Long-range quantum entanglement in noisy cluster states
  We describe a phase transition for long-range entanglement in a three-dimensional cluster state affected by noise. The partially decohered state is modeled by the thermal state of a suitable Hamiltonian. We find that the temperature at which the entanglement length changes from infinite to finite is nonzero. We give an upper and lower bound to this transition temperature. 
Robert Raussendorf, Sergey Bravyi and Jim Harrington