Plaquette Renormalization Scheme for Tensor Network States

We present a method for contracting a square-lattice tensor network in two dimensions based on auxiliary tensors accomplishing successive truncation (renormalization) of the effective 8-index tensors for $2\times 2$ plaquettes into 4-index tensors. The schheme is variational, and thus the tensors can be optimized by minimizing the energy. Test results for the quantum phase transition of the transverse-field Ising model confirm that even the smallest possible tensors (two values for each tensor index at each renormalization level) produce much better results than the simple product (mean-field) state. We also discuss several extensions of the scheme.