  Eigenstate thermalization in quantum many-body systems implies that
eigenstates at high energy are similar to random vectors. Identifying systems
where at least some eigenstates are non-thermal is an outstanding question. In
this work we show that interacting quantum models that have a nullspace -- a
degenerate subspace of eigenstates at zero energy (zero modes), which
corresponds to infinite temperature, provide a route to non-thermal
eigenstates. We analytically show the existence of a zero mode which can be
represented as a matrix product state for a certain class of local
Hamiltonians. In the more general case we use a subspace disentangling
algorithm to generate an orthogonal basis of zero modes characterized by
increasing entanglement entropy. We show evidence for an area-law entanglement
scaling of the least entangled zero mode in the broad parameter regime, leading
to a conjecture that all local Hamiltonians with the nullspace feature zero
modes with area-law entanglement scaling, and as such, break the strong
thermalization hypothesis. Finally, we find zero-modes in constrained models
and propose setup for observing their experimental signatures.
