  We investigate the entanglement entropy (EE) of circular entangling cuts in
the 2+1-dimensional quantum Lifshitz model, whose ground state wave function is
a spatially conformal invariant state of the Rokhsar-Kivelson type, whose
weight 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.
