Generation of a scalar vortex in a rotational frame

TL;DR

The paper investigates vacuum instability toward Bose condensation of a charged scalar in a rapidly rotating frame under external magnetic fields. It formulates a Gross-Pitaevskii–like equation in the rotating frame, analyzes two magnetic-background geometries (an infinitely thin flux tube and a field filling the cylinder), and compares linearized and full nonlinear solutions to determine the critical rotation frequency $\\Omega_{cr}$ and condensate energetics. Key findings show that, for the same input parameters, the condensate energy is lower for the flux-tube background than for a uniform field, and that near threshold the linearized problem closely matches the full GP-like solution while deviations grow at higher rotation rates; approximate analytical solutions provide good intuition, especially for large $\\Omega R$. The results are relevant to pion condensation scenarios in heavy-ion collisions and suggest the potential for flux-tube confinement or lattice formation under rotation, with implications for the structure of rotating QCD-like vacua.

Abstract

We consider generation from the vacuum of a scalar charged field in a rigidly rotating frame. Adding an external magnetic field opens the way to Bose condensation of the field. This phenomenon has been studied for external uniform magnetic field occupying the whole volume of the uniformly rotating cylindrical system of finite radius $R$ with a Dirichlet boundary condition imposed on it. Besides continuing this study, we consider the field formed by a flux tube of small radius. We find numerical solutions of the Ginzburg-Pitaevskii equation for the charged scalar field, the critical rotation frequencies, the mean radii and the condensate energies, and compare them with those found in a linearization scheme and with approximate analytical solutions. We show that for the same input parameters the energy of the condensate in the case of the flux tube is lower than in the case of uniform magnetic field in the whole cylinder.

Figures (10)

• Figure 1: For the infinitely thin flux tube, the condensate ground state energy density for the exact solution, $E_{GP}/(\pi R^2)$, see eq. \ref{['2.2.8']}, and the energy density, $E^{lin}/(\pi R^2)$, see eq. \ref{['2.4.8']}, for the solutions of the linearized equation as functions of the input parameter $\Omega R$. Parameters $\tilde{\Omega}$ and ${\tilde{\epsilon}}$ as functions of $\Omega R$ are shown in the inset. The critical value $\Omega_{cr}R$ is indicated by an arrow. The parameters are $m=1$, $R=10$, $\delta_\phi=50$ and $l=50$.
• Figure 2: For the thin flux tube, at $\nu=0$, the exact solutions $\phi(r)$ of the GP-like equation (solid lines) and the solutions of the linearized equation (dashed lines) for several values of the parameter $\Omega R$, $l=\delta_\Phi=50$, and $R=10, m=1, \lambda =1$. The dash-dotted line shows the approximate solution \ref{['3.2.121']} of the GP-like equation. The dots indicate the mean radius, \ref{['3.3.14']}.
• Figure 3: For the thin flux tube, the condensate energies corresponding to numerical solutions of the exact GP-like equation and the linearized equation for $\tilde{\Omega}=1.52$ ($\Omega=0.036$ at $l=50$ and ${\tilde{\epsilon}}=1.5$) as a function of $\nu$. For the exact solution the minimum of the energy occurs at $\nu=0$, whereas for the linearized solution it occurs at $\nu\approx 1$. The bold dot shows the minimal energy given by \ref{['3.2.12']}, which is found with the approximate analytical solution \ref{['3.2.121']}.
• Figure 4: For the thin flux tube, at $\nu=1$, the exact solutions $\phi(r)$ of the GP-like equation (solid lines), the solutions of the linearized equation (dashed lines) and the interpolating solution \ref{['3.3.C']} (dot-dashed lines) for several values of the parameter $\Omega R$, $l=49$, $\delta_\Phi=50$, and $R=10, m=1, \lambda =1$.
• Figure 5: For the magnetic field in the whole cylinder, the condensate energy density, $\frac{E^{GP}}{\pi R^2}$, see \ref{['2.2.8']}, and $\frac{E^{lin}}{\pi R^2}$, see \ref{['2.4.8']}, as functions of $\Omega R$. The parameters are $m=1$, $R=10$, $\delta_\phi=50$ and $l=45$. The inset shows the dependence of $\tilde{\Omega}$ and ${\tilde{\epsilon}}$ on $\Omega R$.