# Determination of the Critical Manifold Tangent Space and Curvature with Monte Carlo Renormalization Group

## TL;DR

This paper introduces a Monte Carlo Renormalization Group method to accurately determine the tangent space and curvature of the critical manifold in statistical systems, reducing critical slowing down and avoiding truncation errors.

## Contribution

It presents a practical numerical approach using variational bias potentials to analyze the critical manifold's geometry near phase transitions.

## Key findings

- Successfully applied to Ising models on various lattices.
- Reduces critical slowing down in Monte Carlo simulations.
- Free of truncation errors in the analysis.

## Abstract

We show that the critical manifold of a statistical mechanical system in the vicinity of a critical point is locally accessible through correlation functions at that point. A practical numerical method is presented to determine the tangent space and the curvature to the critical manifold with Variational Monte Carlo Renormalization Group. Because of the use of a variational bias potential of the coarse-grained variables, critical slowing down is greatly alleviated in the Monte Carlo simulation. In addition, this method is free of truncation error. We study the isotropic Ising model on square and cubic lattices, the anisotropic Ising model and the tricritical Ising model on square lattices to illustrate the method.