We study Hamiltonians which have Kitaev's toric code as a ground state, and show how to construct a Hamiltonian which shares the ground space of the toric code, but which has gapless excitations with a continuous spectrum in the thermodynamic limit. Our construction is based on the framework of Projected Entangled Pair States (PEPS), and can be applied to a large class of two-dimensional systems to obtain gapless "uncle Hamiltonians".
@article{arxiv.1111.5817,
title = {Gapless Hamiltonians for the toric code using the PEPS formalism},
author = {Carlos Fernández-González and Norbert Schuch and Michael M. Wolf and J. Ignacio Cirac and David Pérez-García},
journal= {arXiv preprint arXiv:1111.5817},
year = {2013}
}