1. Spin correlated dynamics on Bethe lattice Alexander Burin

2. Motivation: to study cooperative dynamics of interacting spins 2of 21

3. Three alternative models 1.Classical model of resonant window E 0 for electronic spins due to nuclear spins: |E| E 0  transition forbidden; P 0 ~E 0 /E d – probability of resonance a.Model on Bethe lattice with z>>1 neighbors b.Model of infinite interaction radius 2. Quantum model: Transverse field  <<E d causes transitions of interacting Ising spins; interaction is of infinite radius 3 of 21

4. Cooperative spin dynamics 4 of 21 Rules for spin dynamic a.All spins are initially random S i =  1/2 b.At every configuration of z neighbors the given neighbor is either resonant (open, probability P 0 <<1) or immobile c.Resonant spins can overturn changing the status of their neighbors

5. 5 of 21 Targets: a.What is the fraction of percolating spins, P *, involved into collective dynamics b.Do percolating spins form infinite cluster?

6. 6 of 21 Non-percolating spins (W * =1-P * ) on Bethe lattice W e is the probability that the given spin is non- percolating at one known non-percolating neighbor

7. 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 z  6 P * jumps to 1 at P 0 ~1/(ez) Infinite cluster of percolating spins is formed earlier at P 0 ~1/(3e 1/3 z)

8. 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)=, 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 correlations

9. Spin lattice with infinite radius: classical model 9 of 21 Rules for spin dynamic a.All spins are initially random S i =  1/2 b.At every configuration of z neighbors the given is either resonant (open, probability P 0 ~E 0 /(u D N 1/2 )<<1) or immobile c.Resonant spins can overturn possibly affecting the status of all N spins

10. Solution: Probability of an infinite number of evolution steps P  =1-W  10 of 21 N-k k WW

11. Results 11 of 21 near threshold

12. 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 manner

13. Quantum mechanical problem: transverse Ising model with infinite interaction radius 13 of 21

14. Qualitative study 14 of 21 Each spin is in the random field of neighbors and in the transverse field Spin is open (resonant) if Probability of resonance Cooperative dynamics exists when each configuration has around one open spin

15. 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

16. 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 

17. Localization transition 17 of 21 Im(  ) gets finite above transition point, so in the transition point one can ignore it in the denominator

18. Localization transition 18 of 21

19. Relaxation rate;  >U 0 /N 1/2 19 of 21

20. 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

21. Acknowledgement