Binder parameter

The Binder parameter or Binder cumulant^[1][2] in statistical physics, also known as the fourth-order cumulant ${\displaystyle U_{L}=1-{\frac {{\langle s^{4}\rangle }_{L}}{3{\langle s^{2}\rangle }_{L}^{2}}}}$ is defined as the kurtosis of the order parameter, s introduced by Austrian Theoretical Physicist Kurt Binder. It is frequently used to determine accurately phase transition points in numerical simulations of various models. ^[3]

The phase transition point is usually identified comparing the behavior of ${\displaystyle U}$ as a function of the temperature for different values of the system size ${\displaystyle L}$. The transition temperature is the unique point where the different curves cross in the thermodynamic limit. This behavior is based on the fact that in the critical region, ${\displaystyle T\approx T_{c}}$, the Binder parameter behaves as ${\displaystyle U(T,L)=b(\epsilon L^{1/\nu })}$, where ${\displaystyle \epsilon ={\frac {T-T_{c}}{T}}}$.

Accordingly, the cumulant may also be used to identify the universality class of the transition by determining the value of the critical exponent ${\displaystyle \nu }$ of the correlation length. ^[1]

In the thermodynamic limit, at the critical point, the value of the Binder parameter depends on boundary conditions, the shape of the system, and anisotropy of correlations. ^[1][4][5][6]

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