  We investigate the scaling of the bipartite entanglement entropy across
Lifshitz quantum phase transitions, where the topology of the Fermi surface
changes without any changes in symmetry. We present both numerical and
analytical results which show that Lifshitz transitions are characterized by a
well-defined set of critical exponents for the entanglement entropy near the
phase transition. In one dimension, we show that the entanglement entropy
exhibits a length scale that diverges as the system approaches a Lifshitz
critical point. In two dimensions, the leading and sub-leading coefficients of
the scaling of entanglement entropy show distinct power-law singularities at
critical points. The effect of weak interactions is investigated using the
density matrix renormalization group algorithm.
