Kadanoff’s block idea pioneers the renormalization group (RG) theory and clarifies the scaling hypothesis in critical phenomena. Nevertheless, it has difficulty as a quantitatively reliable RG method due to uncontrolled approximations when formulated in the spin language. Reformulated in a modern tensor-network language, the block idea is equipped with a natural measure of RG errors. In 2D, the RG errors are typically smaller than 1% and decrease systematically when more coupling constants are retained in the RG map. The relative error of the estimated free energy of the 2D Ising model can easily go down to about 10-9 using a personal computer.
In 3D, due to the linear growth of entanglement entropy, the RG errors are too large for the block-tensor map to be reliable. For the 3D Ising model, the RG errors grow to more than 10% just after one RG step, and then keep growing to more than 30% near the critical fixed point. Even worse, the estimated scaling dimensions fail to converge with respect to the RG step. We propose an entanglement filtering (EF) scheme to cleanse the redundant entanglement. Enhanced by the proposed EF, the RG errors near the critical fixed point goes down to 6%; they decrease slowly to 2% when more couplings are retained. The estimated scaling dimensions become stable respect to the RG step. The relative errors of the first two relevant fields are 0.4% and 0.1% in the best case. The proposed RG is promising as a systematically-improvable real space RG method in 3D.
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Giovanni Rizi, Julio Parra-Martinez
