Thermodynamics and entanglement entropy with Weyl corrections

We consider charged black holes in four-dimensional anti–de Sitter space, in the presence of a Weyl correction. We obtain the solution including the effect of backreaction, perturbatively up to first order in the Weyl coupling, and study its thermodynamic properties. This is complemented by a calculation of the holographic entanglement entropy of the boundary theory. The consistency of results obtained from both computations is established.

Figure 1

$\beta$ as a function of horizon radius in four dimensions with a spherical horizon. Red, green, blue, brown and cyan curves correspond to $\frac{1}{10}$, $\frac{1}{9}$, $\frac{1}{8}$, $\frac{1}{7}$ and $\frac{1}{6}$ respectively. Here $\gamma =0.002$ is kept fixed.

Figure 2

Free energy as a function of temperature in four dimensions with a spherical horizon. Red, green, blue, brown and cyan curves correspond to $\frac{1}{10}$, $\frac{1}{9}$, $\frac{1}{8}$, $\frac{1}{7}$ and $\frac{1}{6}$ respectively. Here $\gamma =0.002$ is kept fixed.

Figure 3

$\beta$ as a function of horizon radius in four dimensions with a spherical horizon. Red, green, blue, brown and cyan curves correspond to $\frac{1}{10}$, $\frac{1}{9}$, $\frac{1}{8}$, $\frac{1}{7}$ and $\frac{1}{6}$ respectively. Here $\gamma =0.01$ is kept fixed.

Figure 4

Free energy as a function of temperature in four dimensions with a spherical horizon. Red, green, blue, brown and cyan curves correspond to $\frac{1}{10}$, $\frac{1}{9}$, $\frac{1}{8}$, $\frac{1}{7}$ and $\frac{1}{6}$ respectively. Here $\gamma =0.01$ is kept fixed.

Figure 5

Free energy as a function of temperature in four dimensions with a spherical horizon. The red, blue and brown curves correspond to $\gamma =0$, $5\times {10}^{-4}$ and $2\times {10}^{-3}$ respectively. Here $q=0.16$ is kept fixed.

Figure 6

Verifying Maxwell’s equal area law construction. Here, we have fixed $\gamma =0.002$. The red line and green dots are the values of the rhs and lhs of Eq. (40) respectively.

Figure 7

$\beta$ as a function of horizon radius in four dimensions with a planar horizon. Red, green, blue, brown and cyan curves correspond to $\frac{1}{10}$, $\frac{1}{9}$, $\frac{1}{8}$, $\frac{1}{7}$ and $\frac{1}{6}$ respectively. Here $\gamma =0.002$ is kept fixed.

Figure 8

Free energy as a function of temperature in four dimensions with a planar horizon. Red, green, blue, brown and cyan curves correspond to $\frac{1}{10}$, $\frac{1}{9}$, $\frac{1}{8}$, $\frac{1}{7}$ and $\frac{1}{6}$ respectively. Here $\gamma =0.002$ is kept fixed.

Figure 9

$\mathrm{\Delta }S$ (scaled by 50) as a function of $\beta$ for $\gamma =0.002$. Red, green, blue, brown and cyan curves correspond to $q=\frac{1}{10}$, $\frac{1}{9}$, $\frac{1}{8}$, $\frac{1}{7}$ and $\frac{1}{6}$ respectively. See text for more details.

Figure 10

$\mathrm{\Delta }S$ (scaled by 50) as a function of $\beta$ for $q=0.16$. The red, blue and brown curves correspond to values of $\gamma =0$, $5\times {10}^{-4}$ and $2\times {10}^{-3}$ respectively. See text for more details.

Figure 11

$\mathrm{\Delta }S$ (scaled by a factor of 50) as a function of inverse temperature $\beta$ with the critical charge $q=q_c\approx 0.137$ keeping $\gamma =0.01$ fixed. The vertical dashed line denotes the second-order transition temperature, $T_c=3.79964$.

Figure 12

$S_{\mathrm{Wald}}$ as a function of inverse temperature $\beta$ with the critical charge $q=q_c\approx 0.137$ keeping $\gamma =0.01$ fixed. The vertical dashed line denotes the second-order transition temperature, $T_c=3.79964$. Note the similarity with Fig. 11.