Spin correlated dynamics on Bethe lattice

Alexander Burin

2

Motivation: to study cooperative dynamics of

interacting spins

of 21

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

3 of 21

Cooperative spin dynamics

4 of 21

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

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 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 neighbor

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 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

9 of 21

D

D

D

D

jjiji

jijiij

uNNuU

g

uuU

Ug

SU

SSUH

22

exp)(

22

exp)(

;

;

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

10 of 21

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

11 of 21

)exp(1)exp(1 0

PnPNPP

res

near threshold

.1 ,12

,1 ,0

2/)exp(1 22

resres

res

res

resresres

nnn

nP

PnPnPPnP

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

Quantum mechanical problem: transverse Ising model with infinite interaction radius

13 of 21

20

2

0

1

2exp

21)(

ˆ

Uu

UuP

SSSuH

ijij

N

i

xi

ji

zj

ziij

Qualitative study

14 of 21

Each spin is in the random field of neighbors

2

0

2

0 2exp

21)(,

NUu

UNPSu ij

i

N

j

zjiji

and in the transverse field xjS

Spin is open (resonant) if || i

Probability of resonance 0

0 2~

UNP

Cooperative dynamics exists when each configuration

has around one open spin NUNP c

00 ~1~

Bethe lattice approach

15 of 21

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

16 of 21

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

17 of 21

N

j jjji

ji

N

j jjii

EE

EE

122

2

1

2

ImReIm

Im

1

Im() gets finite above transition point, so in the transition point one can ignore it in the

denominator

N

j jji

ji EE1

22

ReIm

Im

Localization transition

18 of 21

N

j jji

ji EE1

22

ReIm

Im

NN

j jji

ji EE

tgdt

12

2

ReIm

exp)(Imexp

)0( )(1Imexp ttEEcgt ii

)ln(21)ln()0( 0

NNUNNg cc

Relaxation rate; >U0/N1/2

19 of 21

N

j jjji

ji EE1

222

ImReIm

Im

0

2

122

2

2ImReIm

ImUN

NEE

kN

j jjji

ji

(?)

(Efetov) exp:

cc

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

NNU

c

Acknowledgement