Anyon Bound States and Hybrid Superconductivity

Abstract

The interactions of anyonic quasi-particles (vortices) in the Chern--Simons extension of the Ginzburg--Landau model is investigated and we show that it manifestly realizes a hybridization of type I/II superconductivity. Through Gauss' law, each vortex simultaneously carries a flux quantum and a proportional Noether charge, thereby realizing an anyonic excitation. The Chern--Simons coupling also modifies the screening structure of the gauge fields, producing complex-conjugate masses that yield a common penetration depth with an oscillatory phase. This altered asymptotic behavior breaks the conventional type-I/type-II dichotomy of the Ginzburg--Landau model. As a result, vortex anyons experience short-range repulsion and long-range attraction, enabling the formation of separated multi-vortex bound states with non-monotonic interaction energy.

Figures (2)

• Figure 1: The binding energy per vortex anyon $E_{\textup{bind}}=E_N/N-E_1$ shown for three parameter regimes, corresponding to the value of the Ginzburg--Landau parameter $\lambda$. In the regular Ginzburg--Landau model, the Ginzburg--Landau parameter $\lambda$ controls the type of superconductivity and there is a dichotomy between these types. The first is the type I regime ($\lambda<1$), where the forces between vortices are attractive. Secondly, we have the type II regime ($\lambda>1$), where the intervortex forces are repulsive. At the Bogomolny point ($\lambda=1$), there are no forces between the vortices. This dichotomy is broken in the Chern--Simons extension of the model by introducing an electrostatic repulsion, altering the interaction landscape. Vortex anyons at the Ginzburg--Landau Bogomolny point ($\lambda=1$) now feel a repulsive force. For larger enough Chern--Simons coupling $\kappa$, the type I attractive regime can switch to a repulsive type II regime. Furthermore, a hybrid superconducting state can exist where the vortex anyons experience short-range repulsion and long-range attraction, forming stable bound multi-vortex anyons.
• Figure 2: Density plots of an $N=4$ vortex anyon exhibiting hybrid anyon superconductivity. Each individual vortex carries a magnetic flux $\Phi_0=-2\pi$ and an electric charge $-\kappa\Phi_0$, with $\kappa=\tfrac{1}{2}$. It can be seen that the magnetic flux is centered on the cores of the vortices (bottom left panel), and the electric Noether charge forms a ring of charge around each vortex core (bottom right panel). These vortex anyons are obtained for the Higgs potential \ref{['eq: Higgs potential']}, with parameters $m=1$ and $\lambda=\tfrac{1}{2}$. The binding energy of this $4$-vortex is negative and, so, the vortices form a bound state. They do not collapse to form an axially symmetric state with coincident cores as they experience short-range repulsion.