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 space (the so-called non-injective case), we construct another ‘uncle’ Hamiltonian which is also local and frustration free, has the same ground state space, but is gapless, and its spectrum is (Formula presented.). 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.