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

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.

11 pages, 6 figures and 1 table; add more explanations for TNRG and details of the numerical calculations

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Cited by in corpus (5)

• A Practical Introduction to Tensor Network Renormalization with TNRKit.jl
• Tensor renormalization group approach to critical phenomena via symmetry-twisted partition functions
• Lattice-reflection symmetry in tensor-network renormalization group with entanglement filtering in two and three dimensions
• Essential difference between 2D and 3D from the perspective of real-space renormalization group
• Lattice and PT symmetries in tensor-network renormalization group: Case study of a hard-square lattice gas model