Doubling of entanglement spectrum in tensor renormalization group

We investigate the entanglement spectrum in HOTRG-tensor renormalization group (RG) method combined with the higher order singular value decomposition-for two-dimensional (2D) classical vertex models. In the off-critical region, it is explained that the entanglement spectrum associated with the RG transformation is described by "doubling" of the spectrum of a corner transfer matrix. We then demonstrate that the doubling actually occurs for the square-lattice Ising model by HOTRG calculations up to D = 64, where D is the cutoff dimension of tensors. At the critical point, we also find that a nontrivial D scaling behavior appears in the entanglement entropy. We mention about the HOTRG for the 1D quantum system as well.