Entanglement measures have emerged as one of the versatile probes to diagnose quantum phases and their transitions. Universal features in them expand their applicability to a range of systems, including those with quenched disorder. In this Letter, we show that when the underlying lattice has percolation disorder, free fermions at a finite density show interesting entanglement properties due to massively degenerate ground states. We define and calculate appropriate entanglement measures such as typical, annealed, and quenched entanglement entropy in both one and two dimensions, showing they can capture both geometrical aspects and electronic correlations of the percolated quantum system. In particular, while typical and annealed entanglements show a volume law character directly dependent on the number of zero modes in the system, quenched entanglement is generally an area law albeit showing characteristic signatures of the classical percolation transition. Our work presents an exotic interplay between the geometrical properties of a lattice and quantum entanglement in a many-body quantum system.

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• Entanglement entropy
• Percolation
• Disordered systems
• Random & disordered media
• Tight-binding model