Alexander Burin Motivation to study cooperative dynamics of interacting spins 2 of 21 Three alternative models Classical model of resonant window E 0 for electronic spins due to nuclear spins ID: 363322
Download Presentation The PPT/PDF document "Spin correlated dynamics on Bethe lattic..." is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
Slide1
Spin correlated dynamics on Bethe lattice
Alexander BurinSlide2
Motivation:
to study cooperative dynamics of interacting spins
2
of 21Slide3
Three alternative models
Classical model of resonant window
E
0 for electronic spins due to nuclear spins: |E|<E0 transition allowed, E>E0 transition forbidden; P0~E
0
/E
d
– probability of resonance
Model on Bethe lattice with
z
>>1
neighbors
Model of infinite
interaction radius
2. Quantum model: Transverse field
<<Ed causes transitions of interacting Ising
spins; interaction is of infinite radius
3
of 21Slide4
Cooperative spin dynamics
4
of 21
Rules for spin dynamicAll spins are initially random Si = 1/2At every configuration of z neighbors the given neighbor is either resonant (open, probability P0<<1) or immobile
Resonant spins can overturn changing the status of their neighborsSlide5
5 of 21
Targets:
What is the fraction of percolating spins, P
*, involved into collective dynamicsDo percolating spins form infinite cluster?Slide6
6 of 21
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 neighborSlide7
Solution
for percolating spin density P
*
7 of 21 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
P
0
~1/(
ez
)
Infinite cluster of percolating spins is formed earlier at
P
0
~1/(3e
1/3z) Slide8
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 P
0
~0.09
P
c2
0.07 in the Bethe lattice problem; difference due to correlationsSlide9
Spin lattice with infinite radius: classical model
9
of 21
Rules for spin dynamicAll spins are initially random Si = 1/2At every configuration of z neighbors the given is either resonant (open, probability P0~E0
/(u
D
N
1/2
)<<1
) or immobile
Resonant spins can overturn
possibly affecting
the status of all
N spinsSlide10
Solution: Probability of an infinite number of evolution steps P
=1-W
10 of 21
N-k
k
W
Slide11
Results
11
of 21
near threshold Slide12
Summary of classical approach
12
of 21
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 mannerSlide13
Quantum mechanical problem: transverse
Ising model with infinite interaction radius
13 of 21Slide14
Qualitative study
14
of 21
Each spin is in the random field of neighborsand in the transverse field Spin is open (resonant) if
Probability of resonance
Cooperative dynamics exists when each configuration has around one open spinSlide15
Bethe lattice approach
15
of 21
Interference of different paths In resonant situation i~ or
j
~
so only one term is important because
U
ij
>>~
U
ij
/N
1/2
Slide16
Self-consistent theory of localization
16
of 21
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 Slide17
Localization transition
17
of 21
Im() gets finite above transition point, so in the transition point one can ignore it in the denominatorSlide18
Localization transition
18
of 21Slide19
Relaxation rate;
>U0
/N1/2
19 of 21Slide20
Conclusion
C
lassical 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. 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 21Slide21
Acknowledgement