Zilch vortical effect

We study the question of whether a helicity-transporting current is generated in a rotating photon gas at finite temperature. One problem is that there is no gauge-invariant local notion of helicity or helicity current. We circumvent this by studying not only the optical helicity current but also the gauge-invariant “zilch” current. In order to avoid problems of causality, we quantize the system on a cylinder of finite radius and then discuss the limit of infinite radius. We find that net helicity and zilch currents are only generated in the case of the finite radius and are due to duality-violating boundary conditions. A universal result exists for the current density on the axes of rotation in the high-temperature limit. To lowest order in the angular velocity, it takes a form similar to the well-known temperature dependence of the chiral vortical effect for chiral fermions. We briefly discuss possible relations to gravitational anomalies.

Figure 1

The energy (101) and angular momentum (102) densities as functions of the dimensionless tangential velocity at the boundary $R\mathrm{\Omega }$. We restrict ourselves to relatively low angular velocities $R\mathrm{\Omega }<0.5$ to facilitate numerical evaluation. The plots show that the behavior of the dimensionless quantities $\epsilon /T^4$ (top) and $L/(R{T}^3)$ (bottom) for a temperature range from $TR=0.25$ up to $TR=2.5$ in steps of $\delta (TR)=0.25$. As it can be seen from the plots, both of these dimensionless quantities collapse on a universal high-temperature curve. The arrow in the upper plot marks the Stefan-Boltzmann value $\epsilon =\frac{T^4{\pi }^2}{15}$. At temperature $TR=2.5$ the system is already within 3% of this value at zero rotation. The inset in the lower figure shows the moment of inertia (103) with the corresponding Stefan-Boltzmann value.

Figure 2

The total helicity ($s=1$) and zilch ($s=0$) currents (105) as functions of the dimensionless tangential velocity at the boundary $R\mathrm{\Omega }$. The plots show the behavior of the dimensionless quantities $J_h^{z,\mathrm{tot}}/(R{T}^2)$ and $J_{\zeta }^{z,\mathrm{tot}}/(R{T}^4)$ for the same temperature range as in Fig. 1. The insets show the currents divided by the frequency $\mathrm{\Omega }$.

Figure 3

The dimensionless strengths of the helicity ($s=1$) and zilch ($s=0$) currents (106) in the limit of small angular frequencies $\mathrm{\Omega }\to 0$ as functions of temperature $T$. The insets show the exponential onset of both currents at small temperatures.

Figure 4

Values of the helicity ($s=1$) and zilch ($s=0$) current densities (108) at the axes of rotation $\rho =0$ for very low angular velocities. The results are plotted as functions of $TR$ and as fractions of the result in the unbounded domain (78) and (81) respectively. The insets show the currents at small temperatures.