Phases of Giant Magnetic Vortex Strings

TL;DR

The paper analyzes giant magnetic vortex strings in 3+1D Abelian Higgs models, revealing two sharply distinct large-$n$ phases determined by the scalar potential: bulk strings in the conventional quartic model with $T_n o 2\pi\sqrt{2\beta}\,n$ and a bulk Coulomb core, and domain-wall strings in the degenerate sextic model with $T_n \sim (3/2)\sigma u_n$ and $u_n \sim \sqrt{2}\,n^{2/3}/\sigma^{1/3}$, where the domain wall surrounds a Coulomb core. The authors develop a matched asymptotic framework, combining WKB core solutions with boundary-region analysis to obtain analytic large-$n$ string profiles and tensions, corroborated by numerical solutions. They also analyze stability and inter-string forces, showing type-I/type-II behavior in the conventional model and a stable large-$n domain-wall phase in the degenerate model, with long-range forces governed by Higgs and vector-boson exchange. The results illuminate the phase structure, binding energies, and interactions of vortex strings, and demonstrate how large-$n$ expansions yield precise, predictive descriptions across regimes. The work advances understanding of nonperturbative solitons in gauge theories and their phase-dependent properties, with potential implications for superconductivity analogies and topological defect dynamics.

Abstract

We consider Abrikosov-Nielsen-Olesen magnetic vortex strings in 3+1 dimensional Abelian Higgs models. We systematically analyze the giant vortex regime using a combination of analytic and numerical methods. In this regime the strings are infinitely long, axially symmetric, and support a large magnetic flux n along the symmetry axis in their core that causes them to spread out in the transverse directions. Extending previous observations, we show that the non-linear equations governing giant vortices can essentially be solved exactly. The solutions fall into different universality classes, reflecting the properties of the Higgs potential, that become sharply distinct phases in the large-n limit. We use this understanding to shed light on the binding energies and stability of vortex strings in each universality class.

Figures (14)

• Figure 1: A schematic depiction of an axially symmetric ANO vortex carrying $n-$units of conserved magnetic flux along the $z-$axis. The red arrow illustrates the winding of the Higgs field around the vortex. Requiring finite tension fixes the winding number to be $n$, matching the total magnetic flux carried by the string.
• Figure 2: The dimensionless, rescaled potentials $\widetilde{V}_4(\varphi)$ and $\widetilde{V}_6(\varphi)$, given in \ref{['rescaledV']}, corresponding to the conventional and degenerate models, respectively. The curves are shown for a representative choice of the parameter $\beta=1$.
• Figure 3: The figure depicts numerical solutions for the Higgs field $\varphi(u)$ (top) and the gauge field $a(u)$ (bottom) of the axially-symmetric vortex equations \ref{['eqn:ahmeom']} with conventional and degenerate Abelian-Higgs-model potentials $V_{4,6}$ (shown in solid and dashed lines, respectively). On the left, we fix $\beta=1.2$ and vary $n = 1$ (blue) and $n =3$ (red); on the right, we fix $n = 2$ and consider $\beta=0.7$ (type I, blue) and $\beta=1.3$ (type II, red). The asymptotics at $u = 0, \infty$ are discussed in section \ref{['secasym']}. Note that increasing $n$ or decreasing $\beta$ makes the solutions move to the right, i.e. the vortices become wider. The same occurs at fixed $n, \beta$ if we go from the conventional to the degenerate model. These qualitative observations will be made precise in section \ref{['seclargen']}.
• Figure 4: Higgs and magnetic field profiles for a large string of magnetic flux $n = 100$ in the conventional (left panel) and degenerate (right panel) Abelian Higgs models. The dimensionless mass ratio $\beta = 0.5$ is the same for both models.
• Figure 5: The deep and outer core, boundary, and exterior regions of large-$n$ strings, as well as their shaded overlaps. The coordinate $\xi = u/u_n$ is particularly useful in the deep core. By contrast, the $y$-coordinate interpolates between the outer core, the boundary, and the exterior.