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Quenches across phase transitions :the density of topological defects
Leticia F. Cugliandolo
Université Pierre et Marie Curie – Paris VI
leticia@lpthe.jussieu.fr
www.lpthe.jussieu.fr/ leticia/seminars
In collaboration with
Jeferson Arenzon, Giulio Biroli, Thibault Blanchard, Alan Bray, Federico Corberi,
Ingo Dierking, Asja Jelic, Michikazu Kobayashi, Marcos-Paulo Loureiro, Marco
Picco, Yoann Sarrazin, and Alberto Sicilia.
MECO 39, Coventry UK, April 2014
1
Phenomenon
A talk on a very well-known problem
Dynamics following a quench through a
second order phase transition
• The equilibrium phases are known on both sides of the transition.
• The dynamic mechanism is understood.
Aim : full geometric characterization of the structure at a given time
after the quench.
2
Why is it interesting ?• Learn about real-space configurations:
statistics and morphology of ordered structures.
• Metastability.
• Dynamic scaling hypothesis.
Proven only for one-dimensional systems and mean-field models(e.g. large N limit of a field theory or fully connected spin models).
Can this hypothesis be proven beyond these cases?
Finite size corrections?
• Analytic tools for two-dimensional systems:
Conformal field theory and stochastic Loewner evolution.
Can they be extended to dynamical systems?
3
Settingwith g = T/J the control and ⟨ϕ⟩ the order parameter
Non-conserved order parameter (NCOP, model A) ϕ(t, g) = ct
e.g., development of magnetization in a ferromagnet after a quench.
Locally conserved order parameter (COP, model B) ϕ(t, g) = ct
e.g., phase separation in binary alloys.
4
Instantaneous quenchInitial conditions
Equilibrium at infinite temperature initial condition, T0 → ∞.
The spins take ±1 values with probability 1/2.
Site occupation variable : ni = (si + 1)/2 with ni = 1, 0.
From a site percolation perspective :
pc = 0.65 Kagome lattice.
pc = 0.59 Square lattice.
pc = 0.55 Bow-tie lattice.
pc = 0.5 Triangular lattice.
The triangular lattice is at critical percolations. The others are not.
5
2d square IMInitial distribution of hull-enclosed areas
1e-08
1e-06
1e-04
1e-02
1e00
1e00 1e+01 1e+02 1e+03
n h(A)
A
2 hull-enclosed areas
r1
r2
A1 = πr21
A2 = πr22
nh(A) = limL→∞Nh(A,L) = 2ch A−2 e−A/Ac limp→pc Ac = 0
see, e.g. Stauffer & Aharony 94 ; 2ch Cardy & Ziff 03
6
2d square IMInstantaneous quench to zero temperature
nh(A)
A
In only 8 MCs the distribution got the power-law tail, A−2, with a prefactor
2ch that seem to be the ones of critical percolation.
NB : The percolating clusters have been excluded from the histogram.
nh(A, t) = limL→∞Nh(A,L, t) Arenzon, Bray, LFC & Sicilia 07
7
2d square IMInstantaneous quench to zero temperature
nh(A)
A
How does this arrive ? Is this really critical percolation?
8
2d square IM
t=0.0
9
2d square IM
t=0.57533
10
2d square IM
t=0.94844
11
2d square IM
t=2.00847
12
2d square IM
t=2.57898
13
2d square IM
t=3.99211
14
2d square IM
t=6.58423
15
2d square IM
t=7.46144
16
2d square IMThe percolating structure was decided at tp ≃ 8 MCs
t=7.46144 t=128.0
Banchard, Corberi, LFC & Picco 14
17
Instantaneous quenchInitial condition
Infinite temperature initial condition, T0 → ∞.
The spins take ±1 values with probability 1/2.
ni = (si + 1)/2 with ni = 1, 0.
From a site percolation perspective :
pc = 0.65 Kagome lattice.
pc = 0.59 Square lattice.
pc = 0.55 Bow-tie lattice.
pc = 0.5 Triangular lattice.
Is the system in a percolating stable configuration at t = 0?
18
Triangular 2dIM
t=0.0
19
Triangular 2dIM
t=0.13657
20
Triangular 2dIM
t=0.14539
21
Triangular 2dIM
t=0.15487
22
Triangular 2dIM
t=0.17523
23
Triangular 2dIM
t=0.19880
24
Triangular 2dIM
t=0.21114
25
Triangular 2dIM
t=0.22523
26
Triangular 2dIM
t=0.23981
27
Triangular 2dIM
t=0.28931
28
Triangular 2dIM
t=0.54055
29
Triangular 2dIM
t=0.57534
30
Triangular 2dIM
t=0.78626
31
Triangular 2dIM
t=0.83728
32
Triangular 2dIM
t=2.92228
33
Triangular 2dIMThe percolating structure was decided at tp ≃ 3 MCs
t=2.92228 t=128.0
Banchard, Corberi, LFC & Picco 14
34
Determination of tp(L)Full distribution of hull-enclosed areas
10-14
10-10
10-6
10-2
100 102 104 106
N (A
,t)
A
t=01248
163264
10-1410-1010-610-2
100 102 104 106
Main panel: quench from T0 → ∞. Insert: quench from T0 = Tc.
35
Determination of tp(L)Full distribution of hull-enclosed areas
10-14
10-10
10-6
10-2
100
102
104
106
N (
A,t
)
A
t=01248
163264
10-14
10-10
10-6
10-2
100
102
104
106
The stationary bump is given by the percolating clusters
36
Determination of tp(L)Finite size scaling of the bump
Take A to be the hull-enclosed area or the domain area.
At critical percolation, finite size scaling of the number density of areas
N(A,L) = 2ch A−τ +Np(A/L
D)
with D = d/(τ − 1) the fractal dimension of the percolating clusters.
Stauffer & Aharony 94
For hull-enclosed areas τ = 2 and D = 2
For domain areas τ = 187/91 ≃ 2.05 and D ≃ 1.9
NB the corresponding exponents for critical Ising conditions are very close to these values.
37
Determination of tp(L)Number density of the areas of percolated clusters
0
2
4
6
0.4 0.6 0.8 1
Aτ N
(A,t)
A/LD
t=2
0
2
4
6
0.4 0.6 0.8 1
Aτ N
(A,t)
A/LD
t=4
L=160L=640
L=2560
0
2
4
6
0.4 0.6 0.8 1
Aτ N
(A,t)
A/LD
t=8
0
2
4
6
0.4 0.6 0.8 1
Aτ N
(A,t)
A/LD
t=16 0 2 4 6
0.4 0.6 0.8 1
At t = 16 MCs the bump converged to a stationary form that satisfies the finite
size scaling of critical site percolation. τ = 2.05 and D = 1.9
Insert: failure of collapse if critical Ising exponents are used.
38
Determination of tp(L)The overlap
Quench a system from T0 → ∞ to T = 0 at t = 0.
Let it evolve at T = 0 until tw.
Make a copy of the instantaneous configuration, σi(tw) = si(tw).
Let the two clones evolve with different thermal noises.
Compute the time-dependent overlap
qtw(t, L) =1Ld
∑Ld
i=1⟨si(t)σi(t)⟩
If tw < tp(L) limt≫tw qtw(t, L) = 0
If tw > tp(L) limt≫tw qtw(t, L) > 0
39
Determination of tp(L)The overlap
limt→∞ qtw(L)(t, L) should reach a constant independent of L
0.7
0.8
0.85
20 40 80 160
qt w
L
(a)
L0.45L0.50
tw=L0.55
20 40 80 160
L
(b)
20 40 80 160
L
(c)
L0.30L0.32L0.33
tw=L0.35
Square FBC Kagome FBC Triangular PBC
tp(L) ≃ L0.5 tp(L) ≃ L0.33
40
Determination of tp(L)The overlap
limt→∞ qtw(L)(t, L) should reach a constant independent of L
0.7
0.8
0.85
20 40 80 160
qt w
L
(a)
L0.45
L0.50
tw=L0.55
20 40 80 160
L
(b)
20 40 80 160
L
(c)
L0.30
L0.32
L0.33
tw=L0.35
Square FBC Kagome FBC Triangular PBC
tp(L) ≃ L0.5 tp(L) ≃ L0.33
41
Dynamic scaling
At late times there is a single length-scale, the typical radius of the do-
mains R(t, g), such that the domain structure is (in statistical sense)
independent of time when lengths are scaled by R(t, g), e.g.
C(r, t) ≡ ⟨ si(t)sj(t) ⟩||xi−xj |=r ∼ ⟨ϕ⟩2eq(g) f(
r
R(t, g)
),
C(t, tw) ≡ ⟨ si(t)si(tw) ⟩ ∼ ⟨ϕ⟩2eq(g) fc(
R(t, g)
R(tw, g)
),
etc. when r ≫ ξ(g), t, tw ≫ t0 and C < ⟨ϕ⟩2eq(g).
Review Bray 94
Does tp(L) have an effect on the dynamic scaling ?
42
Dynamic scalingInfinitely rapid quench from T0 → ∞ to T < Tc
⟨ϕ(t)⟩ = 0 Cceq(r) ≃ e−r/ξeq
0
0.2
0.4
0.6
0.8
1
0 20 40 60 80 100 120
y
x
t = 4t = 8
t = 16t = 32t = 64
t = 128t = 256t = 512
t = 1024
• Coloured curves are C(r, t) for different times after the quench.
• The growing length is R(t) ≃ t1/zd with zd = 2
• R(t) is the averaged linear size of the domains.
Review Bray 94
43
Dynamic scalingInfinitely rapid quench from T0 → ∞ to T < Tc
Scaling regime a ≪ r ≪ L, r ≃ R(t) ≃ t1/zd
C(r, t) ≃ m2eq fc
(r
R(t)
)Scaling looks perfect
r/R(t)
44
Dynamic scalingInfinitely rapid quench from T0 → ∞ to T < Tc
Scaling regime a ≪ r ≪ L, r ≃ R(t) ≃ t1/zd
C(r, t) ≃ m2eq fc
(r
R(t)
)Scaling looks perfect
0
0.2
0.4
0.6
0.8
1
1 2 3
C(r,t)
r/R(t)
100
101
102
103
100
101
102
103
R2
t
but · · ·
45
Correction to scalingLinear-Log scale, zoom over C ≲ 0.1
C(r, t, L)
rR(t)
f(
rR(t)
)g(
rR(t)
, L(L)R(t)
)with L(L) = Lαp/zd and
αp = 0.5 for the square & Kagome, and αp = 0.33 for the triangular lattices.
46
Correction to scalingLinear-Log scale, zoom over C ≲ 0.1
C(r, t, L)
rR(t)
f(
rR(t)
)g(
rR(t)
, L(L)R(t)
)The second argument is important for large r where r ≃ R(t) ≃ L(L).Conventional scaling is recovered for L(L)/R(t) → ∞
47
First summaryEarly approach to percolation
• tp(L) ≃ Lαp with, we conjectured, αp = c/zd where c is the (pos-
sibly averaged) coordination number of the lattice and zd is the dyna-
mic exponent.
To be checked in:
systems with conserved order parameter.
systems with weak quenched disorder and interplay with thermal ef-
fects ; e.g., the 2d random bond Ising model.
systems in higher dimensions ; e.g., 3d Ising model.
• Corrections to dynamic scaling.
Blanchard, Corberi, LFC & Picco 14
• Responsible for metastability at T = 0.
48
Stripe blocked statesConsequence : fate of the kinetic Ising model
There exist a plethora of metastable states, such as straight stripes in two
dimensions and more bizarre gyroid or “plumber’s nightmare” states in three
dimensions which are infinitely long lived at zero temperature. Once the system
falls into such a state, the only escape route is via energy-raising spin-flips.
Since such events do not occur at zero temperature, there is no escape to the
ground state.
From Olejarz, Krapivsky & Redner 12
On the statistics of zero temperature metastable states in the 2d and
3d Ising and Potts models quenched from T0 → ∞ or T0 = Tc :
Barros, Olejarz, Krapivsky & Redner 09-14 ; Blanchard & Picco 12-14
49
Artificial spin-ice16 vertex model snapshots
Growth of stripes
Quench to a large a value: black & white vertices energetically favored.
Levis & LFC 11-13
50
Statistics & geometryNon-percolating structures
Basic question : what does R(t, g) really mean ?
Can we compute, e.g. the number of domains with size A at time t per
unit area of the system (L2) ?
An instantaneous configuration t = 32 MCs, T = 1.5
Domains Walls
So typical means...
51
Hull enclosed areasTime evolution
dA
dt= −λ
52
Curvature driven dynamics
(1) the clean non-conserved problem is purely curvature-driven
v(x, t) = − λ2π
κ(x, t)n(x, t)
at T = 0 with κ(x, t) the local geodesic (mean) curvature.
(2) Walls move independently of each other Allen & Cahn 79
The area enclosed by an interface follows dA/dt =∮v ∧ dℓ =∮
vdℓ = − λ2π
∮κdℓ. The Gauss-Bonnet theorem implies
∮κdℓ =
2π for a planar 2d manifold with no holes Thus, all hull-enclosed areas
decrease at constant rate independently of their form: dA/dt = −λ
(3) Domain wall ‘roughening’ at T > 0 is captured by λ(T )
53
StatisticsThe hull-enclosed area pdf evolution
Since all hull-enclosed areas decrease independently of one another, the
number of hulls-enclosed areas A at time t is given by
nh(A, t) =∫dA0 δ(A− A(t, A0))nh(A0, t0)
where nh(A0, t0) is the initial distribution and A(t, A0) is the area at
time t given that the area at the initial time t0 was A0.
For convenience we normalize by the area of the full system, L2.
Thus, we need the time-evolution A(t, A0)
and the initial pdf nh(A0, t0).
54
StatisticsThe hull-enclosed area pdf evolution
• dAdt
= −λ implies A(t, A0) = A0 − λ(t− t0)
• Quench from an infinite temperature ⇔ random initial condition,
critical site percolation after tp ≃ Lαp
Blanchard, Corberi, LFC & Picco 14
• Quench from equilibrium at Tc : critical Ising clusters.
The equilibrium hull-enclosed area distribution at pc and Tc are
nh(A0, t0) ∼(2)chA2
0
, ch =1
8π√3
(a2 ≪ A0 ≪ L2)
Conformal field theory, scaling & numerical checks Cardy & Ziff 03.
55
StatisticsThe predictions
nh(A, t) ≡(2)ch
(A+ λt)2nd(A, t) ≈
(2)cd (λdt)τ−2
(A+ λdt)τ
in the long time limit and for large areas such that a2 ≪ A ≪ L2.
We derived the expected scaling forms, as R(t) = (λt)1/2 :
R4(t)nh(A, t) = fh
(A
R2(t)
)nd(A, t) ≈ (λdt)
−2 fd
(Aλdt
).
The new parameters are cd = ch +O(c2h) and λd = λ+O(ch). Moreover,
the sum rules, Nh(t) = Nd(t) and∫dA And(A, t) = 1 relate ch to τ (or τ ′) !
Arenzon, Bray, LFC & Sicilia 07
56
Simulations vs. theoryNumber density of (finite) hull-enclosed areas per unit area
T = 0 dynamics after a quench from T0 → ∞L = 103 ; 2× 103 samples
10-12
10-10
10-8
10-6
10-4
100 101 102 103 104 105
n(A,
t)
A
t = 4 8
16 32 64
128 256
Solid lines
analytical prediction :
nh(A, t) ≡2ch
(A+ λt)2
57
Simulations vs. theoryNumber density of (finite) hull-enclosed areas per unit area
T = 0 dynamics after a quench from T0 → ∞ vs. T0 = Tc
10-4
10-3
10-2
10-1
10-2 10-1 100 101
(h t)
2 nh(
A,t)
A / h t
t = 4 8
16 32 64
128 256
Solid lines
analytical prediction :
nh(A, t) ≡(2)ch
(A+ λt)2
above T0 → ∞below T0 = Tc
Role of initial condition : Difference between ch and 2ch.
58
Simulations vs. theoryNumber density of (finite) hull-enclosed areas per unit area
T > 0 dynamics after a quench from equilibrium at T0 = Tc
10-8
10-6
10-4
10-2
100 101 102 103
n(A,
t)
A
Equil.T = 0.5
1.5 2.0
Dashed lines: thermal equilibrium distributions
59
Experiments2d liquid crystal after a rapid quench
Domains of two chiralities
• Global constraint : total chirality is conserved.
• Microscopic mechanism for domain growth is not known.
• Sample provided by M. B. Ros (Zaragoza)
• Experimental set-up due to I. Dierking (Manchester).
60
ExperimentsGrowth of chiral domains in a slice of a liquid crystal
10-1210-1010-810-610-410-2100
100 101 102 103 104 105
n h(A,t)
A
10-12
10-10
10-8
10-6
10-4
10-2
100 101 102 103 104 105
n h(A
,t)
A
t = 100t = 200t = 400
Simulations Experiments
Importance of finite visual field.
Sicilia, Arenzon, Bray, Dierking, LFC et al 08
61
GeometryHull enclosed areas and their perimeters
T = 0 dynamics after a quench from equilibrium at T0 = Tc
100
101
102
103
104
101 102 103
A
p
10-2
100
102
104
100 101 102 103 104
A h/t
p/t1/2
t = 48
3264
A
λt∝
(p√λt
)a
a = d/Dh
Large structures keep the critical percolation geometry, a = d/DPh < 2.
Small structures get regular a = d/DIh ≃ 2.
62
Second summary
• We proved that the pdf of hull-enclosed areas in 2d curvature driven
coarsening satisfies dynamic scaling (exact result).
• We argued that the pdf of domain areas and perimeter lengths satisfy
dynamic scaling in 2d curvature driven coarsening and conserved
order parameter dynamics in Ising and Potts model systems (approx-
imate result).
• In all these cases, the large structures (ℓ > R(t, g)) keep the statistics
of the critical percolation (T0 > Tc) or Ising (T0 = Tc) states.
• In all these cases, the small structures (ℓ < R(t, g)) feel the micros-
copic dynamics and the distributions are modified (e.g. Liftshitz-
Slyozov law recovered for COP).
• Extensions to critical quenches. SLE + RG?
63
“Exact results for curvature driven coarsening in two dimensions”,
J. J. Arenzon, A. J. Bray, L. F. Cugliandolo, A. Sicilia. PRL 98, 145701 (2007).
“Domain growth morphology in curvature driven two dimensional coarsening”,
A. Sicilia, J. J. Arenzon, A. J. Bray, L. F. Cugliandolo PRE 76, 061116 (2007).
“Geometric properties of two-dimensional coarsening with weak disorder”,
A. Sicilia, J. J. Arenzon, A. J. Bray, L. F. Cugliandolo. EPL 82, 10001 (2008).
“Experimental test of curvature-driven dynamics in the phase ordering of a two di-mensional liquid crystal",
A. Sicilia, J. J. Arenzon, I. Dierking, A. J. Bray, L. F. Cugliandolo, J. Martínez Perdiguero, I. Alonso, I. Pintre,
PRL 101, 197801 (2008).
“Geometry of phase separation",
Y. Sarrazin, A. Sicilia, J. J. Arenzon, A. J. Bray, L. F. Cugliandolo, PRE 80, 031121 (2009).
“Curvature driven coarsening in the two dimensional Potts model”,
M. P. Loureiro, J. J. Arenzon, L. F. Cugliandolo, A. Sicilia, PRE 81, 021129 (2010).
“Geometrical properties of the Potts model during the coarsening regime",
M. P. Loureiro, J. J. Arenzon, L. F. Cugliandolo, PRE 85, 021135 (2012).
“A morphological study of cluster dynamics between critical points",
T. Blanchard, L. F. Cugliandolo, M. Picco, JSTAT (2012) P05026.
“How soon after a zero-temperature quench is the fate of the Ising model sealed ?",
T. Blanchard, F. Corberi, L. F. Cugliandolo, M. Picco, arXiv:1312.1712
64
GeneralizationPredictions
Generic Potts model with q ≥ 2 with critical initial condition quenched
below Tc(q) evolving with conserved (COP, zd = 2) or non-conserved
(NCOP, zd = 3) order parameter dynamics :
(λ(q)t)2τ/zd nd,h(A, t) ≈(2)c
(q)d,h a2(τ−2)
[A
(λ(q)t)2/zd
]1/2{1 +
[A
(λ(q)t)2/zd
]zd/2}(2τ+1)/zd
τ = 2 for hull-enclosed areas and τ>∼ 2 for domains (critical initial state) ;
an exponential cut-off for q > 2 and T0 → ∞ and q > 4 and T0 = Tc
Sarrazin, Sicilia, Arenzon, Bray & LFC 08 (Ising COP)Loureiro, Arenzon, LFC & Sicilia 09 (Potts NCOP)
65
GeneralizationIsing COP q = 8 Potts NCOP
10-13
10-11
10-9
10-7
10-5
101 102 103 104 105 106
n d(A
)
A
t=41489 88334
183761
10-12
10-10
10-8
10-6
10-4
1 10 100 1000 10000 100000
nh(A
,t)
A
-2
t=64 128 256 512
1024 2048 4096 8192
16384
The Lifshitz-Slyozov-Wagner dilute A−2 is seen as an envelope.
limit n(A, t) ≃ (A/t2/3)1/2 The exponential initial cut-off is
is alright at small areas but crosses seen at all times.
over to the initial pdf at large scales.
66
Critical quenches2dIM quenched from T0 → ∞ to T0 = Tc
Large structures keep the initial condition geometry, a = d/DPh .
Small structures are also fractal a = d/DIh.
Scaling variable p/RDIh(t) = p/tD
Ih/zc
Blanchard, LFC & Picco 12
67
Phase separation in glasses
– Sodium potassium borosilicate SiO2 (70%) B2O3 (20%) BaO (10%)
heterogeneous glass with special mechanical properties.
– Raw material mixed and melted at T ≃ 1500 C.
– Mixture cooled at ambient T : no dynamics (Paris → Grenoble).
– Mixture heated at T = 1000 C phase separates into
one glassy phase : 75% SiO2 + 25% B2O3.
another glassy phase : 60% SiO2 + 20% B2O3 + 20% BaO
roughly c = 50% each.
– 3d X tomography at ESRF (ID 19 line).
– Sample size 700 µml ; pixel size 0.7 µm.
D. Bouttes (ESPCI), E. Gouillart (Saint-Gobain) & D. Vandembroucq (ESPCI)
68
Phase separation in glasses
t = 1 min
69
Phase separation in glasses
t = 2 min
70
Phase separation in glasses
t = 4 min
71
Phase separation in glasses
t = 8 min
72
Phase separation in glasses
t = 16 min
73
Phase separation in glasses
t = 32 min
74
Phase separation in glasses
t = 64 min
75
Phase separation in glassesAreas vs. perimeters
76
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