Nonequilibrium critical dynamics of the two-dimensional
Ising model quenched from a correlated initial state
Ferenc Iglói (Budapest)
in collaboration with
László Környei (Szeged, Budapest)
Michel Pleimling (Virginia Tech)
Leipzig, 27th November 2008
1
AGENDA
• Nonequilibrium relaxation
– Turban model
– ordered (Ti = 0) initial state
– disordered (Ti = ∞) initial state
– analogy with static
infinite criticality
• Models
semi-
• Results
– Relaxation from a critical initial state
– Relaxation from a 1st -order initial state
– Ising model
– Baxter-Wu (BW) model
• Conclusions
2
Nonequilibrium relaxation
• prepare the system in an initial state for t < 0
• quench to the critical point (Tc ) at t = 0
• let evolve with the dynamical rules at Tc for t > 0
• measure
– relaxation of the magnetization:
m(t) = hσ (t)i
– autocorrelation function:
G(t, s) = hσ (s)σ (t)i
s: waiting time t: observation time
in equilibrium:
G(t, s) ∼ (t − s)−2x/z
3
time dependence for t > 0
• ordered initial state (Ti = 0)
relaxation involves only equilibrium critical exponents
position dependence for y > 0
• magnetization profile in a semiinfinite system y ≥ 0 with fixed
boundary condition
m(y) ∼ y−x ,
m(t) ∼ t −x/z ,
Ti = 0
hs = ∞
.
x = β /ν: anomalous dimension of
the magnetization
.
z: dynamical exponent
.
.
correlation function:
.
autocorrelation function:
−2x/z
G(t, s) ∼ (t − s)
g(t/s),
Ti = 0
C(y1 , y2 ) ∼ (y1 − y2 )−2x g(y1 /y2 ), hs = ∞
4
time dependence for t > 0
• disordered initial state (Ti = ∞)
m(t) ∼ mit θ ,
t < ti
position dependence for y > 0
• magnetization profile in a semiinfinite system with small surface field: hs 1
m(y) ∼ yxs −x ,
θ : initial slip exponent
mi → 0: initial magnetization
−z/xi
ti ∼ mi
−1/xs
xi = θ z + x: anomalous dimension
of mi
G(t, s) ∼ t −λ /z ,
t s,
λ = d − θ z = d − xi + x
hs → 0 : surface field
ys ∼ hs
: initial time
Ti = ∞
y < ys
: surface region
xs : anomalous dimension of ms
.
⊥
C(y1 , y2 ) ∼ y−η
1 ,
y1 y2
η⊥ = xs + x
5
Power-law correlations in the initial state
previous studies
• 2d XY model: Ti < Tc < TKT (Berthier, Holdsworth, Sellitto) (Abriet, Karevski)
x(Tc ) → x(Tc ) − x(Ti )
• spherical model (Picone, Henkel)
Our aim is to study nonequilibrium dynamics after a
sudden change of the form of the interaction
Experimental realization: atoms in optical lattices.
6
Models
• 2d Ising model → dynamics at Tc
HI = − ∑i, j J (σi, j σi, j+1 + σi, j σi+1, j )
critical point: sinh(2J/kTc ) = 1
• Baxter-Wu model
HBW = − ∑ JBW (σi, j σi−1, j σi, j−1
i, j
+σi, j σi+1, j σi, j+1 )
critical point:
• Turban model
Hn =
− ∑i, j J2 σi, j σi, j+1 + Jn ∏n−1
k=0 σi+k, j
critical point:
sinh(2J2 /kTc ) sinh(2Jn /kTc ) = 1
4-state Potts universality
– n = 3, 2nd -order
4-state Potts universality
.
– n = 4, 1st -order
sinh(2JBW /kTc ) = 1
7
Relaxation from second-order transition points
[L. Környei, M. Pleimling, F. I.,Phys. Rev. E77, 011127 (2008)]
BW model
n = 3 Turban model
8
Relaxation from a BW critical state
-1.0
0.5
0.4
-2.0
ln(-m)
m(t)
0.3
0.2
0.1
0.0
L=60
L=120
L=240
-4.0
-0.1
-0.2
-3.0
(a)
0
2
(b)
4
ln t
6
8
-5.0
0
2
4
6
8
10
ln t
mi = 0.33, 0.16, 0.08, 0.04, 0.
mi = 0.
(from top to bottom)
θ = 0.18(1)
9
ln(G)+t/τL
ln(G)
L=60
L=120
L=240
-1.0
-2.0
-3.0
-1.5
-2.0
-4.0
-2.5
(a)
0
1000 2000 3000 4000
t
(b)
0
2
4
6
8
ln t
G(t) = At −λ /z exp(−t/τL )
G̃(t) = G(t) exp(t/τL )
λ /z = 0.18(1)
10
Relaxation from an n = 3 critical state
-1.0
-0.5
ln(G)+t/τL
ln(-m)
-1.5
-2.0
-2.5
-3.0
L=60
L=120
L=240
-3.5
-1.0
-1.5
-2.0
(a)
-4.0
0
2
(b)
4
6
ln t
8
10
-2.5
0
2
4
6
8
ln t
magnetization
G̃(t) = G(t) exp(t/τL )
θ = 0.18(1)
λ /z = 0.165(10)
11
Relaxation from a first-order transition point
n = 4 Turban model T < Tc
→ 8-fold degenerate state
• latent heat: ∆/kTc = 0.146(3)
• jump in the magnetization: mc = 0.769(6)
Reminder: nonequilibrium relaxation from Ti = ∞
[M. Pleimling, F. I.,Europhys. Lett. 79, 56002 (2007)]
Autocorrelation
.
.
0.87
(a)
(b)
stretched exponential approach of mc
0.1
C(t)
C(t)-mc
0.83
0.79
C(t) − mc ∼ exp(−bt a )
a = 0.198(3)
L=640
data
fit
L=60
0.75 2
10
4
10
t
6
10
0.01
10
100
t
12
1000
Relaxation from mi > 0
.
.
1.0
1.0
(b)
0.8
0.8
0.6
0.6
m(t)
m(t)
(a)
0.4
0.2
0.0
mi > m∗ = 0.5
0.4
mi < m∗ = 0.5
0.2
0
0.0
1000 2000 3000 4000 5000
0
-5
t
m(t) ∼ t −d/z
m(t) → mc /2
ln t
mi=0.10
mi=0.08
mi=0.06
mi=0.04
0.4
-4.5
-5.0
ln m(t)
0.6
m(t)/mi
m(t) → 0
mi < m∗ = 0.5
5
m(t) → mc
-4.0
0.8
-5.5
-6.0
-6.5
0.2
(c)
0.0
.
0
1
-7.0
2
3 4
ln t
5
6
7
-7.5
7.5
(d)
8
8.5
ln t
discontinuous equilibrium transition
Wetting in time
continuous nonequilibrium transition
13
Autocorrelation of the 2d Ising model
starting with n = 4 initial state
0.0
L=120
L=180
L=240
ln(G)+t/τL
ln(G)
-4.0
-6.0
-8.0
0
-4.0
-6.0
(a)
-10.0
-2.0
10000
t
20000
(b)
0
2
4
6
8
10
ln t
G(t) = At −λ /z exp(−t/τL )
λ /z = 0.475(10)
→ λ = d/2
14
Conclusions
Ising
BW
n=3
n=4
BW
n=3
n=4
(a)
x
1/8
1/8
1/8
0.(a)
z
2.17
2.29(1)
2.3(1)
2.05(10)
: discontinuity fixed-point value
λ /z
0.74(2)
1.13(6)
0.98(2)
∞(b)
0.17(1)
0.165(10)
0.475(10)
(b)
θ
0.187
−0.186(2)
−0.03(1)
−1.00(5)
0.18(1)
0.18(1)
: stretched exponential decay
• a critical initial state has an influence on the nonequilibrium exponents
• initial states in the same universality class have the same effect
• analogy with static critical behavior at an interface separating two critical systems
• if the initial state corresponds to a 1st order transition then λ = d/2
15
Acknowledgment
László Környei (Szeged, Budapest)
Michel Pleimling (Virginia Tech)
16
• 2d random-field Ising model
HRF = HI − ∑i (H + hi, j )σi, j
– homogeneous domains of size:
lb ∼ exp(A/∆2 ):
length
hi, j random field with Gaussian distribution:
h 2i
h
1
P(hi, j ) = √2πh2 exp − 2hi, 2j
breaking-up
H
for H = 0:
P↑
• no magnetic long-range order
• geometrical clusters (with parallel
spins)
(1)
F↑
(3)
DC
(2)
F↓
D
P↓
– percolating for
∆ = J/h < ∆c ≈ 1.65
17
Relaxation from the RFIM initial state
mi = 0.25,
∆ = 1.7,
lattice : 200 × 200
• 1) cluster dissolution → lntmin ∼ ln lb ∼ 1/∆2
−z/xi
• 2) domain growth → ti ∼ mi
• 3) equilibrium relaxation
18
3
ln tmin
(a)
0.045
−1
2
1
ln G(t,0)
m(t)
(b)
0
0.2 0.3 0.4 0.5 0.6
2
1/∆
0.20
−3
−1.5
−5
m(t)
0.040
0.10
−7
0.05
0.035
0
10
20
t
30
0.03 2
10
−4.5
(c)
2
5*10
3
2*10
t
a) ∆ = 1.4, 1.5, 1.6, 1.7, 1.8, 2.0, 2.2, 2.6, 3.0
bottom to top)
−9
∆=1.4
ln G(t,0)
0.050
0
ln t
6
2
9
4
6
8
ln t
(from
∆ = 1.4, 1.6, 1.8, 2.0, 4.0 (from top to bottom),
green: Ti = ∞
c) ∆ = 1.4, 1.6, 1.8, 2.0, 2.2, 2.6, 3.0 (from bottom to
top), dashed: Ti = ∞
inset: L = 128, 200, 500 (from bottom to top)
same nonequilibrium exponents as for Ti = ∞
19