Holographic study of reflected entropy in anisotropic theories

We evaluate reflected entropy in certain anisotropic boundary theories dual to nonrelativistic geometries using holography. It is proposed that this quantity is proportional to the minimal area of the entanglement wedge cross section. Using this prescription, we study in detail the effect of anisotropy on reflected entropy and other holographic entanglement measures. In particular, we study the discontinuous phase transition of this quantity for a symmetric configuration consisting of two disjoint strips. We find that in the specific regimes of the parameter space the critical separation is an increasing function of the anisotropy parameter and hence the correlation between the subregions becomes more pronounced. We carefully examine how these results are consistent with the behavior of other correlation measures including the mutual information. Finally, we show that the structure of the universal terms of entanglement entropy is corrected depending on the orientation of the entangling region with respect to the anisotropic direction.

Figure 1

Left: schematic minimal hypersurfaces for computing $S_{A\cup B}$ in connected configuration. Right: the minimal cross section of the entanglement wedge, $\mathrm{\Sigma }$ in red. Here we only show the connected configuration where the reflected entropy is nonzero.

Figure 2

Finite part of the HEE (left), HMI (middle), and reflected entropy (right) as a function of rotation angle for different values of $\nu$ with $\ell =0.4$ and $h=0.2$. The dashed curve corresponds to isotropic case with $\nu =1$ where the measures are independent of the rotation angle.

Figure 3

Left: the turning point of the RT hypersurface as a function of the width of the boundary subregion for different values of the rotation angle. Middle: the HEE as a function of $\ell$ for the same values of $\beta$. Right: the HMI as a function of $\frac{h}{\ell }$. The solid curves show the anisotropic case with $\nu =2$ and the dashed curve corresponds to isotropic case with $\nu =1$.

Figure 4

Reflected entropy as a function of $\frac{h}{\ell }$ for different values of $\beta$ with $\nu =1.5$ (left) and $\nu =2$ (right). In both plots the dashed curve corresponds to isotropic case with $\nu =1$. Here we set $\ell =0.4$.

Figure 5

Critical separation between the subregions as a function of $\nu$ for different values of $\beta$. For large values of the rotation angle the critical separation is an increasing function of $\nu$ and hence the correlation between the subregions becomes stronger.

Figure 6

The free energy as a function of temperature for different values of the chemical potential with $\nu =2$ (left) and different values of the anisotropy parameter with $\mu =0.05$ (right). The dashed region indicates the instability zone and the point where any curve intersects itself corresponds to the small/large black holes phase transition. Here we set $\ell =0.4$.

Figure 7

The HEE (left) and the turning point of the RT hypersurface (right) as functions of $\ell$ for different values of $\beta$. Here we set $\nu =2$ and the dashed part of each curve corresponds to the parametric region where the HEE becomes multivalued.

Figure 8

The HMI (left) and reflected entropy (right) as functions of $\frac{h}{\ell }$. Right: reflected entropy as a function of $\frac{h}{\ell }$. Here we set $\nu =2$ and $\ell =0.35$. The discontinuity depicted in the inset corresponds to the region where the HEE becomes multivalued.

Figure 9

The turning point of the RT hypersurface (left) and HEE (right) as functions of the rotation angle for different values of the anisotropy parameter. Here we set $\ell =1$.

Figure 10

HMI (left) and reflected entropy (right) as functions of $\beta$ for $\ell =1$, $h=0.2$ and different values of $a$.

Figure 11

Critical separation between the subregions as a function of $\beta$ for different values of $a$. For $\beta \sim \pi /2$ of the rotation angle the critical separation is an increasing function of the anisotropy parameter and hence the correlation between the subregions becomes stronger.

Figure 12

The turning point of the RT hypersurface (left) and HEE (right) as functions of the rotation angle for different values of the anisotropy parameter. Here we set $\ell =1$.

Figure 13

HMI (left) and reflected entropy (right) as functions of $\beta$ for $\ell =0.1,h=0.05,r_h=1$ and different values of $a$.

Figure 14

The turning point of the RT hypersurface (left) and HEE (right) as functions of $\ell$ for different values of the rotation angle. The dashed curves represent the finite temperature results and the solid curves correspond to $T=0$ case. Here we set $\mu =0.079$ for left figure and $\mu =1$ for right figure.

Figure 15

HMI (left) and reflected entropy (right) as functions of $\frac{h}{\ell }$ for different values of $\beta$.Here we set $\mu =1$.

Figure 16

Left: odd entropy as a function of $\beta$ for different values of the anisotropy parameter for model discussed in Sec. 4a. Right: odd entropy as a function of $\frac{h}{\ell }$ for different values of $\beta$ for model discussed in Sec. 5. The dashed curves represent the finite temperature results and the solid curves correspond to $T=0$ case.