Binder cumulant

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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.


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