Binder parameter

The Binder parameter in statistical physics, also known as the fourth-order cumulant $U_{L}=1-{\frac {{\langle s^{4}\rangle }_{L}}{3{\langle s^{2}\rangle }_{L}^{2}}}$ in Ising model, is used to identify phase transition points in numerical simulations. It is defined as the kurtosis of the order parameter. For example in spin glasses one defines the Binder as

${B={\frac {1}{2}}\left(3-{\frac {\overline {\langle q^{4}\rangle }}{{\overline {\langle q^{2}\rangle }}^{2}}}\right)}$

where $\langle \cdot \rangle$ stands for Boltzmann average, ${\overline {\cdot }}$ for average over the disorder and $q$ is the overlap between two identical replicas of the system. The phase transition point is usually identified comparing the behavior of $B$ as a function of the temperature for different values of the system size $L$. The transition temperature is the unique point where the different curves cross. This is based on finite size scaling hypothesis, according to which, in the critical region $T\approx T_{c}$ the Binder behaves as $B(T,L)=b(\epsilon L^{1/\nu })$, where $\epsilon ={\frac {T-T_{c}}{T}}$.