# Exploring Different Extrapolation Approaches for the Critical Temperature of the 2D-Ising Model Based on Exactly Solvable Finite-Sized Lattices

## TL;DR

This paper explores methods to estimate the critical temperature of the 2D-Ising model using small, exactly solvable lattices and compares their effectiveness.

## Contribution

A novel single-parameter model for extrapolating the critical temperature using the N/(N+1)-law is introduced and validated.

## Key findings

- Both scaling models and envelope constructions converge to the same asymptotic critical temperature value.
- The proposed N/(N+1)-law model provides robust convergence with high accuracy.
- Highly accurate finite-size results are sufficient for precise extrapolation of the critical temperature.

## Abstract

The fact that the Ising model in higher dimensions than 1D features a phase transition at the critical temperature Tc despite its apparent simplicity is one of the main reasons why it has lost none of its fascination and remains a central benchmark in modeling physical systems. Building on our previous work, where an approximative analytic free-energy expression for finite 2D-Ising lattices was introduced, we investigate different extrapolation strategies for estimating Tc of the infinite system from exactly solvable small lattices. Finite square lattices of linear dimension N with free and periodic boundary conditions were analyzed, exploiting their exactly accessible density of states to compute the heat capacity profiles C(T). Different approaches were compared, including scaling models for the peak temperature Tmax(N) and an envelope construction across the set of C(T)-profiles. We find that both approaches converge to the same asymptotic value and compare favorably to the established Binder cumulant method. Remarkably, a model for Tmax with a single model parameter following an N/(N+1)-law provides robust convergence, with a physical analogy motivating this proportionality. Our findings highlight that surprisingly few, but highly accurate, finite-size results are sufficient to obtain a precise extrapolation.