Entanglement scaling behaviors of free fermions on hyperbolic lattices

Recently, tight-binding models on hyperbolic lattices (discretized anti–de Sitter 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 a <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>d</a:mi></a:math>-dimensional translation-invariant Euclidean lattice, the EE of band insulators scales as an area law (<b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:mo>∼</b:mo><b:msup><b:mi>L</b:mi><b:mrow><b:mi>d</b:mi><b:mo>−</b:mo><b:mn>1</b:mn></b:mrow></b:msup></b:mrow></b:math>, where <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mi>L</c:mi></c:math> is the linear size of the boundary between two subsystems). Meanwhile, the EE of metals [with a finite density of state (DOS)] scales as the renowned Gioev-Klich-Widom scaling law (<d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mrow><d:mo>∼</d:mo><d:msup><d:mi>L</d:mi><d:mrow><d:mi>d</d:mi><d:mo>−</d:mo><d:mn>1</d:mn></d:mrow></d:msup><d:mo form="prefix">ln</d:mo><d:mi>L</d:mi></d:mrow></d:math>). The appearance of logarithmic divergence, as well as the analytic form of the coefficient <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mi>c</f:mi></f:math>, 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 the inapplicability of these analytic approaches and the potential nontrivial behavior of the EE. Here we make an initial attempt through numerical simulation. Remarkably, we find that both cases adhere to the area law, indicating the effect of background hyperbolic geometry that influences quantum entanglement. To achieve the results, we first apply the vertex-inflation method to generate a hyperbolic lattice on the Poincaré disk, and then apply the Haydock recursion method to compute the DOS. Finally, we study the scaling of the 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 the bulk gap in the gapped case and to the DOS in the gapless case, respectively. Future directions are discussed.