarkably,this system has a weak pyrochlore-like tetrahedral link of V ions 
due to the relatively short interlayer (NNN) distance along the vertical direction.

Consistent with the experimental observation,<cit.>
our calculations indicate that the charge order of alternating V^3+ and V^2+ along the chain
occurs coincidentally with the structure transition from tetragonal to monoclinic.
In the experimentally suggested FI1 state, 
where the spins of the NN V ions are antialigned along the chain,
inclusion of correlation effects leads to a full orbital polarization of 
V1 (t_2g^3↑, S=3/2) 
and V2 (a_1g^1↓e_g^'1↓, S=1),
above the critical value of U_eff^c=3.5 eV for the metal-insulator transition.
This yields the net moment of about 1 μ_B (including the small orbital moments) 
which is 2–3 times larger than the experimentally observed value.
Besides, the energy differences between the few spin-ordered states show 
the strong U-dependence, which is also supported by the FSM calculations.
This implies that a magnetic phase transition is feasible and 
that the NNN superexchange interaction through the pdπ hybridization is sizable.
In addition to the several experimental indications mentioned above,
the significant variations in energy differences, and our calculated tiny orbital moments 
of several hundredth μ_B,
the substantially quenched moment suggests that quantum fluctuation (frustration)
occurs in the pyrochlore-like tetrahedral structure.
 
Our results suggest that this system is a promising candidate to show the vital interplay 
among the charge-, spin-, and lattice-degrees of freedom, and geometrical frustration. 
Further experimental and theoretical researches are required  
to elucidate the complex magnetic properties.
 




§ ACKNOWLEDGMENTS

We acknowledge K.-H. Ahn and Y.-J. Song for useful discussions on calculation methods.
This research was supported by NRF of Korea Grant No. NRF-2016R1A2B4009579. 



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