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Journal of the Physical Society of Japan FULL PAPERS

Impact of Dzyaloshinsky-Moriya Interactions and Tilts of the g Tensors on the

Magnetization Process of a Spherical Kagome Cluster in {W72V30}

Yoshiyuki Fukumoto1 ∗, Yuto Yokoyama1, and Hiroki Nakano2

1Department of Physics, Faculty of Science and Technology, Tokyo University of Science, Noda, Chiba 278-85102Graduate School of Material Science, University of Hyogo, Ako-gun, Hyogo 678-1297

(Received )

In order to clarify why the experimental magnetization curve of the spherical kagome cluster in {W72V30} at 0.5 K

shows no sign of staircase behavior up to 50 T, we study the effects of Dzyaloshinsky-Moriya (DM) interactions and

tilts of the g tensors, both of which lead to the breaking of the total-S z conservation, by using the exact diagonalization

method. It is found that the D vector component parallel to the radiation direction of the polyhedron cancels out the

staircase in a low magnetic field region efficiently. The tilts of the g tensors are inherent to systems defined on the poly-

hedrons and lead to induced magnetic fields varying site by site. This induced magnetic field affects the magnetization

only at high magnetic fields. We also discuss two existing experimental results on the basis of our calculated results.

1. Introduction

The study of resonating valence bond (RVB) states is one of

central issues in frustrated quantum spin systems,1) and over

the past few decades, a considerable number of studies have

been made on the spin-1/2 Heisenberg antiferromagnets in the

kagome lattice,2–13) which is a network of corner-sharing tri-

angles, as well as on the spin-1/2 triangular-lattice Heisen-

berg antiferromagnets,14–31) which include a network of edge-

sharing triangles. In this paper, we study a zero-dimensional

counterpart of the kagome lattice. Icosidodecahedron is a

network of corner-sharing triangles, and can be considered

as a kagome on a sphere.32, 33) Spin-1/2 icosidodecahedrons,

or spherical kagome clusters, are realized in {W72V30} and

{Mo72V30}.34–37) The possible relevance of these spherical

kagome clusters to RVB physics is of great interest.38, 39)

The main Hamiltonians for these spherical kagome clusters

are

H =∑

〈i, j〉

Ji, jSi · S j, (1)

where 〈i, j〉 and Ji, j denote the nearest neighbors and ex-

change couplings, respectively. When Ji, j = J for all 〈i, j〉,

the model has Ih symmetry and will be referred to as an Ih

model (see Fig. 1(a)). The symmetry of both {W72V30} and

{Mo72V30} is not Ih exactly. Both the materials are more or

less distorted and have D5h symmetry, as shown in Fig. 1(b).

We are especially interested in the Ih model from the perspec-

tive of RVB physics, because the distortion tends to eliminate

its characteristic behavior.38)

In 2005, Mo72V30 was synthesized independently by

Muller et al. and Botar et al., and the susceptibility measure-

ment indicated a singlet ground state.34, 35) It was found that

the high-temperature susceptibility was reproduced by the Ih

model with J = 245 K, but the low-temperature part was

∗[email protected]

not.33) As the temperature decreases, the calculated suscep-

tibility of the Ih model vanishes much faster than the experi-

mental susceptibility. In particular, the experimental spin gap

is about one-fourth of that of the Ih model. In order to re-

solve this discrepancy, the distortion in Mo72V30 was taken

into account,38) and a distorted model with the D5h symme-

try was introduced (see Fig. 1(b)). If distortion exists, then

each triangle unit becomes an isosceles triangle, which tends

to stabilize the ferrimagnetic phase with a total spin of S , 0.

By adequately choosing four exchange parameters in the D5h

model , it is possible to reproduce the experimental suscep-

tibility of Mo72V30. The small spin gap of Mo72V30 can be

interpreted to indicate the system being in close proximity to

the ferrimagnetic phase.38)

In 2009, Todae et al. synthesized W72V30 and measured its

magnetic susceptibility.36) This material was independently

synthesized by Li et al., who also carried out a high-energy

Raman scattering experiment.37) Schnack studied the thermo-

dynamics of the Ih model and suggested, via the analysis of

theoretical and experimental susceptibilities, that W72V30 is

well described by the Ih model.40) However, the measure-

ment by Todae et al. was not carried out at very low tempera-

Fig. 1. Schematic illustrations of (a) Ih model and (b) D5h model.

1

http://arxiv.org/abs/1803.06485v1

J. Phys. Soc. Jpn. FULL PAPERS

tures, where the magnetic susceptibility shows activation-type

behavior, and therefore, it gave no information on the spin-

gap energy. The information on spin-gap energy is crucial to

check whether W72V30 is described by the Ih model or not.38)

Subsequently, magnetization experiments were carried out,

in which the measurement temperature was 0.5 K and the

maximum magnetic field was 50 T.41, 42) Surprisingly, the ex-

perimental magnetization processes show no staircase behav-

ior, although the Ih model in (1) shows three steps within the

applied magnetic field region. Schnack et al. reported that the

origin of the luck of the staircase behavior was ascribed to

a broad distribution of the nearest-neighbor exchange inter-

actions, where the width of the distribution was taken to be

about 30% of its average value.41)

In this paper, we study the other two mechanisms that can-

cel out the staircase behavior in the magnetization process.

The symmetry of icosidodecahedron is not high and anti-

symmetric Dzyaloshinsky-Moriya (DM) interactions should

be present.44) For a spin cluster defined on a polyhedron,

the Zeeman interaction is naturally accompanied by an site-

dependent induced magnetic field component, as described

below. We first note that the directions of principal axes of

the g-tensor change site by site in a polyhedron. It is also nat-

ural to expect that the g factor along the radiation direction

of the polyhedron is different from that in the tangent plane.

These factors cause the induced magnetic fields to vary site

by site. Both the DM interaction and the site-dependent mag-

netic field break the conservation of magnetization along the

direction of the applied magnetic field, and can be possible

sources for the luck of the staircase behavior in the magneti-

zation process.

We use the exact diagonalization method to study the ef-

fects of the DM interaction and the site-dependent magnetic

field on the magnetization process. From the result, we find

that the staircase behavior at low magnetic fields, correspond-

ing to the experiment, is eliminated by the modest strength of

DM interaction, if the DM vector contains a certain amount of

radiation components. The site-dependent induced magnetic

field due to the tilts of the g tensors affects the magnetization

only at high magnetic fields.

This paper is organized as follows. The DM interaction and

induced magnetic field are formulated in §2. In §3, we present

our results of exact diagonalization calculations and discuss

two existing experimental results. In §4, we summarize the

results obtained in this study.

2. Model

2.1 DM interaction

We start by defining the direction of a bond 〈i, j〉. As shown

in Fig. 2(a), we order the three sites in each triangle anticlock-

wise from the outside of an icosidodecahedron. In the notation

〈i, j〉, we assume that site i ( j) is the prior (subsequent) site.

Denoting the position vector of site i ( j) as ri (r j), where the

center of the icosidodecahedron is chosen as the origin O, we

introduce the following two unit vectors,

e(i, j)r =

ri + r j

|ri + r j|, and e

(i, j)p =

ri × r j

|ri × r j|, (2)

which form the basis vectors for the plane normal to the bond

direction r j − ri. In Fig. 2(a), the blue and red arrows, respec-

tively, represent e(i, j)r and e

(i, j)p . We call the direction of e

(i, j)r

(e(i, j)p ) as the “radiation (perpendicular) direction” of the bond

〈i, j〉. For a triangle, we have three sets of basis vectors, as

shown in Fig. 2(a). Considering the C3 rotation around the

symmetry axis of the triangle, we can easily observe that the

three sets are replaced by each other due to the C3 rotation op-

eration. In the same way, for the pentagon shown in Fig. 2(b),

the five sets of basis vectors are replaced by each other due

to the C5 rotation operation around the symmetry axis of the

pentagon.

Fig. 2. Schematic representation of e(i, j)r and e

(i, j)p for (a) triangle and (b)

pentagon on the surface of the icosidodecahedron, and for (c) decagon on the

equatorial plane, where O is the center of the icosidodecahedron. The gray

arrows indicate the directions of bonds.

Several magnets with pentagonal structures have attracted

considerable attention45–48) due to the viewpoint that the frus-

tration effect produced by pentagons is different from that pro-

duced by triangles. A pentagon in the present system is sur-

rounded by triangles, which are in an edge-sharing relation-

ship with the pentagons. Some systems including polygons

surrounded by edge-sharing triangles have been investigated.

In the case of distorted kagome lattice, the surrounded poly-

gon is hexagon,4, 11, 12)and in the case of square-kagome lat-

tice, the surrounded polygon is square.49–53) These lattices are

created by a tiling of polygons in a flat plane. Since such tiling

of pentagons in a flat plane without any voids is impossible,

2


the present system forms a sphere. A comparison among the

systems on these lattices is another interesting future issue.

Let us return to the DM interactions in the present system.

We write the DM interaction as

HDM =∑

〈i, j〉

Di, j · (Si × S j), (3)

which is invariant under symmetry operations of Ih. We now

consider the symmetry to determine the direction of Di, j. For

a bond 〈i, j〉, there exists a mirror plane perpendicular to the

bond direction r j−ri. The invariance of the Hamiltonian under

the mirror operation leads to Di, j lying in the mirror plane.

Using {e(i, j)p , e

(i, j)r } as the basis vectors of the mirror plane, we

can write

Di, j = D(i, j)p e

(i, j)p + D

(i, j)r e

(i, j)r . (4)

Next, we use the invariance under the C3 and C5 rotations

around the symmetry axes of both the triangle and the pen-

tagon. Noting that eps and ers are invariant under these ro-

tations, this invariance condition holds if D(i, j)r and D

(i, j)p are

common for all bonds. Thus, we get the final expression

HDM = d∑

〈i, j〉

(cos θ e(i, j)p + sin θ e

(i, j)r ) · (Si × S j), (5)

where d represents the strength of the D vector and θ is the

angle defining its direction.

The symmetry considerations give no further information

on the direction of the D vector. It is instructive to look into

the space inversion operation Ci, where the inversion cen-

ter is the origin O, i.e., the center of the icosidodecahedron.

In Fig. 2(c), we show a decagon on the equatorial plane of

the icosidodecahedron. We can refer to a preceding study on

nanoscale iron rings for more information about the DM in-

teractions in cluster magnets.54) Defining the site indexes 1,

2, 1′, and 2′ as in Fig. 2(c), Ci relates 1 and 2 to 2′ and 1′,

respectively. Noting that the spin operator is an axial vec-

tor, we have (S x1, S

y

1, S z

1) → (S x

2′, S

y

2′, S z

2′) and (S x

2, S

y

2, S z

2) →

(S x1′, S

y

1′, S z

1′) by Ci, and thus

D1,2 · (S1 × S2)→ −D1,2 · (S1′ × S2′). (6)

The invariance condition demands D1′ ,2′ = −D1,2. However,

this relation holds already in eq. (5).

2.2 Zeeman term

We start by introducing a global coordinate system, whose

origin is the center of the icosidodecahedron, and then denot-

ing the unit vectors in the x, y, z directions as e1, e2, e3. We

set the position vector ri as

ri = xie1 + yie2 + zie3. (7)

Next, we introduce a local coordinate system concerned with

site i. We define e(i)

1and e

(i)

2as the unit vectors in the tangent

plane at the position ri, and the unit vector in the radiation

direction as

e(i)

3=

ri

|ri|. (8)

The Zeeman interaction for the whole system is given by

HZemman =∑

i

H(i)

Zemman, (9)

where

H(i)

Zemman= −HgiSi. (10)

It is naturally expected that the g tensor, gi, is in a diagonal

form, if the above-mentioned local coordinate system is used,

H(i)

Zemman= −[e

(i)

1· H, e

(i)

2· H, e

(i)

3· H]

g⊥ 0 0

0 g⊥ 0

0 0 g‖

e(i)

1· Si

e(i)

2· Si

e(i)

3· Si

= −g⊥H · Si − (g‖ − g⊥)(e(i)

3· H)(e

(i)

3· Si)

≡ −g⊥ (H + δHi) · Si, (11)

where

δHi =g‖ − g⊥

g⊥

(H · ri)ri

|ri|2(12)

is a site-dependent induced field in the radiation direction.

The site-dependent magnetic fields break the conservation of

the total S z. It should be stressed that the existence of such ad-

ditional fields is inherent to systems that are defined on poly-

hedrons.

3. Calculation Methods and Results

3.1 Calculation methods

In the magnetization measurements, polycrystalline sam-

ples of {W72V30} were used.41, 42) The orientations of mag-

netic clusters in such samples are distributed randomly with

respect to the magnetic-field direction. In order to repro-

duce this situation in our theoretical calculations, we pre-

pare a sample set of randomly distributed magnetic fields,

{H1,H2, · · · ,HNs}, where |Hl| = H for all l. Then, we apply

Hl on a magnetic cluster and calculate the induced magneti-

zation, Ml, parallel to the field. The observed magnetization

is estimated from

M(H) =1

Ns

Ns∑

l=1

Ml. (13)

Our calculated results shown below are accompanied by er-

ror bars resulting from the statistical error. In general, for the

magnetization steps in a system that has only exchange inter-

actions, we need to use larger values of Ns around the mag-

netic fields . In the following calculated results, the maximum

sampling size is Ns ∼ 50.

As the present system does not conserve the total S z, the

dimension of the Hilbert space is 230 ≃ 1.07 × 109. In our

exact diagonalization calculation via the Lanczos method, we

work with the standard Ising basis, and therefore, the matrix

elements of DM and Zeeman terms are complex. We need

to prepare three complex vectors with a dimension of ∼ 109,

whose memory usage is 48 GB. Our numerical calculations

3


were performed on the SGI Altix ICE 8400EX at the Super-

computer Center, Institute for Solid State Physics, University

of Tokyo, using OpenMP parallelization with up to 24 cores.

3.2 Calculation results

3.2.1 Case of d = 0

Here, we study the effect of tilts of the g tensors. We set

d = 0, and introduce a parameter,

Rg =g‖ − g⊥

g⊥, (14)

which determines the strength of the induced field in (12). The

calculated results for Rg = ±0.1, together with that for Rg = 0

are shown in Fig. 3. In Fig. 3, we find that the effect of Rg , 0

appears only at high magnetic fields. Therefore, the tilts of

the g tensors is not the reason for the absence of staircase

behavior up to 50 T in the experimental observations.

Fig. 3. Magnetization process for the entire magnetic field region, when

the tilts of the g tensors are taken into account.

3.2.2 Case of Rg = 0

Here, we consider the case of g‖ = g⊥ ≡ g, which means

no induced field, i.e. δHi = 0.

After setting the strength of the D vector to d = 0.1J, we

start by studying the direction dependence of the low-field

magnetization process. The calculated results are shown in

Fig. 4, together with a line with the slope corresponding to

the peak value of the magnetic susceptibility, χpeak, for the Ih

model without DM interactions.38) The spin gap at d = 0 is

corresponding to gH/J = 0.218. In the magnetic field region

0 ≤ gH/J ≤ 0.6, which is comparable to the experimental

region, the magnetization process for d = 0.0 contains four

steps. There is a pronounced hint of staircase behavior when

the radiation direction component is small, θ ≃ 0 and π. Con-

versely, it is noticed that the radiation direction component

tends to eliminate the staircase behavior. If the radiation di-

rection component is not very small, then magnetization more

than the spin-gap field does not depends on θ vary much and

the slopes are close to χpeak. On the other hand, magnetiza-

tion lower than the spin-gap field depends on θ, which is quite

natural because the DM interaction determines the value of

zero-temperature susceptibility. On comparing θ = 0.5π and

1.5π, the latter has a weaker cusp around the spin-gap field

and seems to reproduce the experimental result better.

Fig. 4. Low-field magnetization process for various values of θ. The slope

of the gray line corresponds to χpeak for the Ih model with d = 0.

Fig. 5 shows the dependence of low-field magnetization

processes on d up to 0.20J, when the direction is fixed to

be θ = 1.5π. As d increases, the staircase behavior tends to

fade away. It is found, as in Fig. 4, that magnetization more

than the spin-gap field does not depends on d vary much and

the slopes are close to χpeak. However, the dependence on d is

pronounced for magnetization lower than the spin-gap field.

When d increases, the zero temperature susceptibility, χT=0,

increases and we have χT=0 > χpeak for d/J >∼ 0.15.

Fig. 5. Low-field magnetization process for various values of d.

3.2.3 The entire magnetization process

The entire magnetization processes for Rg = 0, ±0.05 up

to the saturation field are shown in Fig. 6, when d = 0.1J and

4


θ = 1.5π. We find that the steps below the one-third plateau,

gH/J <∼ 1, are cancelled out by the radiation component of

the D vector. For the intermediate magnetic field region of

1.4 < g⊥H/J < 2.7, if d = 0 and Rg = 0, then there exist wide

steps at M/Ms = 6/15, 7/15, 8/15, 9/15. The signatures

of these steps are expected to remain, even if we set modest

values of d and Rg. For the high magnetic field region of 2.7 <

g⊥H/J, each step is narrow from the beginning. These steps

are readily smoothed out by modest values of d and Rg.

Fig. 6. Magnetization process for the entire magnetic field region.

3.3 Comparison with experimental data

Here, we use θ = 1.5π and Rg = 0 in our theoretical cal-

culations. In Fig. 7, our calculated magnetization curves are

compared with the existing experimental data from two stud-

ies.41, 42)

The first one was observed at T = 0.5 K in 2013,41) which

exhibits a linear behavior of M with respect to H approxi-

mately. The black line in Fig. 7 represents the linear approx-

imation of their magnetization curve. We find that all over

behaviour of this experimental result is reproduced well by

the theoretical calculation for d = 0.1J, which is shown by

the closed squares in Fig. 7. However, as mentioned before,

the calculated magnetization deviates from the linear approx-

imation below the spin gap field. As for the sample used in

Ref. 41, it was described that there exist two uncorrected ions,

VO2+, per a {W72V30} molecule. It may be plausible that the

subtraction procedure of free V4+ spins from raw magnetiza-

tion data washes out such a structure. In fact, the magnetiza-

tion curve shown in Fig. 3 in Ref. 41 takes negative values

below 5 T.

Another experimental magnetization curve was presented

by Nojiri et al. in 2015,42) where the measurement tempera-

ture T = 1.6 K is somewhat higher than the first one but still

much lower than the spin-gap energy 0.218J = 21.5 K.38)

They treated the number of free V4+ spins as a fitting param-

eter to decompose their raw magnetization curve into a linear

term and Brillouin function.43) In Fig. 7, the red solid line

represents the linear term, which is intrinsic to the spherical

kagome cluster, and the red dotted line represents the Bril-

louin function due to impurity spins. It found that the calcu-

lated data at low magnetic fields are on the linear term. Their

procedure seems to be adequate at low magnetic fields below

10 T.

Fig. 7. Comparison between the theoretical and experimental magnetiza-

tion curves, with J = 115K and g = 1.95.38) The earlier experimental data

from 2013 was reported in Ref. 41, and the latest data from 2015 was reported

in Ref. 42.

4. Summary and Future Problems

We have performed exact diagonalization calculations for

the magnetization process of a spherical kagome cluster, in-

corporating the DM interaction and the tilts of the g-tensors.

It was found that the low-field magnetization steps are effi-

ciently cancelled out by the DM interactions with the D vector

parallel to the radiation direction. When the radiation compo-

nent is not so small, we have the linear magnetization process

with slope of ∼ χpeak at low magnetic fields above the spin

gap field. Below the spin gap field, we have another linear

term with slope of χT=0 which depends on the strength and

direction of D vector. We have compared two existing ex-

perimental magnetization curves with our calculated results

to point out that two different linear magnetization processes

reported in Refs. 41 and 42 may, respectively, correspond to

χpeak × H and χT=0 × H.

The reason why we expected the Ih model to be adequate

for {W72V30} was that the experimental susceptibility down

to 5 K reported in Ref. 36 was consistent with the theoreti-

cal susceptibility of the Ih model.40) However, the experimen-

tal data was provided in the form of χT , and thus it was not

clear whether the theoretical and experimental results agree

with each other at low temperatures. In a later susceptibility

experiment in 2013, measurements down to 2.5 K were car-

ried out and the experimental data in the form of χ were pre-

sented.41) This latest susceptibility data shows a broad peak

5


around 10-20 K and a rapid decrease below about 5 K. On

the other hand, the Ih model leads to a rapid decrease below

about 10-15 K, and thus does not reproduce the experimental

result at low temperatures. The experimental susceptibility in-

dicates that there are more low-energy magnetic components

in {W72V30} than in the Ih model, which seems to be consis-

tent with the existence of DM interactions. Also, we note that

the susceptibility experiment gives χ2.5K/χpeak ≃ 0.7, which

is not contradict to the results shown in Fig. 7.

It is desired to study theoretically how DM interactions af-

fect the magnetic susceptibility quantitatively. However, DM

interactions break the conservation of total spin and get the

evaluation of magnetic susceptibility to be more difficult. As

for kagome-lattice antiferromagnets with DM interactions,

Rigol and Singh calculated magnetic susceptibility up to a

15-site cluster to understand the experimental susceptibility

of Herbertsmithite.55) They calculated all eigenvalues under

magnetic fields H, and evaluated magnetic susceptibility from

the second-order coefficient in H of the free energy. As for

the 30-site spherical kagome cluster, the calculation of free

energy is possible,38) but it is accompanied by statistical er-

ror being an serious obstacle for the numerical evaluation of

the second-order coefficient. Instead, we can use the method

of thermal pure quantum states to calculate magnetizations

at finite temperatures,56) whose first-order coefficients give

the magnetic susceptibility as a function of temperature. The

study along this line is now proceeding.

The main concern in the study of spherical kagome clus-

ters is the exploration of the characteristic behavior of RVB

states. From this perspective, it is desired to make experimen-

tal studies on specific heat and Raman spectrum, by which we

can access low-energy states in the singlet sector directly.38)

Acknowledgments

We acknowledge Dr. K. Kihara and Professor H. Nojiri

for providing their unpublished data and fruitful discussions.

The authors thank the Supercomputer Center, Institute for

Solid State Physics, University of Tokyo for the use of their

facilities. This work was partly supported by JSPS KAK-

ENHI Grant Numbers 17K05519, 16K05418, 16K05419, and

16H01080 (JPhysics).

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