Chiral Hall effect and chiral electric waves

We investigate the vector and axial currents induced by external electromagnetic fields and chemical potentials in chiral systems at finite temperature. Similar to the normal Hall effect, we find that an axial Hall current is generated in the presence of the electromagnetic fields along with an axial chemical potential, which may be dubbed as the “chiral Hall effect” (CHE). The CHE is related to the interactions of chiral fermions and exists with a nonzero axial chemical potential. We argue that the CHE could lead to nontrivial charge distributions at different rapidity in asymmetric heavy ion collisions. Moreover, we study the chiral electric waves led by the fluctuations of the vector and axial chemical potentials along with the chiral electric separation effect, where a density wave propagates along the applied electric field. Combining with the normal/chiral Hall effects, the fluctuations of chemical potentials thus result in Hall density waves. The Hall density waves may survive even at zero chemical potentials and become nondissipative. We further study the transport coefficients including the Hall conductivities, damping times, wave velocities, and diffusion constants of chiral electric waves in a strongly coupled plasma via the AdS/CFT correspondence.

Figure 1

A schematic illustration for (a) CSE and CME, (b) CESE and CME, and (c) Hall and chiral Hall effects. In (b),(c), for simplicity, we have assumed the system has a ${\mu }_A>0$. In those figures, two nuclei collide through the $z$ direction. The strong magnetic and electric fields are at $x$ and $y$ directions. The origin of the frame is set to be the center of the fireball. In (c), we find a possible charge and chirality separation induced by Hall and chiral Hall effects in the $z$ direction.

Figure 2

The green, black, blue, and red (from top to bottom) correspond to ${\mu }_V=0.002T$, $T$, $1.6T$, and $2T$, respectively. The solid and dashed curves correspond to $(B_x,E_y)=(m_{\pi }^2,m_{\pi }^2)=(0.135^2,0.135^2){\mathrm{GeV}}^2$ and $(B_x,E_y)=(0.001^2,0.01^2){\mathrm{GeV}}^2$, where ${\widehat{\mu }}_{V/A}={\mu }_{V/A}/T$.

Figure 3

The color corresponds to the same cases in Fig. 2.

Figure 4

The green, black, blue, and red (from top to bottom) correspond to ${\mu }_A=0.002T$, $T$, $1.6T$, and $2T$, respectively. The solid and dashed curves correspond to $(B_x,E_y)=(m_{\pi }^2,m_{\pi }^2)=(0.135^2,0.135^2){\mathrm{GeV}}^2$ and $(B_x,E_y)=(0.001^2,0.01^2){\mathrm{GeV}}^2$.

Figure 5

The color corresponds to the same cases in Fig. 4.

Figure 6

The color corresponds to the same cases in Fig. 4.

Figure 7

The color corresponds to the same cases in Fig. 4.

Figure 8

Boundary currents normalized by $C$ with $B_x=m_{\pi }^2$, ${\mu }_V=0.2T$, and ${\mu }_A=0.1T$. (a) vector charge density, (b) axial charge density, (c) vector current, (d) axial current, (e) vector Hall current, (f) axial Hall current.

Figure 9

Boundary currents normalized by $C$ with $E_y=m_{\pi }^2$, ${\mu }_V=0.2T$, and ${\mu }_A=0.1T$. (a) vector charge density, (b) axial charge density, (c) vector current, (d) axial current, (e) vector Hall current, (f) axial Hall current.

Figure 10

The transport coefficients for different chemical potentials and fixed electric and magnetic fields. The green, black, blue, and red [from bottom to top in (a)–(e) and from top to bottom in (f)] correspond to ${\mu }_V=0.002T$, $T$, $1.6T$, and $2T$, respectively. Here we take $E_y=B_x=10m_{\pi }^2$. The unit of ${\tau }_{-}$ is in ${\mathrm{GeV}}^{-1}$. In (c), the solid and dashed curves represent $(v_{-}{)}_y$ and $(v_{+}{)}_y$.

Figure 11

The transport coefficients for fixed chemical potentials and different electric and magnetic fields. The green (long-dashed), black (dot-dashed), blue (dashed), and red (solid) correspond to $e{E}_y=m_{\pi }^2$, $5m_{\pi }^2$, $10m_{\pi }^2$, and $20m_{\pi }^2$, respectively. Here we take ${\mu }_V=T$ and ${\mu }_A=0$. The unit of ${\tau }_{-}$ is in ${\mathrm{GeV}}^{-1}$.

Figure 12

The transport coefficients for fixed chemical potentials and different electric and magnetic fields. The green (long-dashed), black (dot-dashed), blue (dashed), and red (solid) correspond to $e{E}_y=m_{\pi }^2$, $5m_{\pi }^2$, $10m_{\pi }^2$, and $20m_{\pi }^2$, respectively. Here we take ${\mu }_V=2T$ and ${\mu }_A=T$. The unit of ${\tau }_{-}$ is in ${\mathrm{GeV}}^{-1}$. In Fig. 10, the solid and dashed curves represent $(v_{-}{)}_y$ and $(v_{+}{)}_y$.