Averaged squared winding angle
We now consider the time evolution of the variance of the winding angle related to spin cluster hulls, the observable h2(x, t)i defined in Sec. 1.3, for a KIM evolving with zerotemperature Glauber dynamics on a lattice with PBC. The goal of this analysis is to see whether, in a certain period of time and over a certain range of curvilinear lengths, h2(x, t)i reaches the form satisfied by the winding angle for 2d hulls in some type of criticality, that is h2i ‘ cst + 4 8+ ln x, with the parameter that identifies the associated SLE family of conformally invariant planar curves. We expect that, as the relaxation dynamics enters in the so-called critical-percolation-like scaling regime, that is, as t ! tp, the dependence of h2(x, t)i on the curvilinear distance x takes the above mentioned form with = 6, the value that corresponds to percolation criticality.
In all the cases that are shown here, the winding angle has been computed for the hulls that make up the interface of the largest spin cluster. More precisely, has been measured for the external hull of the largest spin cluster, or the two wrapping hulls forming its interface in the case in which it is percolating (in only one direction of the lattice or in the so-called“diagonal” topology). The reason behind this choice is that we would like the hulls which we use to measure (x, t) to be as long as possible compared to the lattice unit spacing r0, in order to probe large curvilinear lengths x over which (and in general any other geometrical property of the domain walls) is less affected by the discreteness of the lattice. This is desirable since the relation expressed by Eq. (1.27) is ideally valid in the continuum limit.
This requirement is fulfilled by cosidering the external hull of the largest spin cluster, when it does not percolate, or the hulls that wrap around the system, that most of the time are part of the interface of the largest cluster. Note that, at a sufficiently long time after the quench, even if the largest spin cluster is not percolating it can extend over a distance of order L.
In Fig. A.7, we show h2(x, t)i, measured on the wrapping hulls that are part of the interface of the largest spin cluster, plotted against ln x with x the curvilinear distance along the hull, in the case of the T = 0 Glauber dynamics on a square lattice with PBC and linear size L = 1280. We observe that, for sufficiently large values of x, h2i as a function of ln x has a constant slope. There is, in fact, a crossover length xc(t) such that, for x > xc(t) we observe h2i ‘ a + b ln x, while for x < xc(t) the curve h2i approaches a constant.
Moreover, a fit of the function f(x) = a + 4 (8+) ln x to the data at t ‘ 14.84 for x > 2 yields ‘ 5.90(1), that is rather close to = 6 expected for critical percolation cluster hulls. This means that the domain growth process imposed by the stochastic single spin-flip dynamics is characterised by a separation of length scales (at least at an early stage of the dynamics itself), as already discussed before: on length scales x < xc(t), wrapping domain walls have already the geometrical properties that are typical of the late stages of NCOP coarsening, that is to say, they are mostly smooth and, in the long time limit, they become eventually flat (think about the ”striped” frozen states); instead, on length scales x > xc(t) these domain walls have the geometrical properties of cluster hulls at critical percolation. We expect xc(t) `d(t) t1/zd with zd = 2 since `d(t) is the dynamical length that controls the growth of ordered domains equilibrated at the target temperature of the quench (in this case T = 0).
To test this argument, we plotted h2(x, t)i against ln (x/`G(t)), taking again the characteristic length `G(t), obtained as the inverse of the excess energy, as a measure of `d(t). The plot is shown in an inset in Fig. A.7 and, as one can see, the measurements corresponding to different times collapse one onto the other when performing this scaling.
Size distribution of the percolating clusters.
As explained in Sec. 1.4, the very few large spin clusters that survive the coarsening process after a sufficiently long time are those that signal the onset of the critical-percolation-like scaling regime. At the time tp, these clusters usually span most of the lattice and their geometrical and statistical properties resemble those of the clusters at critical site percolation on the same lattice. Usually, at this time, the two largest spin clusters of opposite spin orientation are both percolating and become “stable” with respect to the coarsening dynamics in the sense explained in Sec. 1.4.2. This is the reason why Np that constitutes the contribution given by the percolating clusters to N, is mainly due to the two largest clusters. For all practical purposes, we will take Np as just the size distribution of the two largest clusters in the system.
The wrapping probabilities h, v, hv, diag defined in Sec. 1.3 can be useful to determine the typical time required for the onset of the critical-percolation-like regime. In the left panel of Fig. 2.5, we show the wrapping probabilities for the Kawasaki dynamics on a square lattice, for different values of the lattice linear size L. The exact values of these probabilities for critical percolation on a torus with unit aspect ratio  are shown with dotted horizontal lines. As done in Sec. 1.5.2, the presence of the pre-percolation regime is exhibited by scaling time t as t/ (L/`d(t)) , with the value of the exponent chosen to make the datasets corresponding to different L collapse on a master curve. The value that gives the best collapse is, in this case, ‘ 2.00. Again, for t/ (L/`d(t)) 1 (or equivalently, for t tp with tp such that `p(tp) = `d(tp) t 1/ p L ) we should observe the time-dependent s approach the constant values corresponding to 2d critical percolation wrapping probabilities.
In our case, we could not perform simulations long enough to observe a clear convergence, even though the data shown suggest that, for sufficiently large system sizes, the critical percolation wrapping probabilities should be the asymptotic values. Nevertheless, the scaling is very good for t/ (L/`d(t)) up to 10−1, while, for larger values of the rescaled time, finite size effects are pronounced. These results can be compared to those for NCOP dynamics, shown in Sec. 1.5.2. The time tp needed to approach the critical percolation wrapping probability values (and thus the time of the onset of the critical-percolation-like regime) can be computed approximately from these data. Concretely, from Fig. 2.5 we can use tp/ (L/`d(tp)) = 1 as a criterium to measure tp. In this way we find tp ‘ 233, 684, 1924, 5212, 13763 for L = 40, 80, 160, 320, 640, respectively.
In Fig. 2.6 we show the same type of scaling on the honeycomb lattice. In this case, the value of that gives the best collapse is ‘ 1.15. Notice also that we plotted the probabilities h and v separately, since the lattice that we implement in our numerical simulations has aspect ratio different from unity, see App. A.2 for more details on this issue. More precisely, the lattice mesh used in the simulations has aspect ratio p3, and we took the vertical side longer than the horizontal one, hence why h > v. Then, our measurements must be compared to the wrapping probabilities for 2d critical percolation on a lattice of aspect ratio p3. In general, one can compute the wrapping probabilities for 2d critical percolation for any aspect ratio r [8, 57, 58]. For an aspect ratio r = p3 their values are given by (p) hv ‘ 0.5120, (p) h ‘ 0.4221, (p) v ‘ 0.0408 and (p) diag ‘ 0.0250.
It is also interesting to check the influence of an unbalance between the densities of the two species and, in particular, to investigate whether clusters retain the critical percolation properties during a certain time regime when the initial concentration of one of the two species is close to the percolation threshold, pc.
Domain area distribution for Kawasaki dynamics on a triangular lattice.
On the triangular lattice the pre-percolating regime is absent since the initial spin configuration is already a realisation of critical percolation, hence Eq. (2.12) should hold without the pre-percolation factor . To check the scaling with `d(t), in Fig. 2.12 we plotted the rescaled cluster size distribution N(A, t) `4 d(t) against the rescaled area A/`2 d(t) for a system of linear size L = 640 evolving at T = Tc/2. We take `d(t) to be proportional to the inverse of the excess-energy, `d(t) = `G(t), with the proportionality constant = 2.78. This value is approximately the one that gives us the best data collapse. The master curve f(x) = 2 cd x1/2 1 + x3/2−(2 A+1)/3, the scaling function for the LCOP domain growth (see Eq. (1.26)), is shown with blue discontinuous line. Deviations from the master curve are expected at very large values of the scaling variable A/`2 d(t), that is to say, for domains with linear size comparable to L. Deviations are also expected at small values of A/`2 d(t) because of the discreteness of the lattice. The master curve from the data collapse slightly differs from the analytic form around the “shoulder”, the point at which there is the crossover between the p A behaviour for small domains and the power law decay A−(2 A+1)/3 for large domains.
Initial configuration with a flat interface
The results shown up to now were obtained using a paramagnetic initial state in equilibrium at infinite temperature. We are now going to consider a “slab” initial state, which consists in a spin configuration where half of the sites take spin +1 and the other half take spin −1, and they are arranged into two “stripes”. For example, a spin configuration where all sites with x-coordinate in [1, L/2] have spin +1, while those with x 2 [L/2 + 1,L] have spin −1 (for L even). Due to PBC, the +1 domain and the −1 domain are separated by two straight (and wrapping) domain walls. Note that this is a frozen state for the zero- temperature Glauber dynamics but it is not for the voter model spin update rule. Under the voter model dynamics, a flat interface can fluctuate.
Table of contents :
I Coarsening and percolation in 2d kinetic Ising models
1 Percolation in the 2d KIM evolving under Glauber dynamics.
1.2 Definition of the KIM equipped with Glauber dynamics
1.4 Percolation phenomena
1.4.2 Largest cluster
1.4.3 Pair connectedness function
1.5 Detailed numerical analysis
1.5.1 Growing length
1.5.2 Wrapping probabilities
1.5.3 Averaged squared winding angle
1.5.4 Largest cluster scaling
1.5.5 Number density of domain areas
2 Coarsening and percolation in the 2d KIM evolving with COP dynamics.
2.2 Definition of the model
2.3 Kawasaki dynamics
2.4 Numerical analysis
2.4.1 The excess-energy growing length
2.4.2 Wrapping probabilities
2.4.3 Average squared winding angle
2.4.4 Largest cluster scaling
2.4.5 Pair connectedness function
2.4.6 Number density of domain areas
3 Coarsening in the 2d Voter Model: hints of a new criticality.
3.2 Definition of the Model
3.3 Numerical analysis
3.3.1 Average squared winding angle
3.3.2 Wrapping probabilities
3.3.3 Largest cluster scaling
3.3.4 Number density of domain areas
II Quench dynamics of the isolated p = 2 spherical spin glass model
4 Quench dynamics of the isolated p = 2 spherical spin glass model.
4.2.1 Definition of the model
4.2.2 The potential energy landscape
4.2.3 The equilibrium behaviour
4.2.4 Relaxation dynamics
4.3 Dynamics of the isolated system after a quench of the disorder strength
4.3.1 Energy change
4.3.2 Asymptotic analysis
4.4 Dynamics of the finite-size system
4.4.1 Formal solution of the mode dynamics
4.4.2 Initial conditions: equilibrium averages for finite N
4.4.3 Behaviour under stationary conditions
4.5 Numerical results
4.5.1 The phase diagram
4.5.2 Constant energy dynamics
4.5.3 Quench dynamics
4.6 Integrals of motion
4.6.1 Gibbs-Boltzmann equilibrium assumption
4.6.2 Fluctuations of the integrals of motion in the equal energy hypersurface
A.1 Finite-temperature effects for KIM evolving with Glauber dynamics
A.2 Ising model evolving with Glauber dynamics on a honeycomb lattice
A.2.1 Percolation phenomena
B.1 Ising model evolving with nonlocal Kawasaki dynamics
B.1.1 The growing length
B.1.2 Critical percolation phenomena
C.1 Generalized 2d KIM
C.2 Some analytic results for the voter model
D.1 Equilibrium measure for the simple harmonic oscillator after a quench
D.2 Neumann’s model, integrability and equilibration