  The entanglement entropy of an annulus is examined in a three-dimensional
system with or without a gap. For a free massive scalar field theory, we
numerically calculate the mutual information across an annulus. We also study
the holographic mutual information in the CGLP background describing a gapped
field theory. We discover four types of solutions as the minimal surfaces for
the annulus and classify the phase diagrams by varying the inner and outer
radii. In both cases, we find the mutual information satisfies the monotonicity
dictated by the unitarity and decays exponentially fast as the gap scale is
increased. We speculate this is a universal behavior in any gapped system.
