Eigenstate thermalization hypothesis in conformal field theory

We investigate the eigenstate thermalization hypothesis (ETH) in d + 1 dimensional conformal field theories by studying the reduced density matrices in energy eigenstates. We show that if the local probes of the finitely excited primary eigenstates satisfy ETH, then any finite energy observable with support on a subsystem of finite size satisfies ETH. In two dimensions, we discover that if ETH holds locally, the finite size reduced density matrix of states created by heavy primary operators is well-approximated by a projection to the Virasoro identity block.