Numerical study on ESR of V15

IIS, U. Tokyo, Manabu MachidaRIKEN, Toshiaki Iitaka

Dept. of Phys., Seiji Miyashita

June 27- July 1, 2005

Trieste, Italy

Nanoscale molecular magnet V15

(http://lab-neel.grenoble.cnrs.fr/)€

K6 V15IVAs6O42 H2O( )[ ] • 8H2O

Vanadiums provide fifteen 1/2 spins.

[A. Mueller and J. Doering (1988)]

Hamiltonian and Intensity

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H = − Jij

v S i ⋅

v S j

i, j

∑ +v D ij ⋅

v S i ×

v S j( )

i, j

∑ − HS Siz

i

∑

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I T( ) = I ω,T( ) dω0

∞

∫ =ωHR

2

2′ ′ χ ω,T( ) dω

0

∞

∫€

′ ′ χ ω,T( ) = 1− e−βω( ) Re M x M x t( ) e−iωt dt

0

∞

∫

[H. De Raedt, et al., PRB 70 (2004) 064401]

[M. Machida, et al., JPSJ (2005) suppl.]

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J = −800K, J1 = −225K, J2 = −350K

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D1,2x = D1,2

y = D1,2z = 40K

The parameter set

Difficulty

– Its computation time is of(e.g. S. Miyashita et al. (1999))

€

M x M x t( ) =Tr e−βH M x M x t( )

Tr e−βH

– Direct diagonalization requires memory of

€

O N 2( )

€

O N 3( )

difficult!

Two numerical methods

• The double Chebyshev expansion method (DCEM) - speed and memory of O(N) - all states and all temperatures

• The subspace iteration method (SIM) - ESR at low temperatures.

DCEM

ESR absorption curves

Typical calculation time for one absorption curve is about half a day.

DCEM

Background of DCEM

The DCEM =a slight modification of the Boltzmann-weighted time-dependent method (BWTDM).

Making use of the random vector technique andthe Chebyshev polynomial expansion

[T. Iitaka and T. Ebisuzaki, PRL (2003)]

DCEM (1)

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M x M x t( ) =Φ e−βH / 2

( )M x M x t( ) e−βH / 2 Φ( )[ ]av

Φ e−βH / 2( ) e−βH / 2 Φ( )[ ]

av

Random phase vector

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Φ ˆ X Φ[ ]av

= n ˆ X nn

∑ + e i θ m −θ n( )−δmn[ ]av

n ˆ X mm,n

∑

= Tr ˆ X + Δ ˆ X ≅ Tr ˆ X €

Φ = n e iθ n

n=1

N

∑

DCEM (2)Chebyshev expansions of the thermal and time-evolution operators.

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e−βH / 2 = I0 − β2( )T0 H( ) + 2 Ik − β

2( )Tk H( )k=1

kmax

∑

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e− i Ht = J0 t( )T0 H( ) + 2 −i( )kJk t( )Tk H( )

k=1

kmax

∑

€

J

€

HS>> small

Temperature dependence of intensity

[Y.Ajiro et al. (2003)]

Our calculation Experiment

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Itot β( ) = I ω,β( )dω0

∞

∫

SIM

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′ ′ χ ω,T( ) =π

e−βEm

m

∑ m,n

∑ e−βEm − e−βEn( ) ψ m M x ψ n

2

×δ ω − En − Em( )( )

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I1 T( ) =πHR

2 HS

8tanh

βHS

2

⎛

⎝ ⎜

⎞

⎠ ⎟

ESR at low temperatures by SIM

We consider the lowest eight levels.

€

R T( ) = I T( ) /I1 T( )

€

I T( ) =ωHR

2

2′ ′ χ ω,T( ) dω

0

∞

∫

Intensity ratio

Temperature dependence of R(T)

With DM Without DM

Triangle model analysis

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J12 = J23 = J31 = J = −2.5K

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D1,2x = D1,2

y = D1,2z = D ≅ 0.25K( )

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E8 = − 34 J + 3

2 HS

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E4 = 34 J + 1

2 HS + 32 D

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E7 = − 34 J + 1

2 HS

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E6 = − 34 J − 1

2 HS

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E5 = − 34 J − 3

2 HS

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E3 = 34 J + 1

2 HS − 32 D

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E2 = 34 J − 1

2 HS + 32 D

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E1 = 34 J − 1

2 HS − 32 D

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HSc0 =

3

2J

Energy levels with weak DM

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E2

€

E5

€

E1

€

O D( )( )

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HS⟨⟨HSc 0

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HSc0⟨⟨HS

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Rtri T( ) T →0 ⏐ → ⏐ ⏐

€

3

€

1+3D

HS

Intensity ratio of triangle model

At zero temperature

SummaryO(N) algorithms for the Kubo formula

DCEM

ESR of V15

■ High to low temperatures by DCEM■ Ultra-cold temperature by SIM■ Triangle model analysis

■ Random vector and Chebyshev polynomials

M. Machida, T. Iitaka, and S. Miyashita, JPSJ (2005) suppl.(cond-mat/0501439)