# Irregular parameter dependence of numerical results in tensor renormalization group analysis

## TL;DR

This paper investigates the irregular parameter dependence in tensor renormalization group results, identifying sharp cutoff truncation as a cause and exploring smooth cutoff alternatives to improve numerical stability.

## Contribution

It reveals the cause of irregular behavior in tensor RG analysis and proposes using smooth cutoff truncation to mitigate this issue.

## Key findings

- Sharp cutoff causes irregular behavior during level crossing
- Smooth cutoff reduces irregularities in numerical results
- Parameter dependence becomes more stable with smooth truncation

## Abstract

We study the parameter dependence of numerical results obtained by the tensor renormalization group. We often observe an irregular behavior as the parameters are varied with the method, which makes it difficult to perform the numerical derivatives in terms of the parameter. With the use of two-dimensional Ising model we explicitly show that the sharp cutoff used in the truncated singular value decomposition causes this unwanted behavior when the level crossing happens between singular values below and above the truncation order as the parameters are varied. We also test a smooth cutoff, instead of the sharp one, as a truncation scheme and discuss its effects.