# Anisotropic Tensor Renormalization Group

## TL;DR

The paper introduces ATRG, an efficient tensor renormalization algorithm for lattice models that reduces computational costs in higher dimensions while maintaining versatility and lattice topology.

## Contribution

It presents ATRG, a novel anisotropic tensor renormalization method that improves efficiency and reduces memory use compared to HOTRG, especially in higher dimensions.

## Key findings

- Successfully applied to square and cubic lattice Ising models.
- Achieves lower computational cost with comparable accuracy to HOTRG.
- Significantly reduces memory footprint in higher-dimensional models.

## Abstract

We propose a new tensor renormalization group algorithm, Anisotropic Tensor Renormalization Group (ATRG), for lattice models in arbitrary dimensions. The proposed method shares the same versatility with the Higher-Order Tensor Renormalization Group (HOTRG) algorithm, i.e., it preserves the lattice topology after the renormalization. In comparison with HOTRG, both of the computation cost and the memory footprint of our method are drastically reduced, especially in higher dimensions, by renormalizing tensors in an anisotropic way after the singular value decomposition. We demonstrate the ability of ATRG for the square lattice and the simple cubic lattice Ising models. Although the accuracy of the present method degrades when compared with HOTRG of the same bond dimension, the accuracy with fixed computation time is improved greatly due to the drastic reduction of the computation cost.