# Binder Cumulant

## Introduction

In Statistical mechanics of many body systems, it is easy to use finite sub-systems from an infinite systems. So, our limitation is the finite size of our lattice and which eventually turns into a problem of recognizing the specific point at which the phase transition occurs.

The Binder cumulant (also known as Binder parameter) was introduced by Austrian theoretical physicist Kurt Binder in the context of finite size scaling. It is an observational tool which helps us to estimate that (critical) point. The main idea of this tool is that, for different sizes of the lattices, the average order parameter curve always passes through a fixed point, which coincides with the critical points. In other words, the intersection between the different cumulant ($U$) curves of different sizes of lattice gives the critical point. Thus, it is a visual characteristics for detection of phase transition.

## Definition

• It is defined as the kurtosis of the order parameter $s$ such that given system size $L$, the binder cumulant

$$U_{L}=1-{\frac {{\langle s^{4}\rangle }_{L}}{3{\langle s^{2}\rangle }_{L}^{2}}}$$

• It’s the fourth-order cumulant of the order parameter.
• Caution is needed in identifying the universality class from the critical value of the Binder cumulant, because that value depends on boundary condition, system shape, and anisotropy of correlations.

Published on Aug 13, 2021

Last revised on Jan 18, 2023

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