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The characterization of ground states among all quantum states is animportant problem in quantum many-body physics. For example, the celebratedentanglement area law for gapped Hamiltonians has allowed for efficientsimulation of 1d and some 2d quantum systems using matrix product states. Amongground states, some types, such as cat states (like the GHZ state) ortopologically ordered states, can only appear alongside their degeneratepartners, as is understood from the theory of spontaneous symmetry breaking. Inthis work, we introduce a new class of simple states, including the $W$ state,that can only occur as a ground state alongside an exactly degenerate partner,even in gapless or disordered models. We show that these states are never anelement of a stable gapped ground state manifold, which may provide a newmethod to discard a wide range of 'unstable' entanglement area law states inthe numerical search of gapped phases. On the other hand when these degeneratestates are the ground states of gapless systems they possess an excitationspectrum with $O(1/L^2)$ finite-size splitting. One familiar situation wherethis special kind of gaplessness occurs is at a Lifshitz transition due to azero mode; a potential quantum state signature of such a critical point. Weexplore pathological parent Hamiltonians, and discuss generalizations to higherdimensions, other related states, and implications for understandingthermodynamic limits of many-body quantum systems.

$W$ state is not the unique ground state of any local Hamiltonian