Edge states and thermodynamics of rotating relativistic fermions under magnetic field

We discuss free Dirac fermions rotating uniformly inside a cylindrical cavity in the presence of background magnetic field parallel to the cylinder axis. We show that, in addition to the known bulk states, the system contains massive edge states with the masses inversely proportional to the radius of the cylinder. The edge states appear at quantized threshold values of the fermion mass, which depend on the details of (chiral) MIT boundary conditions imposed at the surface of the cylinder. In the limit of infinite fermion mass, the masses of the edge states remain finite but, generally, nonzero as contrasted to the masses of the bulk states which become infinitely large. The presence of magnetic field affects the spectrum of both bulk and edge modes, and the masses of the edge states may vanish at certain quantized values of magnetic field. The moment of inertia of Dirac fermions is nonmonotonically increasing, oscillating function of magnetic field. The oscillations are well pronounced in a low-temperature domain and disappear at high temperatures. We also show that the edge modes alone do not support the anomalous transport phenomena such as chiral magnetic and chiral vortical effects.

Figure 1

(a) Solutions of the eigenvalue Eq. (27) as the function of the fermion mass $M$: (a) the real-valued solutions $q\ge 0$ corresponding to the bulk modes (from Ref. [15]) and (b) the purely imaginary solutions $q^{\mathrm{edge}}=i{\nu }^{\mathrm{edge}}$ with $\nu \ge 0$ corresponding to the edge modes. The inset shows $({\nu }^{\mathrm{edge}}{)}^2$ vs $M$.

Figure 2

The masses (44) of the edge states (the solid blue lines) and the masses (32) of the lowest ($l=1$) bulk states (the dashed magenta lines) as the function of the fermion mass $M$ in the absence of magnetic field, $B=0$. Four lowest states are shown. The critical points (34) are marked by the red dots (and the thin gray lines). The asymptotic masses of the edge states (76) in the limit $M\to -\infty$ are shown by the green arrows. Four lowest $l>1$ bulk states are shown by the dotted lines.

Figure 3

An example of the density of the edge modes for $k=0$ and $m=0$ at various fermionic masses $M$ in the absence of the background magnetic field ($B=0$). For the sake of convenience, in this figure, we show the density normalized to unity at $\rho =R$.

Figure 4

Five lowest bulk states (the dashed lines) and the edge mode (the solid line) for $MR=-5$ and $m=0$ at zero magnetic field. The modes are normalized according to Eq. (30).

Figure 5

The bulk $q_{ml}^B$ solutions of Eqs. (122) and (123) vs the fermion mass $M$ in the background of magnetic flux (124) ${\varphi }_B=7.5{\varphi }_0$ for (a) $m=0$ and (b) $m=-1$ orbital numbers and various radial excitation numbers $l$.

Figure 6

The edge ${\nu }_m^B$ solutions (131) of Eqs. (122) and (123) vs the fermion mass $M$ in the background of different magnetic fluxes (124) ${\varphi }_B$ for (a) $m=0$ and (b) $m=-1$ orbital numbers.

Figure 7

The masses of the lowest bulk ($l=1$) and edge states vs the mass of the fermion $M$ for various values orbital angular momenta $m$ and magnetic field $B$. The bulk (edge) modes are shown by the thicker (thinner) lines, while the positions where the bulk modes are converted to the corresponding edge modes are marked by the red points.

Figure 8

The values of the fermion masses $M=M_c^{(m)}$ at which the masses of the edge modes vanish (132) vs the magnetic flux ${\varphi }_B$ for various values of orbital momentum $m$.

Figure 9

The masses of the edge modes (76) as functions of magnetic field $B$ in the limit $M\to -\infty$.

Figure 10

Densities of (a) the angular momentum (139) and (b) moment of inertia (141) of the cylinder in the limit an infinite fermion mass $M\to \infty$ as the function of angular frequency $\mathrm{\Omega }$ at various temperatures $T$ and zero magnetic field.

Figure 11

Density of the moment of inertia (141) at $\mathrm{\Omega }=0$ vs temperature $T$ at vanishing magnetic field $B=0$.

Figure 12

Angular momentum $L$ of the edge modes per unit height of the cylinder vs the angular frequency $\mathrm{\Omega }$ and magnetic flux ${\varphi }_B$ at temperatures $TR=0.05$, 0.1, 1 in the limit of infinite fermionic mass $M\to -\infty$ (the bulk modes are absent).

Figure 13

Moment of inertia (divided by temperature squared) per unit height of the cylinder vs the flux ${\varphi }_B$ of the background magnetic field at (a) vanishing angular frequency $\mathrm{\Omega }=0$ and (b) nonzero frequency $\mathrm{\Omega }R=0.5$.

Figure 14

Moment of inertia per unit height of the cylinder vs magnetic flux ${\varphi }_B$ and temperature $T$ at (a) vanishing ($\mathrm{\Omega }=0$) and (b) nonzero ($\mathrm{\Omega }R=0.5$) angular frequency.