Cornering Gapless Quantum States via Their Torus Entanglement

The entanglement entropy (EE) has emerged as an important window into the structure of complex quantum states of matter. We analyze the universal part of the EE for gapless systems on tori in 2D and 3D, denoted by chi. Focusing on scale-invariant systems, we derive general nonperturbative properties for the shape dependence of chi and reveal surprising relations to the EE associated with corners in the entangling surface. We obtain closed-form expressions for chi in 2D and 3D within a model that arises in the study of conformal field theories (CFTs), and we use them to obtain Ansatze without fitting parameters for the 2D and 3D free boson CFTs. Our numerical lattice calculations show that the Ansatze are highly accurate. Finally, we discuss how the torus EE can act as a fingerprint of exotic states such as gapless quantum spin liquids, e.g., Kitaev's honeycomb model.