quantum dynamics in low dimensional spin systems

Quantum dynamics in low dimensional spin systems

University of TokyoSeiji MIYASHITA

YKIS2007 Kyoto29 Nov. 2007

Topics• Quantum dynamics under time-dependent fields

Quantum hysteresis in single molecular magnetsLandau-Zener process + Magnetic Foehn effects (Sweep)

Nontrivial Resonance and Coherent Destruction of Tunneling (AC)Quantum mechanical reentrant phenomena Quantum annealing

• Quantum dynamics between macroscopic statesQuantum spinodal phenomena of quantum phase transitionNagaoka magnetism

• Quantum ResponseESR in pure quantum dynamicESR in dissipative dynamics

• Related topicsOrigin of the energy gap and Gap control Potential trap

Quantum dynamics of magnetization

Molecular magnets

V6 Cu3 Ni4

V15Mn12 Fe8

Temperature dependence

Quantum tunneling+

Thermal effects

Resonant tunnelingL. Thomas, et al. Nature 383 (1996) 167.

Resonance tunneling

Control of quantum states inDiscrete energy structure

(Non)-adiabatic transitionLandau-Zener-Stueckelberg Mechanism

( )⎟⎟⎠

⎞⎜⎜⎝

⎛

−Δ

−−=vMM

Epoutin

2

2exp1 π

C. Zener, Proc. R. Soc. (London) Ser. A137 (1932) 696.

SM, JPSJ 64(1995) 3207, 65(1996) 2734.H. De Raedt et al, PRB56 (1997) 2734

Sweeping velocity dependence

W. Wernsdorfer et al. EPL 50 (2000) 552JPSJ 69 Suppl. 375.

Quantum interferenceBerry phase

W.Wernsdorfer & R. Sessoli:Science 284 (1999) 133

NMR (H) measurement on Fe8 Effect on the resonant tunneling

K1066.9

K1052.37

9,10

710,10

−−

−−

×=Δ

×=Δ

( )⎟⎟⎠

⎞⎜⎜⎝

⎛

−Δ

−−=vMM

Epoutin

2

2exp1 π

M. Ueda, S. Maegawa and S. Kitagawa:Phys. Rev. B66 (2002) 073309

Landau-Zener transitions in magnetization process

I. Chiorescu, et alPhys. Rev. Lett. 84 (2000) 3454.

I. Rousouchazakis, et al. PRL 94 (2005) 147204

K.Y. Choi, et al. PRL 96 (2006) 107202

V15

V6 Cu3


Fe8

Chiorescu, W. Wernsdorfer, A. Mueller, H. Boegge, B. Barbara,Phys. Rev. Lett. 84 (2000) 3454.

Heat bath

sample

Phonon Bottleneck phenomenain V15

Plateau induced by thermal effect

Heat flow

Quantum Master Equation[ ]

( )

)0(1(e )(1(e

eTr/e ,Tr

tenvironmen ofReduction

,

,

,

1)((

0

1)((

BeqBeq

B

I

BI0

BI0

00

ρρσσ

ρρρρσ

ω

λ

ρρρ

ββ

κ

p)ppp)pp

p

p)iLp)iL −+−+=∂∂

===

+=

++=

++=

++==∂∂

−−−−

−−

∫

∑∑

stt st

HH

kkk

k kkk

iLdstiLiLLit

bbH

XbbH

HHHH

HHHi

iLt η

１

HB

H0

HI

e.g. Photon dissipation and pumping : (SM., H. Ezaki, and E. Hanamura PRA 57 (1998) 2046)

[ ] ( )bbbbbbHit

+++ +−−=∂∂ σσσκσσ 2,0η

１

Lindblad form Stochastic Schrodinger Equation (antibunching, squeezingphoto emission)

General formulation

K. Saito, S. Takesue and SM. Phys. Rev. B61 (2000) 2397

[ ] [ ] [ ]( )++−−= ρρλρρ RXRXHidtd ,,,

( ) ( ) ( )ωωωζ

ζ β

−−=

−⎟⎠⎞

⎜⎝⎛ −

=

II

mXkEEnEEmRk mkmk ,)(

η

[ ]

{

}

( ) ( ) ( )

( ) ( ) ( ) density spectral the:0

1e)(e)(

)( operators sreservoie' theoffunction n correlatio time)()()()(e

)()(e)()()(e

,1

02

2

02

2

>==

−−−

=Φ=Φ

Φ−−−+

−−−Φ−

=

∫

∫ ∫

∞

∞−

−

∞

∞−

ωωωωγω

ωωωγω

ρρ

ρρωωλ

ρρ

α

ωβω

ωβ

ωβω

IDI

DDtdt

tXtsXXsXt

sXtXtsXXdds

Hidt

d

ti

tti

　　

　　　　　　　　　

　　

η

η

η

η

η

η

No feedback effects

Adiabatic transition andRelaxation

K. Saito, SM, H.de Raedt, Phys. Rev. B60 (1999) 14553

0→T

Pure Quantum

+ Thermal Bath

Field sweeping with thermal bathFast sweeping Slow sweeping

vv <AD ADTH vvv <<ADv

K. Saito & SM. JPSJ (2001) 3385.

MagneticFoehn EffectLZS

Nonadiabatic Tr. & Heat-inflow

Magnetic Foehn Effect

LZ transition

Fe-rings

H. Nakano & SM, JPSJ 70(2001) 2151

Y. Ajiro & Y. Inagaki

Y. Narumi & K. Kindo

Fe2 Y. Shapira, et al PRB59 (1999) 1046

dHdM

dHdM

Fast Magnetization Tunneling in Tetranicke(II) SMM

En-Che Yang,et al: Inorg. Chem. 45 (2006) 529

V=0.002, ..... , 0.28T/s

[Ni(hmp)(dmb)Cl]4

LZ transition + Thermal relaxation + MFE

v=0.0512, ...., 0.0002

[ ] [ ] [ ]( )++−−= ρρρρ RXRXzHidtd ,,,

Two different types of sites ?Adiabatic change

Thermal relaxation

x 3/4

x 1/4

Possible magnetic process

Quantum dynamics under an AC field Non-trivial Resonance

( )∑−=i

ziSthtH ωcos)( W

( )( )

( ) ⎟⎠⎞

⎜⎝⎛

+Ω=

Δ

Δ−−=

−=Ω

Mc

Ep

p

tMtM

4

2exp1

cos12

0 ,cos)(

π

απω

δ

SM, K. Saito, H. De Daedt, Phys. Rev. Lett. 80 (1998) 1525.

)(tH

Amplitude dependence(Coherent destruction of tunneling)

F. Grossman, et al. Phys. Rev. Lett. 67 (1991) 516.Y. Kayamuma, PRB 47 (1993) 9940.SM, K. Saito, H. De Daedt, Phys. Rev. Lett. 80 (1998) 1525.

Eigenstates of Floquet operator and Hamiltonian

( ) ( )

jjj

ji

j

EH

TeTF

dssHiF

j

φφ

ε

ωπ

=

Ψ=Ψ

⎟⎠⎞⎜

⎝⎛ −=

−

∫/2

0)(expT

Switching by AC field

Y.Teranishi and H. Nakamura:PRL81(1998) 2032

0 1000

0

1

0 1000

0

1

Time

1p2p

Probability of the Level k Energy level as a function of time

With appropriate oscillation,We may change the state bya single operation.

K.Saito and Y. KayanumaPRB 70 201304(R) (2004)

AC field trap by Coherent Destruction of Tunneling (CDT)

∑∑ −−= +

iii

ij

tiji nxEecctH ω

E=0 diffusion ＣＤＴ localization

Y. Kayanuma and K. Saito: arXiv:0708.3570

tieE ω

Reentrant behavior in quantum fluctuation

Ground state search:Quantum annealing

Quantum dynamics in TI model

XH=Γ

quantum tunneling: LZ in TI modelquantum nucleationquantum spinodal decomposition?collective motion?

Quantum fluctuation[ ] [ ] ,0, ,0,

)(

)()(1

≠≠

=

Γ−−−= ∑∑∑ +

HM

tHdtdi

tthJH

xi

zi

ii

x

ii

zi

z

i

zi

σσ

ψψ

σσσσ

η

∑∑ Γ−−= xi

z

jziJH σσσ

T

Γ

Ground state Phase transition

Thermal fluctuation

T. Ikegami, SM, H. Rieger: JPSJ 67 (1998) 2671

Quantum fluctuation

dim-)1(dim- +⇔ dd

Order phase

Phase diagram of Transverse Ising model

Ｈｘ

Ｈｚ

Ｈｘｃ

Quantum disorder

Quantum Critical Point

Symmetry broken ordered state

M > 0

M < 0

T=0Field sweepMetastability,nucleationSpinodal decomposition

Order-disorder transition

J. Dziarmaga: PRL 95 (2005) 245701

Remaining DWs after quench

η/221

τπ Jn =

Critical phenomena in Energy spectrum E(H)

small gap: quantum tunneling-M M E1

level crossings: nucleation

collective motion?

symmetry breakinggap: E2

Landau-Zener-Stueckerberg scatteringat each crossings

H. De Raedt, S. Miyashita, K. Saito, D. Garcia-Pablos and N. Garcia:Phys. Rev. B56 (1997) 11761

Non-adiabatic transition at the avoided level crossing points

Field sweep

Hz-jump

Size-independent phenomenaa kind of collective motion(?)

Hsp

Hx=0.5 L=10,12,14,16

E2

E1 Hzc Hzc

HzcHzc

General structure

Densely populated levels

LEH /2ZC Δ=

Sweep velocity dependence

Dependence on Hx

Metastability and Spinodal decomposition

Mean field theory : classical spin

Quantum spinodal decomposition

( )

( ) 2/33/2SP

2/13/2

2

22

12

2J

,1 0

01

1

γ

γ

γσσ

σσσ

σ

σσσ

−=

Γ=

−=⇒=

=+−

Γ+−=

+−Γ−−=

JH

ddH

HJddE

HJE

0 1 20

1

2 Hsp(Hx) J=1 M(Hsp)=−0.5

Hx

Itinerant ferromagnetism and its dynamics

Y. Watanabe and SM: JPSJ 68 (1999) 3086.66 (1997) 2123,

Transition between AF and Nagaoka-Ferromagnetic state

Ground state change

Dynamics after decimation

( )

23

)22(:

0)22(:

2

2

=

=

=++

⇓

=+

−

↓↑∑

tS

initialet

initialGcc

GS

iHti

ii

( ) ( )( )'

')22(:11

0)22(:

'

2

initialet

initialGcc

GS

iHti

iiii

−

↓↑

=

=++++

⇓

=+

∑ δδ

Adiabatic decimation

( ) ( )↓↑↓↑++ +−++−= ∑ ∑ 55 nnnnUcccctH

ij iiiijjiij μ

σσσσσ

Quantum response inpure quantum and dissipative

environments

collaborators: Akira Ogasahara, Keiji Saito,Chikako Uchiyama, and Mizuhiko Saeki

ESR line shape in strongly interacting spin systemsTemperature-dependence of the shift and width in low-dimensional quantum spin systems

Y. Ajiro, et al: JPSJ 63 (1994) 859.

Spin trimer: 3CuCl2 ・2Dioxane

F F AF

Microscopic expression of the line shape from Hamiltonian

Kubo Formula

dttMM tixxxx

ωβωωχ −∞

∞−

− ∫−= e )()0()e1(21)("

R. Kubo: JPSJ 12 (1957) 570R. Kubo & K.Tomita JPSJ (1954) 888

ηη //)( )0()( iHtiHttiL MeeMetM −⇒=>>= mEmH m ||

( ) ))(()(" mnmn

mn EED −−= ∑ ωδωωχ

( ) ( ) )( ,ee

2

mn

x

EEmn EE

Z

nMmD nm −=−= −− ωπω ββ

Pure quantum dynamics

Shift from the PMR

( )∑∑

∑−−

⋅−=><

i

xi

i

zi

jiij

ij

StHSH

JH

ωcos

2

10

SS

Β== μγγω gη2

1 ,R HParamagnetic Resonance

Isotropic models

( )( )

Λ

Λ

+×⋅+

+⎟⎟⎠

⎞⎜⎜⎝

⎛ ⋅⋅−

⋅+

++−=

∑

∑

∑

><

><

><

ijji

mn mn

mnnmnm

mn

nm

zj

ziz

yj

yi

xj

ij

xi

rrD

SSJSSSSJH

SSD

rSrSSS53

onperturbati

3

])([2

Perturbation

Studies on the line shape• F. Bloch: PR 70 (1946) 460. Nuclear Induction (Bloch equation)• J. H. Van Vleck: PR 74 (1948) 1168.

Dipolar broadening, and exchange narrowing• N. Bloembergen, E. M. Purcell and R. V. Pound: PR 73 (1948) 679.

Relaxation Effects in Nuclear Magnetic Resonance Absorption.• I. Solomon: PR 99 (1955) 559.

Relaxation processes in a system of two spins• F. Bloch: PR 105 (1957) 1206. General theory of relaxation

• A. Abragam: The principles of Nuclear Magnetism, Oxford Univ. Press (1978)

Shift & Width

Peak position

Peak width ( ) ( ))(

,ee

2

mn

x

EEmn

EEZ

nMmD nm

−=

−= −−

ω

πω ββ

τω /2 0)()0( ttixx emtMM −−≅

( )( ) 2Γ+−

Γ∝ 2

0

"ωω

ωχ

δωωω += R0

Nagata-Tazuke Dependence

(J. Kanamori & M.TachikiJPSJ 48 (1962) 50)

K. Nagata and Y. Tazuke: JPSJ 32 (1972) 337

1D Heisenberg model withDipole-dipole interaction

Frequency sweep abd Field sweep

( ) ( )( )00

00

" of smany value

"given :,"0

HH

HH

xx

Hxxxx

ωχ

ωχωχ

⇒

⇒

Line shape as an ensemble

of delta-function

( ) ( ))(

,ee

2

mn

x

EEmn

EEZ

nMmD nm

−=

−= −−

ω

πω ββ

N=8

Shift1D Heisenberg AF

Temperature Dependence

Angle Dependence

SM, T. Yoshino, A. OgasaharaJPSJ 68 (1999) 655

2/πθ =

0=θ

Width

Magic Angle R.E. Dietz, et al. PRL 26 (1971) 1186.T.T. Cheung, et al. PRB 17 (1978) 1266

SM, T. Yoshino, A. OgasaharaJPSJ 68 (1999) 655

parallelmagic angleperpendicular

Zigzag Chain

A. Ogasahara and S. MiyashitaJ. Phys. Soc. Jpn. Suppl. B 72,44-52 (2003).

Spiral structure Dipole-dipole interaction

r

r=0.1 parallel r=0.2

Response in dissipative dynamics

dttMM tixxxx

ωβωωχ −∞

∞−

− ∫−= e )()0()e1(21)("

pure quantum dynamicsηη //)( iHtxiHtx eMetM −=

quantum dynamics with dissipationRelaxation effects:

I. Solomon: PR 99 (1955) 559.Relaxation processes in a system of two spins

F. Bloch: PR 105 (1957) 1206.General theory of relaxation

Y. Hamano and F. Shibata: JPSJ 51 (1982) 1727,2721,2728.M. Saeki: Prog. Theor. Phys. 67 (1982) 1313. : relaxation method

Prog. Theor. Phys. 115 (2006) 1. : TCLE method

Dissipative dynamicsQuantum Master equation method

dttMM tixxxx

ωβωωχ −∞

∞−

− ∫−= e )()0()e1(21)("

Quantum master equation

( ) ( ) ηη //BTr)( tHHHixtHHHix BISBIS eMetM ++++=quantum dynamics with dissipation

[ ] [ ] [ ]( )++−−= ρρπλρρ RXRXHidtd ,,,

2

ηηF. Bloch: PR 105 (1957) 1206.S. Nakajima: PTP 20 (1958) 987, R. Zwanzig: J. Chem. Phys. 33 (1960) 1338.A. G. Redfield: Adv. Magn. Reson. 1 (1965) 1.H. Mori: PTP 33 (1965) 423. M. Tokuyama and H. Mori: PTP 55 (1976) 411.N. Hashitsume, F. Shibata and M. Shingu: J. Stat. Phys. 17 (1977) 155 & 171.T. Arimitsu and H. Umezawa: PTP 77 (1987) 32.

Formulation of line-shape with dissipative dynamics

( ) ( ) ( )

( ) ( )

[ ] [ ] [ ]( )

( )[ ] ( )[ ]

( )[ ] ( )[ ] )0(e)(

)(

),(,),(,,1

cf.

TrTr

00

00

/0

/0

/0

/0

//

tAttA

tALttAt

LtRXtRXHit

etett

etAAetAAeeAtA

Lt

iHtiHt

iHtiHtiHtiHt

ρρ

ρρ

ρρργρρ

ρρ

ρρ

=

=∂∂

≡+−=∂∂

=+

==

+

−

−−

η

ηη

ηηηη

K. Saito, S. Takesue and SM. Phys. Rev. B61 (2000) 2397.

Eigenmode of time-evolution operator

[ ] [ ] [ ]( )

( )

[ ]

)0( , ,)(

)0(e)( )( )(

)1,),(( , vector

)1,,( , ),(matrix

),(,),(,,1

21

2

→→

→→→→

→

==

=

=

==∂∂

=

=

+−=∂∂

∑

+

ρφφφφρ

φφ

φεφ

ρρρρ

ρρ

ρ

ρργρρ

ε

ε

cect

et

iL

ttLtt

Nkk

Njiji

tRXtRXHit

Mmti

mm

mti

m

mim

Lt

i

i

Λ

Λ

Λη

I. Knezevic and D. K. Ferry: Phys. Rev. E66(2003) 016131,Phys. Rev.A 69 (2004) 012104.

S. Miyashita and K. Saito: Physica B 329-333 (2003) 1142.

Explicit form of the autocorrelation

( ) ( )[ ] ( ) ikMmti

M

mm

M

ikik

ki

M

ikik

iecAttAAAtA +−

→

∑∑∑ =⎟⎠⎞

⎜⎝⎛= )1(0 )(

2

φρ ε

( )( )

( ) ( )

( ) ( ) ikMm

M

mmik

i

M

ik

ikMm

M

mmik

i

iM

ik

ti

cAi

cAi

edteAtAi

+−

+−

∞−∞ −

∑∑

∑∑∫

−−=

−−

=

)1(

)1(0

1

1

2

2

φωε

φωε

ωεω

( )[ ] [ ] ( )[ ]→→

== ∑ 0210 , )( tAcecttA Mmti

mm

i ρφφφφρ ε Λ

( ) ( )[ ]

( ) ( ) ( )

( ) ( ) ( )( ) ikMm

M

mmmik

i

M

ikAA

ikMm

M

mmik

i

M

ik

ti

tiAA

dcA

dAidtetAA

dtetAAAtAi

+−

+−

∞ −

∞ −

∑∑

∑∑∫

∫

−−

=

−−=

−=

)1(

)1(0

0

1ReIm

1

2

2

φωε

χ

φωε

χ

ω

ω

η

Line shape

Paramagnetic Resonance

( ) ( )ωωω

γβα

βωβω

−Φ=−

=Φ

++=

−=

−

∑

∑

eeI

SSSX

SHH

zii

yii

i

xii

i

zi

1

20

1.02 =λ

01.02 =λ

( )ωχ xx

ω

∑ ∑ ⋅+−=i i

jizi SSJSHH

1.02 =λ

01.02 =λ

( )ωχ xx

ω

Exchange narrowing

Dipole-dipole interaction

1.02 =λ

01.02 =λ

( )ωχ xx

ω

zii

yii

i

xii

i iji

zi

SSSX

SSJSHH

γβα ++=

+⋅+−=

∑

∑ ∑ DD

(Motional narrowing)Quantum narrowing effect

H. Onishi and SM: JPSJ 72(2003) 392

H = J 1+ α ui+1 − ui( )[ ]Si ⋅ Si+1i=1

N

∑ +1

2mpi

2 +k2

ui +1 − ui( )2⎡ ⎣

⎤ ⎦ i=1

N

∑

◆ effects of quantum lattice fluctuationbecomes small when m small

uniform

dim erizat ion

Spin-Peierls systems

0

0.05

0.1

0.15

0.2

0 0.2 0.4 0.6 0.8 1 1.2

magnetic susceptibility

adiabaticm=10000m=100m=1uniform

χ / N

T

N=64

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

magnetic excitation spectrum

m=10000m=100m=1uniform

q / 2 π

E(q)

N=64

-1 0 1 2 3 4 5 6 70

10

20

30

40

50

lattice position i

imag

inar

y tim

e τ

m=1, T=0.02, N=64

Origin of the adiabatic change

etc.))()(( ))()((

)(

44

22

2

+−+

−+

−−=

−+ SSCSSEhSSDH

yx

zz

S: even Large S (S=10) Mn12, Fe8

S: odd (S=1/2) V15 No anisotropy & Kramers doublet

Dzyloshinskii-Moriya interaction

SM, &. N. Nagaosa, Prog. Theor. Phys. 106 (2001) 533

( ) D ∑ ×=ij

jiij SSDH

Energy structure with DM

|3/2,3/2>

|3/2,-1/2>

|1/2,a>

|1/2,b>

No adiabatic changeat H=0

ο0=θ

ο45=θ

ο90=θ

H. De Raedt, SM, K. Michielsen & M. Machida: PRB 70 (2004) 064401

I.Chiorescu, W. Wernsdorfer, A. Mueller, SM, and B. Barbara: PRB 67 (2003) 020402

Anisotropy of DM interaction

Nontrivial coherence

V=0.01 V=0.001

K.Y. Choi, et al. PRL 96 (2006) 107202

Cu3

H. De Raedt, SM, K. Michielsen,M. Machida: PRB 70 (2004) 064401

Directionally independent energy gap due toHyperfine interaction

SM, H.de Raedt and K. Michielsen:Prog. Thor. Phys.. 110 (2003) No.11

hgSgSAH zNB

zB )'( σμμσ +−⋅−=

Triangle case

hgSgSAH zNB

zB )'( σμμσ +−⋅−=

M(t) from the ground state

2

0 )()( ftHGP Ψ=

π/)1log(2

)()(2

0

PvE

tHGP f

−−=Δ

Ψ=

Apparent LZS relation

Gap control using hidden symmetries

Quantum interferenceBerry phase

Nontrivial control( ) ( )( ) xxyxz ShSSCSSEDSH −++−+−= −+ 44222 )(


|'|2 mmxxxz hEShDSH −∝Δ⇒−−=

Transverse field

Non-monotonic gap due to Hx

( ) ( ) StHSSCSSEDSH yxz ⋅−++−+−= −+ )(2 44222

( )

( )( )

2,0

220 ,5.0

)1(2

0 ,5.0

22

2

222

±=Δ

−−+=

=+=+−−=

−+−−=

==

x

yxxx

xxx

xxyxz

M

SSHSDHCDE

SDSSHDS

SHSSSDHCDE

δδ

δ

Gaps open at crossing remain at (M,M+1) etc. => 2S crossings

E=0.50

E=0.40

Gap with the C term( ) ( ) StHSSCSSEDSH yxz ⋅−++−+−= −+ )(2 44222

Collapse of degeneracy?

P. Bruno: PRL 96 (2006) 117208

Temporal symmetry-breaking induced DM interaction

Charge transfer, Phonon,Orbital degree of freedom, etc.

NaV2O5 : charge fluctuation reduces the symmetry => virtual DM ESR Nojiri, et al.: JPSJ 69 (2000) 2291

Fe12 : configuration fluctuation reduces the symmetry => virtual DM M(H) H. Nakano and SM: JPSJ 71 (2002) 2580

SrCu2(BO3)2 : configuration fluctuation reduces the symmetry => Raman,ESRCepas and Zimann cond-mat 0401240SM & Ogasahara: JPSJ 72 (2003) 2350

Fluctuating DM interaction model

( ) ( )

[ ]

? 0tripletsinglet

0 ,

21

20

22112121

≠

=

==

++++×⋅+⋅=

iH

zz

e

xipx

xdd

pm

xkSSHSSdSSJH

η

Smooth magnetization process

ポテンシャル移動による粒子運搬における量子効果

Particle trap by potential well--quantum dynamics for particle

conveyance--

S. Miyashita, Conveyance of quantum particles by a moving potential-wellJ. Phys. Soc. Jpn. {¥bf 76} (2007) 104003.

Eigenstates in moving frame

Sweep velocity dependence (flat)fast

Slow

1/V

Trap probability

Sweep velocity dependence (carry-up)fast medium slow

Adiabatic energy level as a function of the potential well

Successive Landau-Zener scattering

Adiabatic trap vs. tunneling

Y. N. Demkov and V. I. Osherov, Sov. Phys.-JEPT 26 1211 (1968)

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Multiple free particles (fermion)N=0.1L

N=0.3L

Uniform acceleration

Adiabatic acceleration

0/

→=

aact

Thank you very much

quantum dynamics in low dimensional spin systems

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