Higher-curvature corrections to holographic mutual information

Abstract

In this paper, we study some non-local measurements of quantum correlations in extended gravities with higher-order curvature terms, including conformal gravity. Precisely, we consider higher-curvature correction on holographic mutual information in conformal gravity. There is in fact one deformation in the states because of the higher-curvature corrections. Here by making use of the holographic methods, we study the deformation in the holographic mutual information due to the higher-curvature terms. We also address the change in the quantum phase transition due to these deformations.

Appendix: some useful mathematical relations

Here in this appendix, we present some useful relations that we have used in this paper. Let us choose a five-dimensional metric with coordinate t, r, x, y, z as follows

$$\begin{aligned} \left( \begin{array}{ccccc} -\frac{f(r)}{r^2} & 0 & 0 & 0 & 0 \\ 0 & \frac{1}{r^2 f(r)} & 0 & 0 & 0 \\ 0 & 0 & \frac{1}{r^2} & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{r^2} & 0 \\ 0 & 0 & 0 & 0 & \frac{1}{r^2} \\ \end{array} \right) \end{aligned}$$

The determinant of the induced metric reads as

$$\begin{aligned} \frac{1}{r^6 f(r)}+\frac{x'(r)^2}{r^6} \end{aligned}$$

Therefore, two normal vectors are obtained as

$$\begin{aligned}&\left\{ \frac{1}{\sqrt{\frac{r^2}{f(r)}}},0,0,0,0\right\} ,\\&\quad \left\{ 0,-\frac{x'(r)}{\sqrt{r^2 f(r) x'(r)^2+r^2}},\frac{1}{\sqrt{r^2 f(r) x'(r)^2+r^2}},0,0\right\} \end{aligned}$$

The nonzero component of the extrinsic curvature ten reads as

$$\begin{aligned} {{{\mathcal {K}}}}_{11} &= \frac{-r f'(r) x'(r)+2 f(r)^2 x'(r)^3+2 f(r) \left( x'(r)-r x''(r)\right) }{2 r^2 f(r) \left( f(r) x'(r)^2+1\right) ^{5/2}}\\ {{{\mathcal {K}}}}_{12} &= \frac{x'(r) \left( -r f'(r) x'(r)+2 f(r)^2 x'(r)^3+2 f(r) \left( x'(r)-r x''(r)\right) \right) }{2 r^2 \left( f(r) x'(r)^2+1\right) ^{5/2}}\\ {{{\mathcal {K}}}}_{21} &= \frac{x'(r) \left( -r f'(r) x'(r)+2 f(r)^2 x'(r)^3+2 f(r) \left( x'(r)-r x''(r)\right) \right) }{2 r^2 \left( f(r) x'(r)^2+1\right) ^{5/2}}\\ {{{\mathcal {K}}}}_{22} &= \frac{f(r) x'(r)^2 \left( -r f'(r) x'(r)+2 f(r)^2 x'(r)^3+2 f(r) \left( x'(r)-r x''(r)\right) \right) }{2 r^2 \left( f(r) x'(r)^2+1\right) ^{5/2}}\\ {{{\mathcal {K}}}}_{33} &= \frac{f(r) x'(r)}{r^2 \sqrt{f(r) x'(r)^2+1}}\\ {{{\mathcal {K}}}}_{44} &= \frac{f(r) x'(r)}{r^2 \sqrt{f(r) x'(r)^2+1}} \end{aligned}$$

and also one finds

$$\begin{aligned}&{R_{\mu \nu }}n_i^\mu n_i^\nu \\&\quad =\frac{r f'(r)-4 f(r)}{f(r) x'(r)^2+1}\\&\qquad -\frac{f(r) x'(r)^2 \left( r^2 f''(r)-5 r f'(r)+8 f(r)\right) }{2 \left( f(r) x'(r)^2+1\right) }\\&\qquad +\frac{1}{2} \left( -r \left( r f''(r)-5 f'(r)\right) -8 f(r)\right) \\&{R_{\mu \nu \alpha \beta }}n_i^\mu n_i^\alpha n_j^\nu n_j^\beta \\&\quad =-2 \left( \frac{f(r) x'(r)^2 \left( r^2 f''(r)-2 r f'(r)+2 f(r)\right) }{2 \left( f(r) x'(r)^2+1\right) }\right. \\&\qquad \left. -\frac{r f'(r)-2 f(r)}{2 \left( f(r) x'(r)^2+1\right) }\right) \end{aligned}$$