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

• URL: http://arxiv.org/abs/2412.13758v1
• Date: Wed, 18 Dec 2024 11:53:05 GMT
• Title: Three-dimensional real space renormalization group with well-controlled approximations
• Authors: Xinliang Lyu, Naoki Kawashima,
• Abstract summary: We make Kadanoff's block idea into a reliable 3D real space renormalization group (RG) method. The proposed RG is promising as a systematically-improvable real space RG method in 3D.
• Score: 0.10742675209112622
• License:
• 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 imposed. 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.

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