  A class of (2+1)-dimensional quantum many body system characterized by an
anisotropic scaling symmetry (Lifshitz symmetry) near their quantum critical
point can be described by a (3+1)-dimensional dual gravity theory with negative
cosmological constant along with a massive vector field, where the scaling
symmetry is realized by the metric as an isometry. We calculate the
entanglement entropy of an excited state of such a system holographically,
i.e., from the asymptotic perturbation of the gravity dual using the
prescription of Ryu and Takayanagi, when the subsystem is sufficiently small.
With suitable identifications, we show that this entanglement entropy satisfies
an energy conservation relation analogous to the first law of thermodynamics.
The non-trivial massive vector field here plays a crucial role and contributes
to an additional term in the energy relation.
