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Higher-curvature corrections to holographic mutual information

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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.

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The author would like to thank Mohsen Alishahiha, for his helpful comments and discussions. MRT also wishes to acknowledge A. Akhvan and F. Omidi for some their comments. This work has been supported in parts by IAUCTB.

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Correspondence to M. Reza Tanhayi.

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Appendix: some useful mathematical relations

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 trxyz 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

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

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Bagheri, H., Tanhayi, M.R. Higher-curvature corrections to holographic mutual information. J Theor Appl Phys (2020).

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  • Holographic mutual information
  • Higher-curvature correction