Post on 28-Jul-2020
Fate of the Kinetic Ising Model
Basic question:
How does magnetic order emerge in a kinetic Ising model when an initially disordered state is cooled to very low temperature?
Sid Redner, Physics Department, Boston University, physics.bu.edu/~rednercollaborators: P. L. Krapivsky, V. Spirin, K. Barros, J. Olejarz + NSF support
The System
H = !!
!i,j"
!i!j !i = ±1
Ising Hamiltonian
Initial state: disordered, zero magnetizationFinal state: ???
Dynamics of the System
Pick a random spin and compare the outcome after reversing the spin
if ∆E < 0 flip spin
if ∆E > 0 don’t flip
if ∆E = 0 flip with prob. 1/2
time
+ + + + + + +_ _ _ _ __ space------+++++++---------------++++++++++----------------------++++++++++-----------------+++++++++----------+++++-------------------++++++++-----------+++++
Domain Wall Picture in 1d
ξ ∼ t1/2
------+++++++---------------++++++++++----------------------++++++++++-----------------+++++++++----------+++++-------------------++++++++-----------+++++
time
+ + + + + + +_ _ _ _ __ space
• no. interfaces: ∝ t−1/2 (Glauber, 1963)• time to ground state: T ∝ L2
Domain Wall Picture in 1d
t=4
Two Dimensions
coarsening of 256x256 system
t=4 t=16
Two Dimensions
coarsening of 256x256 system
t=4 t=16 t=64
Two Dimensions
coarsening of 256x256 system
t=4 t=16 t=64 t=256
coarsening of 256x256 system
Two Dimensions
Evolution to the Ground State
Evolution to Stripe State
Evolution to Diagonal Stripe State
Two DimensionsQuestion: what is the final state?
0.0 0.1 0.2 0.3 0.41/log2L
0.0
0.1
0.2
0.3
0.4
strip
e pr
obab
ility
Answer from simulations:ground state with probability ≈ 2/3stripe state with probability ≈ 1/3
Basic result: ground state is never reached!typical 20x20x20 system
Features:1. Swiss cheesy2. Zero average curvature
Three Dimensions
Basic result: ground state is never reached!typical 20x20x20 system
Features:1. Swiss cheesy2. Zero average curvature
Three Dimensions
Evolution from Antiferromagnetic State
Evolution from Antiferromagnetic State
blinker spin
Basic result: ground state is never reached!typical 20x20x20 system
Features:1. Swiss cheesy2. Zero average curvature3. Non-static
Three Dimensions (Olejarz, Krapvisky, & SR)
Blinker Evolution in Three Dimensions
Blinker Evolution in Three Dimensions
Genus Distribution
0 1 2 3 4g/〈g〉
0.0
0.2
0.4
0.6
0.8
1.0
P(g/〈g〉)
L=54
L=76
L=90
g=3
g=16
�g� ∼ L1.7
Slow Relaxation of Blinkers
deflated inflatedintermediate
synthetic blinker configuration
101
ln t10−2
10−1
100
S(t)
100 102 104 106 108t
10−6
10−4
10−2
100
S(t)
Slow Relaxation in 3d
46
8
10
14
20
3040
40
30
20
S(t) ∼ (ln t)−3 ?
slope -3
final state: the ground statecompletion time: L²domain length distribution still unsolved
d=1: almost, but not quite, completely soluble
d≥3: topologically complex final statetopological connection between energy & genusperpetually blinking spinsultra-slow relaxation whose functional form is unknownfinite temperaturecorner geometry: partial result
rich state space structure
d=2: final state: usually the ground stateconnection to percolation crossing probabilitiescompletion time: usually L², sometimes L finite temperaturecorner geometry: exactly soluble
ground state/metastable stripe states
3.5
Summary & Open Problems