Massive Dirac states bound to vortices by a boson-fermion interaction

TL;DR

This work shows that a scalar vortex in a flat-space Maxwell-Klein-Gordon-Dirac system can trap massive Dirac fermions through a repulsive boson-fermion interaction. By formulating a covariant, dimensionless model and analyzing stationary axisymmetric solutions, the authors map the existence and stability of bound states across a broad parameter space, supported by approximate Gaussian-trap analytics and full numerical solutions. Time evolution confirms stability up to a critical back-reaction on the scalar vacuum, while head-on vortex collisions reveal new phenomena, including coalescence into pseudo-mass-2 states and the generation of non-topological, bubble-like bound states carrying fermions. The results connect to quasiparticle physics in gapped Dirac materials and offer insights for Bose-Fermi mixtures and phase-transition dynamics in related theories.

Abstract

Results are presented from numerical simulations of the flat-space nonlinear Maxwell-Klein-Gordon-Dirac equations. The introduction of a boson-fermion interaction allows a scalar vortex to act as a harmonic trap that can confine massive Dirac bound states. A parametric analysis is performed to understand the range of boson-fermion coupling strengths, Ginzburg-Landau parameters, and fermion effective masses that support the existence of bound state solutions; results are shown to be comparable to quasiparticle bound states in gapped Dirac materials. Solutions are time-evolved and are observed to be stable until the fermion field ψ becomes large enough to collapse the spontaneously broken vacuum of the condensate. Head-on scattering simulations are performed, and traditional vortex right-angle scattering is shown to break down with increased fermion field strength. For sufficiently large ψ and low velocity, the collision of two m = 1 vortices results in a pseudostable m= 2 bound state that eventually becomes unstable and decays back into two m= 1 vortices. For large ψ and collision velocity, vortex scattering is observed to produce nontopological (zero winding number) scalar bound states that are ejected from the collision. The scalar bubbles contain coherent fermion bound states in their interiors and interpolate between the spontaneously broken vacuum of the bulk and the modified vacuum induced by the boson-fermion interaction.

Figures (18)

• Figure 1: Plots of scalar field $\phi$ (top, red), fermion field components $\psi_1$ (top, green), $\psi_4$ (top, blue), electric field $E_R$ (bottom, red), and magnetic field $B_z$ (bottom, blue) for a bound state solution with effective mass $\kappa_d^{-1}=1$, and GL parameter values $\kappa=1,10,100$ in dashed, dotted, and solid lines, respectively. The confinement strength, $\kappa_m$, is chosen to give a fermion radius of unity for $\kappa=1$.
• Figure 2: Plots of fermion bound state radius $R_\psi$ as a function of boson-fermion interaction strength, $\kappa_m$, for $\kappa_d^{-1}=1, 10, 10^2, 10^3, 10^4$ in red, orange, yellow, green, and blue x's, respectively. The solid lines represent best-fit approximations to the small-$R$ solutions that are in good agreement with the approximate closed-form solutions (see Table \ref{['table:RpsiBestFit']}).
• Figure 3: Plots of boson-fermion interaction strength, $\kappa_m$ required to keep a fermion bound state of mass $\kappa_d^{-1}$ confined to a particular radius, $R_\psi$. From top to bottom, plots for $R_\psi=0.75$ and $\kappa=100, 10, 1$ are in red, magenta, and orange, respectively; plots for $R_\psi=1$ and $\kappa=100, 10, 1$ are in yellow, olive, and green, respectively; and plots for $R_\psi=1.25$ and $\kappa=100, 10, 1$ are in blue, purple, and black, respectively. Plots are bounded above and below by curves demonstrating $\kappa_m\propto (\kappa_d^{-1})^{-2}$ (dotted gray).
• Figure 4: Plots of fermion bound state radius, $R_\psi$, and vortex core radius, $R_\phi$, as a function of fermion field strength for $\kappa_d^{-1}=1, 10, 100$ (top, middle, and bottom, respectively). Each plot contains a curve (black) with the boson-fermion interaction strength $\kappa_m$ that confines the fermion bound state to $R_\psi = 1$ for field strength $\psi_{1,0} = 1 \times 10^{-3}$. Additional solutions are provided by increasing $\kappa_m$ by a factor of 2, 4, 8, and 16 (dotted red, orange, green, and blue, respectively) and decreasing $\kappa_m$ by a factor of 2, 4, 8, and 16 (dashed red, orange, green, and blue, respectively). A $\psi_{1,0} = \kappa_d^{-1}$ vertical dotted line is drawn on each graph and all plots have GL parameter $\kappa=100$.
• Figure 5: Plots of fermion energy (solid lines) and total energy (points) as a function of fermion field strength $\psi_{1,0}$ for $\kappa = 100$ and $\kappa_d^{-1}=1,10, 100$ in red, green, and blue, respectively. The fermion energy is observed to exceed the uncharged vortex energy when $\psi_{1,0} \gtrsim 1$, and and for $\kappa_d^{-1}=100$, total energies exceeding $10^4$ times the uncharged energy were observed.