Simulating the symmetron: Domain walls and symmetry-restoring impurities

In this paper we study the dynamics of relativistic domain walls in the presence of static symmetry-restoring impurities. The field theory is precisely the same as what is known to cosmologists as the “symmetron model,” whereby the usual ${\mathbb{Z}}_2$ symmetry breaking potential is appended with a space-varying mass term (the space variation is set by the profile of the impurity, which we take to be a “tanh” function). After presenting the outcomes of a suite of different numerical experiments we have three main results: (1) domain walls pin to impurities, (2) domain wall necklaces can be energetically preferred configurations, and (3) impurities significantly modify the usual $N_{\mathrm{dw}}\propto t^{-1}$ scaling law for random networks of domain walls.

Figure 1

Scalar field and energy density profiles of one-dimensional static solutions of the symmetron equation with a tanh impurity density profile (1.8). The colors correspond to different values of the internal density ${\rho }_0$ as shown in the legend in panel (a). The dashed line in (a) corresponds to the shape of the impurity density profile, and the dotted line in both panels denotes the profiles in the absence of impurities.

Figure 2

Evolution of the energy density for a scenario which illustrates the phenomenon of domain wall pinning. The impurity has radius $r_0=3$ and the domain wall is placed along the $y$-axis at $x_0=r_0+2$ (with zero velocity). It is clear that the domain wall is attracted by the impurity, and then the domain wall pinches off to pin to the distribution: these ends then travel towards the poles of the impurity. As explained in the main text, gradient flow is used until $t=50$, after which the second order time derivative equation of motion is used; as such one should not take the actual value of the times shown too seriously.

Figure 3

Steps to constructing the energetic bound for the domain wall necklace. The impurities are the small circles of radius $r_0$ that live on the large circle of radius $R_0$. The coordinates of intersection are $(x_I,y_I^{\pm })$, and the angle subtended by the arc of the large circle through them is $\vartheta$.

Figure 4

Plots showing the anecdotal conditions for the allowed configurations of domain wall necklaces. We remind the reader that there are $N$ impurities, each of size $r_0$, evenly distributed on a circle of radius $R_0$. The blue/light shaded region shows where the preferred energy condition (2.10) holds on the circle of radius $r_{\mathrm{w}}=R_0-r_0$, and the red/dark shaded region shows where both the preferred energy (2.10) and no-touch conditions (2.12) hold. There is no overlap between the regions in the equivalent plots for $N=3$.

Figure 5

Evolution of the energy density during an attempt to construct an $N=3$ domain wall necklace. It is clear that the domain wall loop collapses after it reconnects in the interior of the circle which the impurities sit on.

Figure 6

Finding an $N=5$ necklace solution: this is a plot of the energy density. It is clear that the initial domain wall loop collapses until it connects onto the impurities, after which the wall straightens out and stays connected to the impurities.

Figure 7

Evolution of the energy density during the construction of an $N=7$ domain wall necklace. We have skipped out images of the intermediate time steps for brevity.

Figure 8

Evolution of the number of domain walls from random initial conditions in the presence of impurities for two different box sizes. The physical lengths of the boxes, $L$, are shown in the captions, and for clarity we have given the number of grid points in each direction, $P$. These systems have $N=10$ impurities, each with $r_0=10,s=0.1$, equally spaced on a circle of radius $R_0=50$. For each physical length of the box there are four sets of simulations, each with a different value of the internal density ${\rho }_0$ as shown in the legend. For a given value of ${\rho }_0$ we have performed 20 realizations, as shown by the thin lines: the solid lines are the averages over the realizations. The dot-dashed line is the expected $N_{\mathrm{dw}}\propto t^{-1}$ scaling law (the system with ${\rho }_0=0$ agrees with this scaling law).

Figure 9

Evolution of the scaling exponents for the same distribution of impurities described in the caption of Fig. 8, with $1\sigma$-error bars and the results for a larger box included for completeness. To avoid cluttering the plots, the horizontal position of every other line is shifted slightly to the right to aid viewing, as well as skipping out some error bars. We have computed the average of the evolution of the number of domain walls from 20 realizations; the scaling exponent $\gamma$ is computed from the average via ${\overline{N}}_{\mathrm{dw}}\propto t^{-\gamma }$, in bins of width ${\delta t}_{\text{bin}}=10$. The different colors correspond to different values of the internal densities, as shown in the legend in the third panel (the color schemes are consistent across each of the three panels). It is clear that increasing ${\rho }_0$ has quite a significant effect on the scaling exponent, as well as inducing a rather substantial evolution when the box size is increased. These plots also show that the critical exponent is reached by ${\rho }_0=15$.

Figure 10

Images of the energy density at various times (the times are logarithmically spaced) for a system evolving from random initial conditions with a single impurity of radius $r_0=10$ at the origin, with internal density ${\rho }_0=5$. This simulation has relatively small physical length $L=128$, but this is enough to illustrate our point.

Figure 11

Images of precisely the same simulation we presented in Fig. 10, but deliberately zoomed in to focus on the impurity. The important thing we want to draw attention to is the changing energy density inside the impurity.

Figure 12

Evolution of the energy inside impurities of size $r_0=10$ and with various internal densities as shown in the legend; there is only one impurity at the origin. The thin lines denote the evolution of each of the members of the 20 realizations, and the thick lines are the averages.

Figure 13

Figures from two members of the realizations at $t=4t_{\mathrm{lx}}$, from the evolution of domain walls with random initial conditions. On the left panels we give the scalar field, and on the right the energy density. Each of the $N=6$ impurities have radius $r_0=3$, and sit on a circle of radius $R_0=10$. We have deliberately zoomed in to focus on the region containing the impurities which are marked on with blue circles. On the top row we present a system which has not formed a domain wall necklace, and on the bottom row we present an example of a system which has formed a domain wall necklace from random initial conditions.

Figure 14

Examples of necklaces formed from random initial conditions, for randomly located impurities. We have marked the locations of the impurities with blue circles onto these plots of the energy density. We have also applied a minimum threshold for the color scheme: everywhere the energy density is lower than 0.2, the color is set to black.