Impact of domain walls on the chiral magnetic effect in hot QCD matter

The chiral magnetic effect (CME)—the separation of positive and negative electric charges along the direction of the external magnetic field in quark-gluon plasma and other topologically nontrivial media—is a consequence of the coupling of electrodynamics to the topological gluon field fluctuations that form metastable $CP$-odd domains. In phenomenological models it is usually assumed that the domains are uniform and the influence of the domain walls on the electric current flow is not essential. This article challenges the latter assumption. A simple model consisting of a uniform spherical domain in a uniform time-dependent magnetic field is introduced and analytically solved. It is shown that (i) no electric current flows into or out of the domain; (ii) the charge separation current, viz. the total electric current flowing inside the domain in the external field direction, is a dissipative Ohm current; (iii) the CME effect can be produced either by the anomalous current or by the boundary conditions on the domain wall; and (iv) the charge separation current oscillates in plasma long after the external field decays. These properties are qualitatively different from the CME in an infinite medium.

Figure 1

Left panel: $B_z$ inside a spherical domain at a representative point $r=R/2,\theta =\pi /3$. Right panel: $B_z$ outside the domain at a representative point $r=3R/2,\theta =\pi /3$. The domain radius is $R=1$ fm (black dots) or $R=2$ fm (brown triangles). Other parameters: $B_0=1{\mathrm{fm}}^{-2}$, $\sigma =1/\left(36\text{fm}\right)$ [37, 38, 39, 40, 41], ${\sigma }_{\chi }=1/\left(100\mathrm{fm}\right)$, and ${\mathrm{\Theta }}_0=2\pi$. Solid red line represents the external field $B^{\text{ext}}$ of Eq. (34).

Figure 2

Snapshot of the magnetic field inside a spherical domain of radius $R=1$ fm at $t=2$ fm. Left panel: $\mathit{B}$, right panel: $B_z$ (zoomed in). Other parameters: $B_0=1{\mathrm{fm}}^{-2}$, $\sigma =1/\left(36\text{fm}\right)$, ${\sigma }_{\chi }=1/\left(100\mathrm{fm}\right)$, ${\mathrm{\Theta }}_0=2\pi$.

Figure 3

Electric current flowing inside a domain in the $z$ direction through a cross section at $z=R/2$. $B_0=1{\mathrm{fm}}^{-2}$, and $\sigma =1/\left(36\text{fm}\right)$. Black circles: $R=1$ fm, ${\sigma }_{\chi }=0.01/\mathrm{fm}$, and ${\mathrm{\Theta }}_0=2\pi$. Brown triangles: $R=2$ fm, ${\sigma }_{\chi }=0.01/\mathrm{fm}$, and ${\mathrm{\Theta }}_0=2\pi$. Blue squares: $R=1$ fm, ${\sigma }_{\chi }=0.01/\mathrm{fm}$, and ${\mathrm{\Theta }}_0=-2\pi$. Green stars: $R=1$ fm, ${\sigma }_{\chi }=-0.01/\mathrm{fm}$, and ${\mathrm{\Theta }}_0=2\pi$.