ARTICLE IN PRESS

Physica E 29 (2005) 538–540
www.elsevier.com/locate/physe

Spectral statistics and the Dzyaloshinsky–Moriya
interaction of nanomagnet V15Manabu Machidaa,, Seiji Miyashitaba
Institute of Industrial Science, The University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan
Department of Physics, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japanb

Available online 22 July 2005

Abstract
The spectral statistics of nanomagnet V15 is studied. We study the dependence of the Dzyaloshinsky–Moriya
interaction in V15 on the spectral statistics.
r 2005 Elsevier B.V. All rights reserved.
PACS: 75.30.Gw; 75.50.Xx; 05.45.Mt
Keywords: Molecular magnet; Spectral statistics

Nanoscale molecular magnets, or nanomagnets,
have been studied intensively in recent years. The
novel structure of discrete energy levels of nanomagnets stimulates not only fundamental researchsuch as quantum dynamics but also appliedresearch of quantum device. Indeed, a quantum
computation by nanomagnets with the help of theESR technique is proposed [1].
Nanomagnet V15 is the complex of formulaK6 ½VIV
15 As6 O42 ðH2 OÞ  8H2 O [2,3]. In the cluster ofthe complex, 15 vanadium ions, each of which has
a 12 spin, are placed almost on a sphere. The threeCorresponding author.

E-mail addresses: machida@iis.u-tokyo.ac.jp (M. Machida),
miya@spin.phys.s.u-tokyo.ac.jp (S. Miyashita).

spins forming a triangle are sandwiched by 12
spins forming 2 hexagons.
The existence of the Dzyaloshinsky–Moriya(DM) interaction [4–6] in the Hamiltonian of V15
is implied by the following experiments. If wesweep the magnetic ﬁeld adiabatically at lowtemperatures, the magnetization changes smoothly
at zero ﬁeld from 12 to 12 [7]. This fact implies that
there exists a ﬁnite gap between the ground state
and the lowest excited state at zero ﬁeld. The
origin of this gap is attributed to the DM
interaction [8]. In addition, the ground statemagnetization changes smoothly from 12 to 32 at
the ﬁeld around 2:8 T [9]. This broadness of thechange is also considered to be caused by the DM
interaction [10]. However, the DM vectors of theDM interaction are not fully understood yet. In1386-9477/$ - see front matter r 2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.physe.2005.06.022

ARTICLE IN PRESS
M. Machida, S. Miyashita / Physica E 29 (2005) 538–540

this paper, we study the DM interaction of V15
from the viewpoint of the spectral statistics.
We consider V15 in the magnetic ﬁeld applied
parallel to the c-axis of the molecule. The total
Hamiltonian H is given byXH ¼ H0  H SSzi
H0 ¼ Xi

J ij S i  S j þ HDMhi;ji

HDM ¼

XDij  ½S i

S j .

ð1Þhi;ji

We adopt the same notation as Ref. [12] fornumbering the sites in the cluster, spins, and DMvectors. The interactions J ij in H0 take threedifferent values J, J 1 , and J 2 (jJj4jJ 2 j4jJ 1 j)
[11,12]. Here we take J ¼ 800 K, J 2 ¼ 350 K,
and J 1 ¼ 225 K [10]. We also take H S ¼ 4 T.
We note that the Hamiltonian H has the C 3
symmetry.
Let us determine the DM vectors in the DM
interaction HDM . Since the system has C 3
symmetry, HDM is invariant with the unitarytransformationX2p zU ¼ T L ei 3 Stot ; S ztot ¼Szi ,(2)iwhere T L is an operator acting on the spatial
degrees of freedom and it rotates the position ofspin S i by 2p=3. Let us assume that position 5 isdifferent from position 1 by 2p=3 rotation in thex–y plane. Then, S 1 is transformed by U as
2p
2p
þ S y5 sin
,
332p
2p
US y1 U 1 ¼  Sx5 sin
þ S y5 cos,
33US z1 U 1 ¼ S z5 .

US x1 U 1 ¼ S x5 cosð3ÞThe DM vectors are considered to exist in thebonds with interaction J on the upper and lowerhexagons. That is, we consider D1;2 , D3;4 , and D5;6
on the upper hexagon and D10;11 , D12;13 , and D14;15
on the lower hexagon.
In order to satisfy ½HDM ; U ¼ 0, the DMvectors on the upper hexagon are obtained byrotating the reference DM vector D1;2 by 2p=3 and539

4p=3. For example, D3;4 is given as
1! 0!
Dx3;4
Dx1;2
sin 2pcos 2p
33@A¼,Dy3;4
Dy1;2
 sin 2pcos 2p33Dz3;4 ¼ Dz1;2 .

ð4ÞSince the V15 cluster itself has the D3 symmetry [2],D14;15 on the lower hexagon is obtained by rotating
D1;2 by p=6 in the x–y plane and changing thesign of Dz1;2 . The other two DM vectors on thelower hexagon are obtained by rotating D14;15 by
2p=3 and 4p=3.
We obtain the states jfyi which are eigenstatesof U by applying the projection operator P^ y on thebasis vectors jfi [13].
jfyi ¼1P^ y jfi,
^kPy jfik(5)

where the projection operator is deﬁned byP^ y ¼X

exp in¼0;1y ¼ k þ S ztot ;2p kn þ Sztot
3k ¼ 0; 1.

T nL ,
ð6Þ

We classify subspaces of the total Hilbert space byy. The dimension of the total Hilbert space is215 ¼ 32 768. For example, the space of y ¼ 32contains 4821 levels.
We study the spectral statistics of V15 in thesubspace of y ¼ 32. We use from the 1500th to2500th energy levels. We consider the following
ﬁve cases of the reference DM vector D1;2 :(i) Only Dx1;2 and Dy1;2 exist (Dx1;2 ¼ Dy1;2 ¼
40:0 K; Dz1;2 ¼ 0).
(ii) Only Dz1;2 exists (Dx1;2 ¼ Dy1;2 ¼ 0; Dz1;2 ¼ 40 K).
(iii) All of Dx1;2 , Dy1;2 , and Dz1;2 exist (Dx1;2 ¼
Dy1;2 ¼ Dz1;2 ¼ 40 K).
(iv) Same as (iii) but the magnitude is half
(Dx1;2 ¼ Dy1;2 ¼ Dz1;2 ¼ 20 K).
(v) Same as (iii) but the magnitude is twice larger(Dx1;2 ¼ Dy1;2 ¼ Dz1;2 ¼ 80 K).
We obtain the energy levels E i by diagonalizing
the Hamiltonian in the subspace. The normalizedARTICLE IN PRESSM. Machida, S. Miyashita / Physica E 29 (2005) 538–540

540

0.002

Hamiltonian, and the ﬁve cases align according to(i)o(ii)o(iv)o(iii)o(v)ξ0.0015in the order of the strength of level repulsion.

0.001

0.0005

0(i)

(ii)

(iii)

(iv)

(v)

Fig. 1. The values of x are shown with respect to the ﬁve cases.

nearest-neighbor spacing S i is given bySi ¼ r

E iþ1  E i
ðE iþ1  E i Þ,2

We would like to thank Dr. T. Iitaka forvaluable discussions on V15 . One of the authors(M. M.) also would like to thank Professor N.
Hatano. This work is supported by the Grant-inAid from Ministry of Education, Culture, Sports,
Science and Technology, and also by the NAREGI Nanoscience Project, Ministry of Education,Culture, Sports, Science and Technology, Japan.
The simulation was partially carried out by using
the computational facilities of the Super Computer
Center of Institute for Solid State Physics, theUniversity of Tokyo.

(7)
Referenceswhere rðEÞ is the density of states. Let us denoteby PðSÞ dS the probability that S i Ris in the range
1between
R 1 S and S þ dS. Note that 0 PðSÞ dS ¼ 1
and 0 SPðSÞ dS ¼ 1. The cumulative spacing
distribution IðSÞ is written asZ SIðSÞ ¼
PðS0 Þ dS0 .
(8)0

If there is no correlation between energy levels, thecumulative spacing distribution is given by [14]I 0 ðSÞ ¼ 1  eS .

(11)

(9)

If there exists the effect of level repulsion, IðSÞ will
deviate from I 0 ðSÞ. In order to study this deviationquantitatively, we introduce a parameter x.
Z 1½IðSÞ  I 0 ðSÞ2 dS.
x
(10)0The parameter x measures distance between IðSÞand I 0 ðSÞ. Fig. 1 shows x for each case. We
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