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

Area of Science:

• Condensed Matter Physics
• Statistical Mechanics
• Computational Physics

Background:

• Kadanoff's block concept provides qualitative insight into critical scaling behavior.
• Traditional real-space renormalization group (RG) methods struggle with quantitative accuracy in 3D due to approximations.
• Tensor-network formulations offer a path to quantify RG errors.

Purpose of the Study:

• To develop a reliable and systematically improvable three-dimensional (3D) real-space renormalization group (RG) method.
• To enhance Kadanoff's block idea into a quantitative RG tool using tensor networks.
• To numerically obtain 3D critical fixed points in high-dimensional tensor spaces.

Main Methods:

• Reformulated Kadanoff's block idea using tensor networks to enable error measurement.
• Developed an entanglement filtering scheme for enhancing the block-tensor map in 3D.
• Exploited lattice reflection symmetry within the RG framework.
• Applied the method to the cubic-lattice Ising model.

Main Results:

• Achieved a significant reduction in RG errors, down to approximately 2%, by retaining more couplings.
• Estimated scaling dimensions for two relevant fields with high accuracy (0.4% and 0.1% error).
• Successfully obtained a 3D critical fixed point in a high-dimensional tensor space.

Conclusions:

• The proposed tensor-network-based 3D real-space RG method is a promising, systematically improvable approach.
• This method offers a quantitative and reliable alternative to conventional techniques for analyzing critical systems.
• The fixed-point tensor contains richer information than traditional observables, enabling deeper insights into critical phenomena.