  Recently, tight-binding models on hyperbolic lattices (discretized AdS
space), have gained significant attention, leading to hyperbolic band theory
and non-Abelian Bloch states. In this paper, we investigate these quantum
systems from the perspective of quantum information, focusing particularly on
the scaling of entanglement entropy (EE) that has been regarded as a powerful
quantum-information probe into exotic phases of matter. It is known that on
$d$-dimensional translation-invariant Euclidean lattice, the EE of band
insulators scales as an area law ($\sim L^{d-1}$; $L$ is the linear size of the
boundary between two subsystems). Meanwhile, the EE of metals (with finite
Density-of-State, i.e., DOS) scales as the renowned Gioev-Klich-Widom scaling
law ($\sim L^{d-1}\log L$). The appearance of logarithmic divergence, as well
as the analytic form of the coefficient $c$ is mathematically controlled by the
Widom conjecture of asymptotic behavior of Toeplitz matrices and can be
physically understood via the Swingle's argument. However, the hyperbolic
lattice, which generalizes translational symmetry, results in inapplicability
of the Widom conjecture and thus presents significant analytic difficulties.
Here we make an initial attempt through numerical simulation. Remarkably, we
find that both cases adhere to the area law, indicating that the logarithmic
divergence arising from finite DOS is suppressed by the background hyperbolic
geometry. To achieve the results, we first apply the vertex inflation method to
generate hyperbolic lattice on the Poincar\'{e} disk, and then apply the
Haydock recursion method to compute DOS. Finally, we study the scaling of EE
for different bipartitions via exact diagonalization and perform finite-size
scaling. We also investigate how the coefficient of the area law is correlated
to bulk gap and DOS. Future directions are discussed.
