# Tensor Renormalization Group Centered About a Core Tensor

## TL;DR

This paper introduces a modified tensor renormalization group algorithm that reduces computational cost while maintaining accuracy, enabling more efficient analysis of classical models on 2D lattices.

## Contribution

A new tensor renormalization group method that coarse-grains only adjacent rows and columns, reducing computational complexity from O(χ^6) to O(χ^4).

## Key findings

- Cost scales as O(χ^4), cheaper than TRG.
- Accuracy comparable to TRG at same bond dimension.
- Enables analysis of larger or more complex models.

## Abstract

We propose a modified form of a tensor renormalization group algorithm for evaluating partition functions of classical statistical mechanical models on 2D lattices. This algorithm coarse-grains only the rows and columns of the lattice adjacent to a single core tensor at each step, such that the lattice size shrinks linearly with the number of coarse-graining steps as opposed to shrinking exponentially as in the usual tensor renormalization group (TRG). However, the cost of this new approach only scales as $O(\chi^4)$ in terms of the bond dimension $\chi$, significantly cheaper than the $O(\chi^6)$ cost scaling of TRG, whereas numerical benchmarking indicates that both approaches have comparable accuracy for the same bond dimension $\chi$. It follows that the new algorithm can allow for more accurate investigation of classical models on modestly sized lattices than is possible with standard TRG.