Based on:

• arXiv:1607.0076
• arXiv:1611.08456
• arXiv:1809.03384

With:

• Mark Hindmarsh
• Anna Kormu
• Asier Lopez-Eiguren
• Kari Rummukainen

Motivation

• Necklaces are networks of strings carrying monopoles.
• Models can be embedded in GUTs, including $\mathrm{SO}(10)$.
• Might behave differently to strings - role of monopoles?
• Observations:
  • Cosmic ray and $\gamma$-ray signals? astro-ph/9704257
  • Gravitational waves? (or lack thereof)

The model

• "$\mathrm{SU}(2)$ Georgi-Glashow model with two adjoint Higgses" $$ \begin{multline} V(\Phi_1,\Phi_2) = - m_1^2\mathrm{Tr}\Phi_1^2 - m_2^2\mathrm{Tr}\Phi_2^2 \\ +\lambda (\mathrm{Tr}\Phi_1^2)^2 + \lambda (\mathrm{Tr}\Phi_2^2)^2 +\kappa(\mathrm{Tr}\Phi_1\Phi_2)^2. \end{multline} $$ with $\Phi_n^a = \phi_n^a\sigma^a/2$ and the usual action.
• If $m_1^2 > m_2^2$, get necklaces with monopoles, charge $\{\pm 1\}$.
• If $m_1^2 = m_2^2$, behaviour depends on $\kappa/2\lambda$:
  • If $\kappa = 2\lambda$, there's a global $\mathrm{U}(1)$ - superfluidity.
  • If $\kappa \neq 2\lambda$, monopoles split into two semipoles with complex charge $\{1,i,-1,-i\}$.

Necklaces

• When $m_1^2 > m_2^2$, $\mathrm{U}(1)$ breaks to $Z_2$
• Strings form, joining up the monopoles

Semipoles 1

• When $m_1^2 = m_2^2$ and $\kappa/2\lambda < 1$, get semipoles.
• Four types of pole, with complex charge $\{1, i, -1, -i\}$.
• Linked by four types of string, with complexified flux $\Phi_B^{(1)} + i \Phi_B^{(2)} \in \{1 + i, -1 + i, -1 - i, 1 - i\}$.

Semipoles 2

• 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...)

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.

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.

Or in other words

• ✅ 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,
  • either $\mathbf{B}^{(1)}$ and $\mathbf{B}^{(2)}$ from $\Phi_1$ and $\Phi_2$
  • or $\mathbf{B}^{(\pm)}$ from $\Phi^{(\pm)} = \frac{1}{\sqrt{2}}(\Phi_1 \pm \Phi_2)$.

String and pole position movie

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.

Movie interlude

(We also have 3D and 360 movies!)

Necklaces

Semipoles 1 ($\kappa/2\lambda < 1$)

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

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

Dynamics: key quantities

• Lattice projectors give the positions of monopoles (total $N$) and plaquettes with winding (total $L$).
• Average defect separations in volume $V$ $$\xi_\mathrm{m} = \left(\frac{V}{N}\right)^{1/3}; \qquad \xi_\mathrm{s} = \left(\frac{V}{L}\right)^{1/2}.$$
• Comoving pole density $$ n = \frac{\xi_\mathrm{s}^2}{\xi_\mathrm{m}^3}. $$
• Energy, Lagrangian-derived measures less successful 🤷‍♂️.

$\xi_\mathrm{s}$ and $\xi_\mathrm{m}$ - necklaces

• String separation: linear scaling with conformal time $\tau$.
• Monopole separation: $\xi_m \mathrm \propto{\tau}^{2/3}$.
• No real dependence on initial $\xi_\mathrm{m}$.

$\xi_\mathrm{s}$ and $\xi_\mathrm{m}$ - semipoles

• Same general story as for necklaces.
• Some hints of different $\kappa/2\lambda$ behaviour...

Monopole linear density - necklaces and semipoles

• Note 'pseudopoles': projecting wrong complexified $\mathbf{B}$ fields.
• Everything physical goes to constant $n$ (more or less).