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, 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 estimatation 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)