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

The characterization of ground states among all quantum states is an important problem in quantum many-body physics. For example, the celebrated entanglement area law for gapped Hamiltonians has allowed for efficient simulation of 1d and some 2d quantum systems using matrix product states. Among ground states, some types, such as cat states (like the GHZ state) or topologically ordered states, can only appear alongside their degenerate partners, as is understood from the theory of spontaneous symmetry breaking. In this 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 an element of a stable gapped ground state manifold, which may provide a new method to discard a wide range of 'unstable' entanglement area law states in the numerical search of gapped phases. On the other hand when these degenerate states are the ground states of gapless systems they possess an excitation spectrum with $O(1/L^2)$ finite-size splitting. One familiar situation where this special kind of gaplessness occurs is at a Lifshitz transition due to a zero mode; a potential quantum state signature of such a critical point. We explore pathological parent Hamiltonians, and discuss generalizations to higher dimensions, other related states, and implications for understanding thermodynamic limits of many-body quantum systems.