# Ratio of critical quantities related to Hawking temperature-entanglement entropy criticality

## TL;DR

This paper investigates the critical ratios involving Hawking temperature and entanglement entropy in charged AdS black holes, revealing their similarities and differences and how they depend on dimension and entangling region size.

## Contribution

It introduces and compares the ratios T_c   S_c / Q_c and T_c S_c / Q_c, highlighting their dependence on dimension and entangling region size, and explores their relation to black hole entropy.

## Key findings

- Both ratios are independent of the characteristic length scale l.
- The ratios depend on the spacetime dimension d.
- The ratios differ even under the same parameters, due to entanglement entropy properties.

## Abstract

We revisit the Hawking temperature$-$entanglement entropy criticality of the $d$-dimensional charged AdS black hole with our attention concentrated on the ratio $\frac{T_c \delta S_c}{Q_c}$. Comparing the results of this paper with those of the ratio $\frac{T_c S_c}{Q_c}$, one can find both the similarities and differences. These two ratios are independent of the characteristic length scale $l$ and dependent on the dimension $d$. These similarities further enhance the relation between the entanglement entropy and the Bekenstein-Hawking entropy. However, the ratio $\frac{T_c \delta S_c}{Q_c}$ also relies on the size of the spherical entangling region. Moreover, these two ratios take different values even under the same choices of parameters. The differences between these two ratios can be attributed to the peculiar property of the entanglement entropy since the research in this paper is far from the regime where the behavior of the entanglement entropy is dominated by the thermal entropy.