/
Spin correlated dynamics on Bethe lattice Spin correlated dynamics on Bethe lattice

Spin correlated dynamics on Bethe lattice - PowerPoint Presentation

natalia-silvester
natalia-silvester . @natalia-silvester
Follow
415 views
Uploaded On 2016-06-15

Spin correlated dynamics on Bethe lattice - PPT Presentation

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

spin spins resonant lattice spins spin lattice resonant infinite bethe percolating dynamics transition cooperative probability radius localization transverse interaction

Share:

Link:

Embed:

Download Presentation from below link

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.


Presentation Transcript

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 z6 P* jumps to 1

at

P

0

~1/(

ez

)

Infinite cluster of percolating spins is formed earlier at

P

0

~1/(3e

1/3z) 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