Entanglement entropy and mutual information of circular entangling surfaces in the 2 + 1-dimensional quantum Lifshitz model

We investigate the entanglement entropy (EE) of circular entangling cuts in the 2 + 1-dimensional quantum Lifshitz model. The ground state in this model is a spatially conformal invariant state of the Rokhsar-Kivelson type, whose amplitude is the Gibbs weight of 2D Euclidean free boson. We show that the finite subleading corrections of EE to the area-law term, as well as the mutual information, are conformal invariants and calculate them for cylinder, disk-like and spherical manifolds with various spatial cuts. The subtlety due to the boson compactification in the replica trick is carefully addressed. We find that in the geometry of a punctured plane with many small holes, the constant piece of EE is proportional to the number of holes, indicating the ability of entanglement to detect topological information of the configuration. Finally, we compare the mutual information of two small distant disks with Cardy's relativistic CFT scaling proposal. We find that in the quantum Lifshitz model, the mutual information also scales at long distance with a power determined by the lowest scaling dimension local operator in the theory.