  For every Matrix Product State (MPS) one can always construct a so-called
parent Hamiltonian. This is a local, frustration free, Hamiltonian which has
the MPS as ground state and is gapped. Whenever that parent Hamiltonian has a
degenerate ground state (the so-called non-injective case), we construct
another 'uncle' Hamiltonian which is local and frustration free but gapless,
and its spectrum is $\R^+$. The construction is obtained by linearly perturbing
the matrices building up the state in a random direction, and then taking the
limit where the perturbation goes to zero. For MPS where the parent Hamiltonian
has a unique ground state (the so-called injective case) we also build such
uncle Hamiltonian with the same properties in the thermodynamic limit.
