On the mean Euler characteristic and mean Betti numbers of the Ising model with arbitrary spin

TL;DR

This paper investigates the topological invariants, specifically the mean Euler characteristic and Betti numbers, of the Ising model with arbitrary spin on a 2D lattice, revealing sharp transitions at the critical temperature through Monte Carlo simulations.

Contribution

It provides the first detailed analysis of topological invariants in the Ising model with arbitrary spins, highlighting their behavior at phase transitions.

Findings

Topological invariants exhibit sharp transitions at the critical point.

Mean Euler characteristic and Betti numbers vary significantly with temperature.

Topological measures differ across spin states, reflecting phase changes.

Abstract

The behaviour of the mean Euler-Poincar\'{e} characteristic and mean Betti's numbers in the Ising model with arbitrary spin on $\mathbbm{Z}^2$ as functions of the temperature is investigated through intensive Monte Carlo simulations. We also consider these quantities for each color $a$ in the state space $S\_Q = \{- Q, - Q + 2, ..., Q \}$ of the model. We find that these topological invariants show a sharp transition at the critical point.