Hall viscosity and angular momentum in gapless holographic models

We use the holographic approach to compare the Hall viscosity ${\eta }_H$ and the angular momentum density $\mathcal{J}$ in gapless systems in $2+1$ dimensions at finite temperature. We start with a conformal fixed point and turn on a perturbation which breaks the parity and time-reversal symmetries via gauge and gravitational Chern-Simons couplings in the bulk. While the ratio of ${\eta }_H$ and $\mathcal{J}$ shows some universal properties when the perturbation is slightly relevant, we find that the two quantities behave differently in general. In particular, ${\eta }_H$ depends only on infrared physics, while $\mathcal{J}$ receives contributions from degrees of freedom at all scales.

Figure 1

Hall viscosity for the Gao-Zhang potential (3.16) with $\alpha =1.1$, 1.5, $\sqrt{3}$ (dashed lines) and for a quadratic potential with $m^2=-2$ (solid line).

Figure 2

Angular momentum density for the Gao-Zhang potential (3.16) with $\alpha =1.1$, 1.5, $\sqrt{3}$ (dashed lines) and for a quadratic potential with $m^2=-2$ (solid line).

Figure 3

Ratio of angular momentum density and Hall viscosity for the Gao-Zhang potential (3.16) with $\alpha =1.1$, 1.5, $\sqrt{3}$ (dashed lines) and for a quadratic potential with $m^2=-2$ (solid line).

Figure 4

${\eta }_H/{\vartheta }_0{T}^{2-{\mathrm{\Delta }}_{-}}$ as a function of $m^2$ for ${\vartheta }_0/T^{{\mathrm{\Delta }}_{-}}=1/6$.

Figure 5

$\mathcal{J}/{\vartheta }_0{T}^{2-{\mathrm{\Delta }}_{-}}$ as a function of $m^2$ for ${\vartheta }_0/T^{{\mathrm{\Delta }}_{-}}=1/6$.

Figure 6

Angular momentum density to Hall viscosity ratio as a function of $m^2$ for ${\vartheta }_0/T^{{\mathrm{\Delta }}_{-}}=1/6$.

Figure 7

The angular momentum density generated by the gauge Chern-Simons coupling as a function of ${\mu }^2/T^2$ for the Gaussian potential $m^2=-2$ and dilatonic coupling $\alpha =0.5$ (upper solid red line) and 0.9 (lower solid orange line) and for Gao-Zhang potentials with $\alpha =0.5$, 1.5 and $\sqrt{3}$ (dashed lines).

Figure 8

The angular momentum density generated by the gauge Chern-Simons coupling as a function of ${\mu }^2/T^2$ for dilatonic coupling $\alpha =0.5$ and Gaussian potentials with $m^2=-2$ (upper line, red), 1.4 (orange), $-1$ (green), $-0.6$ (blue) and $-0.2$ (lower line, purple).

Figure 9

Hall viscosity as a function of ${\mu }^2/T^2$ for Gao-Zhang potentials with $\alpha =0.5$, 1.5 and $\sqrt{3}$ (dashed lines) and for quadratic potentials with $m^2=-2$ and dilatonic coupling $\alpha =0.5$ (upper solid red line) and 0.9 (lower solid orange line).

Figure 10

Gravitational angular momentum density as a function of ${\mu }^2/T^2$ for Gao-Zhang potentials with $\alpha =0.5$, 1.5 and $\sqrt{3}$ (dashed lines) and for quadratic potentials with $m^2=-2$ and dilatonic coupling $\alpha =0.5$ (lower solid red line) and 0.9 (upper solid orange line).

Figure 11

Gravitational angular momentum density to Hall viscosity ratio as a function of ${\mu }^2/T^2$ for Gao-Zhang potentials with $\alpha =0.5$, 1.5 and $\sqrt{3}$ (dashed lines) and for quadratic potentials with $m^2=-2$ and dilatonic coupling $\alpha =0.5$ (lower solid red line) and 0.9 (upper solid orange line).

Figure 12

${\eta }_H/{\mu }^2$ (blue, upper line on the right) and $\mathcal{J}/{\mu }^2$ (black, lower line on the right) as a function of $m^2$ for ${\mu }^2/T^2=0.1$.

Figure 13

Angular momentum density to Hall viscosity ratio as a function of $m^2$ for ${\mu }^2/T^2=0.1$.