Numerical determination of inner-vortex molecule separations near the critical coupling

Determine the equilibrium separation of inner-vacuum vortex molecules throughout the parameter interval in which the interaction energy has a finite-separation minimum, including values of the coupling constant close to the upper critical value where the minimum becomes shallow.

Background

The modified Abelian–Higgs model admits inner-vacuum vortices whose interaction energy can develop a minimum at a nonzero separation. This minimum corresponds to a bound vortex molecule and persists analytically across the interval between the lower and upper critical couplings.

The authors obtain the molecule numerically using gradient flow over only part of this interval. Near the upper critical coupling, the minimum becomes increasingly shallow, causing the numerical evolution to separate the vortices instead of converging to the bound configuration. An analytical asymptotic estimate is provided, but the equilibrium separation is not reliably determined numerically across the full parameter range.

References

We remark that, although the interaction energy exhibits a minimum at a finite separation throughout the range \lambda_{c1} < \lambda < \lambda_{c2}, we are unable to numerically obtain the vortex molecule over the entire parameter range.

— Vortex clusters and vortex-filled bubbles in a modified Abelian-Higgs model  (2610.21930 - Miguélez-Caballero et al., 15 Oct 2026) in Section 3, immediately following Fig. 8 (Fig. \ref{f:dVsLambda_Inner})