Real-space renormalization group in 3D with well-controlled approximations*

Real-space renormalization group (RG) often has uncontrolled approximations. Agreement of the estimated critical exponents with other methods can serve as a posterior justification for a scheme, without explaining why it works.

Inspired by quantum-information concepts, tensor-network RG (TNRG) is a type of real-space RG with definite approximation errors and free of Monte-Carlo sign problem. In 2D criticality, like Ising model, those errors decay exponentially while the dimensionality of RG flows increases algebraically. This provides a promising framework for an accurate 3D real-space RG. However, block-spin idea using tensor-network method in 3D has RG errors more than 20% near Ising critical fixed point. Estimations of critical exponents from linearized RG are unstable with respect to RG steps, with errors over 10%.

We design an entanglement-filtering (EF) scheme for eliminating lattice-scale physics in 3D TNRG. Reducing RG errors to about 5%, the EF-enhanced TNRG leads to more stable tensor RG flows near the Ising criticality. With a few minutes on a desktop computer, the new scheme gives stable estimations of thermal and spin exponents with errors 1% and 5%. This is an important step towards a systematically-improvable 3D real-space RG method.