Slicing the three-dimensional Ising model: Critical equilibrium and coarsening dynamics

We study the evolution of spin clusters on two-dimensional slices of the three-dimensional Ising model in contact with a heat bath after a sudden quench to a subcritical temperature. We analyze the evolution of some simple initial configurations, such as a sphere and a torus, of one phase embedded into the other, to confirm that their area disappears linearly with time and to establish the temperature dependence of the prefactor in each case. Two generic kinds of initial states are later used: equilibrium configurations either at infinite temperature or at the paramagnetic-ferromagnetic phase transition. We investigate the morphological domain structure of the coarsening configurations on two-dimensional slices of the three-dimensional system, compared with the behavior of the bidimensional model.

Figure 1

Distribution of the spin cluster sizes in the 2D IM at its critical temperature. Scaling of the number density of finite areas, $A^{{\tau }_A}N\left(A\right)$ (inset), and full distribution of all areas (main panel) vs $A/M_L$, with $M_L$ given by Eq. (11), for various system sizes $L$ given in the legend, together with the values of ${\tau }_A$ used. Inset: The dashed horizontal line is $c_d\simeq 0.025$ [12], to which the plateau should asymptotically converge.

Figure 2

Distribution of the percolating spin cluster areas in the 2D IM at its critical temperature. Scaling of the number density of percolating areas, $A^{{\tau }_A}[\mathcal{N}\left(A\right)/L^2-N\left(A\right)]$, vs $A/M_L$, with $M_L$ given by Eq. (11), for various system sizes $L$ given in the legend. Inset: Values of ${\tau }_A$ from Fig. 1 as a function of $1/L$ showing the convergence to the asymptotic value ${\tau }_A^{\left(2\mathrm{D}\right)}=379/187$.

Figure 3

Distribution of spin cluster sizes for slices of the 3D IM at its critical temperature. Scaling of the number density of finite areas, $A^{{\tau }_A}N\left(A\right)$ (inset), and full distribution of all areas (main panel) vs $A/M_L$, with $M_L\simeq L^{1.86}$, for various system sizes $L$ given in the legend, together with the values of ${\tau }_A$ used. Inset: The dashed horizontal line is $2c_d\simeq 0.05$ [12], twice the value for the 2D IM.

Figure 4

Distribution of the size of the largest cluster, $A^{{\tau }_A}\left[\mathcal{N}\left(A\right)/L^2-N\left(A\right)\right]$, vs $A/M_L$, with $M_L\sim L^{1.86}$ for spin cluster areas on slices of the 3D IM at its critical temperature. Differently from Fig. 2, the curves here do not scale.

Figure 5

Effective value of ${(\beta /\nu )}_s(L,L^{\prime })$ vs $(L+L^{\prime })/2$ for the 3D IM at its critical temperature as extracted from the analysis of the averaged size of the largest spin cluster on 2D slices; see Eq. (12).

Figure 6

Time evolution of a 2D circular domain (top row) and an equatorial slice of a 3D spherical domain (bottom row) at $T=0$ (left column) and $T=T_c/2$ (right column). In both cases the initial radius is $R_0=40$ (for finite temperatures, the box length must be large enough to prevent the circle from growing and percolating). Different colors correspond to different times (in Monte Carlo steps, starting from the background: 0, $500,...,$ 2500). Although at zero temperature both areas decrease at the same rate, ${\lambda }_{\mathrm{sl}}\left(0\right)={\lambda }_{2\mathrm{D}}\left(0\right)\simeq 2$, in three dimensions the domains stay closer to their circular initial shape at all times.

Figure 7

The parameter $\lambda \left(T\right)$, obtained from the shrinking of a single disk, ${\lambda }_{2\mathrm{D}}$, or an equatorial slice of a single sphere, ${\lambda }_{\mathrm{sl}}$, as a function of the temperature.

Figure 8

Evolution of a toroidal domain of one phase immersed in a sea of the opposite phase at $T=T_c/2$ for two initial conditions. The snapshots of the cross section of a single-ring torus are shown with different colors at times $t=0$, 150, 500, and 1000 Monte Carlo steps (from back to front). Due to the shrinkage of the torus, there appears to be a small attraction between the domains as their centers get slightly closer with time. In both cases the minor radius is $r=20$ and, depending on the value of the major radius, $R=40$ (a) and $R=30$ (b), the two initially separated circles may merge. This merging mechanism, which slows down the change in area, is only present in the slices of a 3D system because the domains are connected along the orthogonal direction.

Figure 9

Collapsed equal time correlation $C(r,t)$, for several times after the quench from $T_0\to \infty$, indicated in the legend, as a function of the rescaled distance, $r/\sqrt{t}$. As expected, dynamical scaling is observed. Inset: length scale $R\left(t\right)$ obtained from $C(R,t)=1/2$. The straight line has exponent 0.5.

Figure 10

The number density of geometric domain areas (inset) and its rescaled form (main panel), in a 2D slice of the 3D IM (including the spanning clusters), per unit area of the system after a quench from $T_0\to \infty$ to $T=2$. Averages are over 7000 configurations (700 samples with 10 slices each) of an $N={200}^3$ system. Note that since the linear system size is much smaller than the ones used in Ref. [11], the distributions have smaller cutoffs. Albeit the 3D system is far from the percolation threshold, the distributions on the slice soon approach a power-law distribution with an exponent that is compatible, asymptotically and for large systems, with ${\tau }_p=187/91\simeq 2.055$ (we use, indeed, the value obtained in Sec. 2 for $L=200$: ${\tau }_A\simeq 1.93$). The lines are the 2D result, Eq. (14), with $c_d\simeq 0.02,{\lambda }_{3\mathrm{D}}\left(2\right)\simeq 1.04$, obtained from the single sphere, and the slope ${\tau }_p$ above. For small areas with respect to the typical one, $A/t<10$, the inset shows that the distribution of thermal fluctuations approaches its equilibrium form.

Figure 11

The same as Fig. 10, but for the hull-enclosed areas. Differently from the geometric domains whose exponent ${\tau }_A$ has a strong size dependence, the power-law exponent here is 2 and is attained much more rapidly. The lines are the 2D result, Eq. (13), with ${\lambda }_{3\mathrm{D}}\left(2\right)\simeq 1.04$, obtained from the single sphere. Note that for long times, the distribution develops a bump, even though the contribution from percolating clusters has been removed.

Figure 12

Rescaled hull-enclosed areas against the corresponding rescaled perimeters for slices of a 3D system quenched from $T_0\to \infty$ to $T=2$. For small sizes with respect to the typical one, since domains get round, the rescaled areas are simply the square of the rescaled lengths, $y\simeq x^2$. For large sizes, the rescaled quantities are linked by an exponent that is close to the one of critical 2D percolation, $8/7\simeq 1.14$ [12, 46]. Numerically we find $y\simeq x^{1.2}$. Both behaviors, for small and large rescaled areas, are shown by straight dotted lines. The times at which the data are gathered are shown in the legend.

Figure 13

Collapsed equal time correlation $C(r,t)$ for several times after the quench from $T_0=T_c$, indicated in the legend, as a function of the rescaled distance, $r/\sqrt{t}$. As expected, dynamical scaling is observed. Inset: Length scale $R\left(t\right)$ obtained from $C(R,t)=1/2$. The straight line has exponent 0.5.

Figure 14

Number density of geometric domains per unit area after a quench to $T=2$ in a 2D slice of a 3D IM evolving from a $T_0=T_c\simeq 4.5115$ initial condition. Averages are over more than 7000 configurations. The solid line on top of the $t=0$ data (inset; open symbols) is $2c_d/A^{1.93}$, while for $t>0$ we use Eq. (14) without the factor 2 in the coefficient. The main panel shows the collapsed version after proper rescaling of both axes. The solid line has the same exponent as the $t=0$ distribution but half the coefficient, $c_d/A^{1.93}$. Although the data could be well enveloped by a power law with an exponent slightly smaller than 1.93, we remark that at later times, due to the small size of the slice, the power-law regime is barely observed in the simulation.

Figure 15

The same as Fig. 14, but for hull-enclosed areas. The solid line on top of the $t=0$ data (inset; open symbols) is $2c/A^{1.92}$, while for $t>0$ the distribution no longer has the factor 2 in the coefficient. The main panel shows the collapsed version after proper rescaling of both axes. Again, the solid line has the same exponent as the $t=0$ distribution but half the coefficient, $c_d/A^{1.92}$.

Figure 16

Rescaled hull-enclosed areas against the corresponding rescaled perimeters for slices of a 3D system quenched from $T_0=T_c$ to $T=2$. For small rescaled domains $y\sim x^2$ (dotted red line), while for larger ones $y\sim x^{1.3}$ (straight dotted blue line). This exponent is compatible with that for the geometric domains on 2D slices of the 3D IM at equilibrium and $T_c,\delta \simeq 1.23$ [26], while it is well below the critical 2D exponent, ${\delta }^{\left(2d\right)}\simeq 1.45$ [25] (dotted green line).