Numerical study on ESR of V15 IIS, U. Tokyo, Manabu Machida RIKEN, Toshiaki Iitaka Dept. of Phys.,...
-
date post
20-Dec-2015 -
Category
Documents
-
view
214 -
download
0
Transcript of Numerical study on ESR of V15 IIS, U. Tokyo, Manabu Machida RIKEN, Toshiaki Iitaka Dept. of Phys.,...
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
€
H = − Jij
v S i ⋅
v S j
i, j
∑ +v D ij ⋅
v S i ×
v S j( )
i, j
∑ − HS Siz
i
∑
€
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.]
€
J = −800K, J1 = −225K, J2 = −350K
€
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.
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)
€
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
€
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.
€
e−βH / 2 = I0 − β2( )T0 H( ) + 2 Ik − β
2( )Tk H( )k=1
kmax
∑
€
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
€
Itot β( ) = I ω,β( )dω0
∞
∫
€
′ ′ χ ω,T( ) =π
e−βEm
m
∑ m,n
∑ e−βEm − e−βEn( ) ψ m M x ψ n
2
×δ ω − En − Em( )( )
€
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
€
E8 = − 34 J + 3
2 HS
€
E4 = 34 J + 1
2 HS + 32 D
€
E7 = − 34 J + 1
2 HS
€
E6 = − 34 J − 1
2 HS
€
E5 = − 34 J − 3
2 HS
€
E3 = 34 J + 1
2 HS − 32 D
€
E2 = 34 J − 1
2 HS + 32 D
€
E1 = 34 J − 1
2 HS − 32 D
€
HSc0 =
3
2J
Energy levels with weak DM
€
E2
€
E5
€
E1
€
O D( )( )
€
HS⟨⟨HSc 0
€
HSc0⟨⟨HS
€
Rtri T( ) T →0 ⏐ → ⏐ ⏐
€
3
€
1+3D
HS
Intensity ratio of triangle model
At zero temperature