Shapes of quantum entanglement*

Persistent homology is a relatively new computational tool to study shapes that are present at different length scales in discrete data. We use this method to study the structure of entanglement entropy in quantum states (obtained through exact diagonalization). The shapes in the entanglement structure are summarized in a barcode that reveals geometric and topological information. We show that abrupt changes in the barcode indicate a quantum phase transition for the example of a transverse-field Ising chain. Beyond this basic demonstration, we also analyze the XXZ spin chain in a random tranverse field, providing a new angle to study the elusive many-body localization transition. Finally, we discuss the promising future applications of this modern computational approach.