quantum dynamics in low dimensional spin systems
TRANSCRIPT
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
W.Wernsdorfer & R. Sessoli:Science 284 (1999) 133
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 )(
W.Wernsdorfer & R. Sessoli:Science 284 (1999) 133
|'|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
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Multiple free particles (fermion)N=0.1L
N=0.3L
Uniform acceleration
Adiabatic acceleration
0/
→=
aact
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