Three-dimensional real space renormalization group with well-controlled approximations

TL;DR

This paper develops a reliable 3D real space renormalization group method using tensor networks and entanglement filtering, achieving high accuracy in critical exponent estimation for the cubic-lattice Ising model.

Contribution

It introduces a tensor-network reformulation with entanglement filtering for 3D RG, enabling systematic error control and high-precision critical point analysis.

Findings

RG errors reduced to about 2% with more couplings

Scaling dimensions estimated with errors of 0.4% and 0.1%

Method can numerically obtain 3D critical fixed points in tensor space

Abstract

We make Kadanoff's block idea into a reliable three-dimensional (3D) real space renormalization group (RG) method. Kadanoff's idea, expressed in spin representation, offers a qualitative intuition for clarifying scaling behavior in criticality, but has difficulty as a quantitative tool due to uncontrolled approximations. A tensor-network reformulation equips the block idea with a measure of RG errors. In 3D, we propose an entanglement filtering scheme to enhance such a block-tensor map, with the lattice reflection symmetry exploited. When the proposed RG is applied to the cubic-lattice Ising model, the RG errors are reduced to about 2% by retaining more couplings. The estimated scaling dimensions of the two relevant fields have errors 0.4% and 0.1% in the best case, compared with the accepted values. The proposed RG is promising as a systematically-improvable real space RG method in 3D. The unique feature of our method is its ability to numerically obtain a 3D critical fixed point in a high-dimensional tensor space. A fixed-point tensor contains much more information than a handful of observables estimated in conventional techniques for analyzing critical systems.