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Three alternative models
1. Classical model of resonant window E0 for electronic spins due to nuclear spins: |E|<E0 transition
allowed, E>E0 transition forbidden; P0~E0/Ed – probability of resonance
a. Model on Bethe lattice with z>>1 neighbors
b. Model of infinite interaction radius
2. Quantum model: Transverse field <<Ed causes transitions of interacting Ising spins; interaction is of
infinite radius
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Cooperative spin dynamics
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Rules for spin dynamic
a. All spins are initially random Si = 1/2
b. At every configuration of z neighbors the given
neighbor is either resonant (open, probability
P0<<1) or immobile
c. Resonant spins can overturn changing the
status of their neighbors
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Targets:
a. What is the fraction of percolating spins, P*, involved into collective dynamics
b. Do percolating spins form infinite cluster?
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Non-percolating spins (W*=1-P*) on Bethe lattice
We is the probability that the given spin is non-percolating at one known non-percolating neighbor
Solution for percolating spin density P*
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For z<6 the density of percolating spins, P*,
continuously increases to 1 with increasing
the density of open spins.
For z6 P* jumps to 1 at P0~1/(ez)
Infinite cluster of percolating spins is
formed earlier at P0~1/(3e1/3
z)
8 of 21
Comparison to Monte-Carlo simulations in 2-d
Problem: dynamic percolation for randomly interacting spins with z=4, or 8 neighbors
Parameter of interest K(t)=<S(t)S(0)>, t, W*=1-P*K()
Results: continuous decrease of W* to 0 for z=4, discontinuous vanishing of W* at P0~0.09
Pc20.07 in the Bethe lattice problem; difference due to correlations
Spin lattice with infinite radius: classical model
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D
D
D
D
jjiji
jijiij
uN
NuU
g
u
uU
Ug
SU
SSUH
2
2exp
)(
2
2exp
)(
;
;
2
2
2
2
,
Rules for spin dynamic
a. All spins are initially random Si = 1/2
b. At every configuration of z neighbors the given is
either resonant (open, probability
P0~E0/(uDN1/2
)<<1) or immobile
c. Resonant spins can overturn possibly affecting the
status of all N spins
Solution: Probability of an infinite number of evolution steps P=1-W
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N-k k
W
kk0 W)W(1 kN
0W
k)!(Nk!
N!
)exp(1)1(1
))1(1(
00
0
PNPPPP
WPWN
N
N
1k
kk0
kN0 W)W(1W
k)!(Nk!
N!W
Results
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)exp(1
)exp(1 0
Pn
PNPP
res
near threshold
.1 ,1
2
,1 ,0
2/)exp(1 22
resres
res
res
resresres
nn
nn
P
PnPnPPnP
Summary of classical approach
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Exact solution on Bethe lattice shows that at small resonant window there is no cooperative dynamics;
increase of resonant window turns it on in either continuous or discontinuous manner
Quantum mechanical problem: transverse Ising model with infinite interaction radius
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20
2
0
1
2exp
2
1)(
ˆ
U
u
UuP
SSSuH
ijij
N
i
xi
ji
zj
ziij
Qualitative study
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Each spin is in the random field of neighbors
2
0
2
0 2exp
2
1)(,
NU
u
UNPSu ij
i
N
j
zjiji
and in the transverse field
xjS
Spin is open (resonant) if || i
Probability of resonance 0
02
~UN
P
Cooperative dynamics exists when each configuration
has around one open spin N
UNP c
00 ~1~
Bethe lattice approach
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Interference of different paths zj
zi
zj
zi -SSSS , ,
)()(
22
ijjijijjii UU
In resonant situation i~ or j~ so only one term is important because Uij>>~Uij/N1/2
zj
zj
zi
zi -SSSS , z
izi
zj
zj -SSSS ,
zi
zi SS
Self-consistent theory of localization
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N
j jjii
iiii EEEEG
1
2 1 ,
1
Abou-Chacra, Anderson and Thouless (1973)
i is some Ising spin state, j enumerates all N states formed by single spin overturn from
this state caused by the field
Localization transition
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N
j jjji
ji
N
j jjii
EE
EE
122
2
1
2
ImRe
ImIm
1
Im() gets finite above transition point, so in the transition point one can ignore it in the
denominator
N
j jji
ji
EE12
2
Re
ImIm
Localization transition
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N
j jji
ji
EE12
2
Re
ImIm
NN
j jji
ji
EEtgdt
12
2
Re
Imexp)(Imexp
)0( )(1Imexp ttEEcgt ii
)ln(
21)ln()0( 0
NN
UNNg cc
Relaxation rate; >U0/N1/2
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N
j jjji
ji
EE122
2
ImRe
ImIm
0
2
122
2
2ImRe
ImIm
UNN
EEk
N
j jjji
ji
(?)
(Efetov) exp:
c
c
Ck
Conclusion
(1) Classical cooperative dynamics of interacting spins is solved exactly on Bethe lattice and for the
infinite interaction radius of spins. At small resonant window there is no cooperative dynamics. It
turns on in discontinuous manner on Bethe lattice with large coordination number and continuously
for small coordination number in agreement with Monte-Carlo simulations in 2-d.
(2) Transverse Ising model with infinite interaction radius is resolved using self-consistent theory of
localization on Bethe lattice. There exists sharp localization-delocalization transition at transverse field
20 of 21
)ln(
2 0
NN
Uc