Spin correlated dynamics on Bethe lattice

Alexander Burin

2

Motivation: to study cooperative dynamics of

interacting spins

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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)

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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

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)ln(

2 0

NN

Uc

Acknowledgement