Rotating Dirac fermions in a magnetic field in $1+2$ and $1+3$ dimensions

We consider the effects of an external magnetic field on rotating fermions in $1+2$ and $1+3$ dimensions. The dual effect of a rotation parallel to the magnetic field causes a net increase in the fermionic density by centrifugation, which follows from the sinking of the particle lowest Landau level in the Dirac sea for free Dirac fermions. In $1+d=2n$ dimensions, this effect is related to the chiral magnetic effect in $2n-2$ dimensions. This phenomenon is discussed specifically for both weak and strong interfermion interactions in $1+2$ dimensions. For QCD in $1+3$ dimensions with Dirac quarks, we show that in the strongly coupled phase with spontaneously broken chiral symmetry, this mechanism reveals itself in the form of an induced pion condensation by centrifugation. We use this observation to show that this effect causes a shift in the chiral condensate at leading order in the pion interaction, and to discuss the possibility for the formation of a novel pion superfluid phase in off-central heavy-ion collisions at collider energies.

Figure 1

The particle ($+M$) and antiparticle ($-M$) LLL for $\mathrm{\Omega }=0$ are shown in panel (a) each with degeneracy $N$. For $\mathrm{\Omega }\ne 0$ the degeneracy is lifted. In panel (b) we illustrate how the centrifugation lifts the degeneracy on the states with angular momentum $N$ by shifting them down by $\pm M-\mathrm{\Omega }(N+\frac{1}{2})$. The rotating vacuum now includes the particle LLL which needs to be filled.

Figure 2

Effective potential $\mathcal{V}$ as a function of $\sigma$ in units of $\sqrt{eB}$ at $T=0$: $\mathrm{\Omega }=0.0001\sqrt{eB}$ (top), $\mathrm{\Omega }=0.00049\sqrt{eB}$ (middle), and $\mathrm{\Omega }=0.0005\sqrt{eB}$ (bottom).

Figure 3

Effective mass as a function of $\sqrt{eB}$ and $\mathrm{\Omega }$ in units of $\mathrm{\Lambda }$. The mass gap disappears for $\mathrm{\Omega }\ge {\mathrm{\Omega }}_c$ as given by Eq. (35) through a first order transition.

Figure 4

Finite-temperature effective potential $\mathcal{V}$ as a function of $\sigma$ in units of $\sqrt{eB}$: $\beta =100/\sqrt{eB}$ and $\mathrm{\Omega }=0.0003\sqrt{eB}$ (top); $\beta =43/\sqrt{eB}$ and $\mathrm{\Omega }=0.0001\sqrt{eB}$ (bottom).

Figure 5

Effective mass as a function of $\beta$ and $\mathrm{\Omega }$ in units of $\sqrt{eB}$ at $T\ne 0$.

Figure 6

The current density in the weak-coupling regime with $g_r=1$, as a function of $x=\frac{eB{r}^2}{2}$ in units of $\frac{eB}{2\pi }$ at $T=0$ and $\mathrm{\Omega }=0.0005$ (in units of $\sqrt{eB}$) (first), $\beta =100$, $\mathrm{\Omega }=0.0001$ (second), $\beta =100$, $\mathrm{\Omega }=0.0005$ (third), and $\beta =40$, $\mathrm{\Omega }=0.0001$ (fourth).

Figure 7

Finite-temperature effective potential $\mathcal{V}(\sigma )$ at $\beta =80/\sqrt{eB}$ and $\mu =0.007/\sqrt{eB}$ as a function of $\sigma$ in units of $\sqrt{eB}$: $\mathrm{\Omega }=0$ (top) and $\mathrm{\Omega }=0.0003\sqrt{eB}$ (bottom).

Figure 8

Mass gap $\sigma /\mathrm{\Lambda }$ in the strong-coupling regime with $g_r=-4$, as a function of $\mathrm{\Omega }/({10}^{-4}\mathrm{\Lambda })$ for $\mathrm{\Lambda }/\sqrt{eB}=5$ (top) and $\mathrm{\Lambda }/\sqrt{eB}=3$ (bottom).

Figure 9

The charge distribution (55) in the LLL in a closed volume $V$ with overall charge neutrality, for $N=1000$ as a function of $r$ and in units of $eB$.

Figure 10

The mean number of superfluid pions ${\mathbb{N}}_{{\pi }^{\pm }}$ in the range $0.5m_{\pi }\le T\le 1.5m_{\pi }$, ${\mu }_f=0.5m_{\pi }$ and $0.03m_{\pi }\le \mathrm{\Omega }\le 0.04m_{\pi }$.

Figure 11

The mean number of superfluid pions ${\mathbb{N}}_{{\pi }^{+}}$ in the range $0.5m_{\pi }\le T\le 1.5m_{\pi }$ and $0.03m_{\pi }\le \mathrm{\Omega }\le 0.08m_{\pi }$, for ${\mu }_f=0.8m_{\pi }$.

Figure 12

The mean number of superfluid pions ${\mathbb{N}}_{{\pi }^{\pm }}$ in the range $0.5m_{\pi }\le T\le 1.5m_{\pi }$ and $0.03m_{\pi }\le \mathrm{\Omega }\le 0.05m_{\pi }$, for case ($a$) (upper) and case ($b$) (lower).

Figure 13

Effective potential $\mathcal{V}(\sigma )$ at $T=0$ and $\mu =-0.031/\sqrt{eB}$ in units of $\sqrt{eB}$: $\mathrm{\Omega }=0$ (top), $\mathrm{\Omega }=0.00012\sqrt{eB}$ (middle), and $\mathrm{\Omega }=0.001\sqrt{eB}$ (bottom).

Figure 14

Effective mass at $T=0$ and $\mu =-0.03\sqrt{eB}$ in units of $\sqrt{eB}$ as a function of $\mathrm{\Omega }$ in units of ${10}^{-5}\sqrt{eB}$.