Extra stuff

Implementing the simulations ...

• Real-time lattice simulation, temporal gauge ($A^0 = 0$).
• Initial conditions and evolution must satisfy Gauss law.
• Fundamental quantities: links $U_i \in \mathrm{SU}(2)$ and fields $\Phi_n$.
• Our key measurements are the location of monopoles and plaquettes with winding number.

Do the simulations work?

• ✅ Yes, we can obtain the residual $\mathrm{U}(1)$ field.
• ✅ Yes, we can measure the vortex winding associated with the other Higgs field.
• ✅ Yes, they're both gauge invariant.
• For necklaces, the heavier field (wlog $\Phi_1$) forms monopoles.
• For semipoles, we have to make two magnetic fields (both scalar fields have a vev in places).

More semipoles

• When $m_1^2 = m_2^2$ and $\kappa/2\lambda > 1$, slight difference...
• Vacua rotated by 45° so string fields are rotated, too, $$\Phi_\pm = (\Phi_1 \pm \Phi_2)/\sqrt{2}.$$

Special case

• When $m_1^2 = m_2^2$ and $\kappa/2\lambda = 1$, there is a global $\mathrm{U}(1)$.

(We won't look at this any further today...)

Getting the monopoles

• Make projectors $\Pi_\pm = \frac{1}{2} (1 \pm \hat{\Phi}_1)$ where $$\hat{\Phi}_1 = \Phi_1\sqrt{2/\mathrm{Tr}\, \Phi_1^2}$$
• Get the $\mathrm{U}(1)$ gauge field corresponding to $\Phi_1$,
$$ u_\mu(x) = \Pi_+(x) U_\mu(x) \Pi_+(x+\hat{\mu}). $$
• Construct an effective "field strength tensor" hep-lat/0009037 $$ \alpha_{\mu\nu} = \frac{2}{g} \; \mathrm{arg} \; \mathrm{Tr}\; u_\mu(x) u_\nu(x+\hat{\mu}) u_\mu^\dagger(x+\hat{\nu}) u_\nu^\dagger(x). $$
• From which effective magnetic field and charge is $$B_i = \frac{1}{2} \epsilon_{ijk} \alpha_{jk}; \qquad \rho_{\mathrm{M}}= \sum_{i=1}^3 [B_i(x+\hat{\imath}) - B_i(x)] $$

Getting the winding

• Similar to Abelian Higgs hep-ph/9809334
• Difference in phase angle for $\Phi_2$ hep-lat/0009037 $$\begin{multline}\delta_i(x) = \mathrm{arg} \; \mathrm{Tr} \; \big[ \hat{\Phi_2}(x) \Pi_-(x) U_i(x) \Pi_-(x+\hat{\imath}) \\ \hat{\Phi_2}(x +\hat{\imath}) \Pi_+(x+\hat{\imath}) U_i^\dagger(x) \Pi_+(x) \big]. \end{multline}$$
• Winding number through a plaquette in the $ij$-plane at $x$ is then $$Y_{ij}(x) = \delta_i(x) + \delta_j(x+\hat{\imath}) - \delta_i(x+\hat{\jmath}) - \delta_j(x) - g \alpha_{ij}(x). $$
• We can trace this (and the monopoles) through the lattice.

Running the simulations

• "Naive" random initial conditions (memory soon lost).
• Simulate physical equations of motion, but first:
  • Some smoothing $$ \Phi_n(\mathbf{x}) \to \frac{1}{12} \sum_i \left[ \Phi_n(\mathbf{x}-\hat{\imath}) + 2\Phi_n(\mathbf{x}) + \Phi_n(\mathbf{x} + \hat{\imath}) \right]. $$
  • Then some heavy damping in a Minkowski background.
  • And 'core growth' (run equations with $s=-1$).
• $1920^3$ lattices, run for one light crossing time.
• Gauss Law OK, energy conservation < 1%.

Limitations of the simulations

(in addition to the usual caveats for field theory strings)

• Mass scales $m_1^2$ and $m_2^2$ will never be that different 😕
  • Largest we have is $m_2^2/m_1^2 = 0.04$.
  • Decrease $m_2$? Fatter strings reduce statistics.
  • Decrease $m_2$? Defect formation dynamics happens on time $1/m_2$ - need longer simulations. Light crossing time?
  • Increase $m_1$? Smaller monopoles risk pinning on lattice.

Extra: Semipoles ($\kappa/2\lambda > 1$)

(note, isosurfaces of unrotated $\Phi_1$, $\Phi_2$)

Extra: string velocities

Extra: monopole velocities

Extra: $r$

• Define the linear monopole density $n$ in units of $d_\mathrm{BV}$, $$ r = d_\mathrm{BV} \frac{n}{a}, \quad \text{where} \quad d_\mathrm{BV} = \frac{M_\mathrm{m}}{\mu} $$
• and the network energy density is $$ \rho_n \simeq \frac{\mu}{\xi_\mathrm{s}^2}(1+r). $$
• So $r$ gives the ratio of energy density in strings to monopoles.
• If $\xi_\mathrm{s} \propto \tau$ and $r$ constant, monopoles are a constant fraction of the total energy.

Extra: $r$ for necklaces

Extra: $r$ for semipoles

Extra: $r$ vs $v_\mathrm{m}$

Extra: $r$ vs $v_\mathrm{s}$