Frustration free gapless Hamiltonians for Matrix Product States
  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. 
Carlos Fern\'andez-Gonz\'alez, Norbert Schuch, Michael M. Wolf, J.   Ignacio Cirac and David P\'erez-Garc\'ia
quant-ph cond-mat.str-el