Binder cumulant

The Binder cumulant was introduced by Kurt Binder in the context of finite size scaling. It is a quantity that is supposed to be invariant for different system sizes at criticality. For an Ising model with zero field, it is given by

$U_{4}=1-{\frac {\langle m^{4}\rangle }{3\langle m^{2}\rangle ^{2}}}$

where m is the order parameter. It is therefore a fourth order cumulant, related to the kurtosis. In the thermodynamic limit, where the system size $L\rightarrow \infty$, $U_{4}\rightarrow 0$ for $T>T_{c}$, and $U_{4}\rightarrow 2/3$ for $T<T_{c}$. Thus, the function is discontinuous in this limit. The useful fact is that curves corresponding to different system sizes (which are, of course, continuous) all intersect at approximately the same temperature, which provides a convenient estimate for the value of the critical temperature.

References

1. K. Binder "Finite size scaling analysis of ising model block distribution functions", Zeitschrift für Physik B Condensed Matter 43 pp. 119-140 (1981)