General relation between energy spectrum and entanglement spectrum

We demonstrate that the bipartite density matrix, arising from a spatial bipartitioning of a gapped topological state which possesses gapless edge modes in the form of a conformal field theory (CFT ) (when terminated against a topologically trivial state/vacuum), such as e.g. a general quantum Hall state, is the density matrix of a the chiral edge state CFT at a finite temperature. We obtain this result by applying a physical instantaneous cut of the gapped system, and by viewing the cutting process as a sudden ``quantum quench'' into a CFT, using the tools of boundary conformal field theory. In particular, we obtain a general relation between the Hamiltonian spectrum of gapless theories and the entanglement spectrum of the gapped theory obtained from coupling two gapless theories.