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Local topological order inhibits thermal stability in 2D

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We study the robustness of quantum information stored in
the degenerate ground space of a local, frustration-free Hamiltonian with
commuting terms on a 2D spin lattice. On one hand, a macroscopic energy barrier
separating the distinct ground states under local transformations would protect
the information from thermal fluctuations. On the other hand, local topological
order would shield the ground space from static perturbations.

Here we demonstrate that local topological order implies
a constant energy barrier, thus inhibiting thermal stability. Joint work with
David Poulin.