### Frustration free gapless Hamiltonians for Matrix Product States

**Carlos Fernández-González, Norbert Schuch, Michael M. Wolf, J. Ignacio Cirac, David Pérez-García**

(2012)

**arXiv.org**: 1210.6613

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**Abstract:**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.