Binder cumulants and critical-point estimation

Let m be total spin divided by the number of sites. Define U as one minus the fourth moment divided by three times the square of the second moment. A centered Gaussian has fourth moment three times variance squared and therefore gives zero. An equal mixture concentrated at two nonzero opposite magnetizations has fourth moment equal to second moment squared and gives two thirds. These are limiting distribution checks, not universal values for every finite temperature and lattice size.

The ratio of moments often called B four equals three times one minus U in this convention; it is not the same numerical observable. At finite zero-field volume the signed magnetization expectation vanishes by symmetry if both sectors are sampled, while its second moment stays positive. Replacing moments of m by central moments around a trapped sample mean changes the quantity. The fourth and second moments must also come from the same equilibrium ensemble, including any consistent reweighting factors.

For a short-range system in an ordinary finite-size scaling regime, a dimensionless observable is a function of reduced coupling times size raised to the inverse correlation-length exponent, with corrections from irrelevant fields. Fix lattice shape and boundary conditions across sizes before comparing its curves. At the critical coupling, the leading scaling variable vanishes, so the leading Binder value is size independent. The correction terms generally remain nonzero and shift the crossing of any finite pair away from the infinite-volume critical point.

Expanding around criticality gives one term linear in reduced coupling and another proportional to a negative power of size. Equating the expansions at L and a fixed multiple of L shows that the crossing shift decays with the sum of the correction exponent and inverse nu, provided their leading amplitudes are nonzero and no additional singular terms dominate. Improved actions can suppress an amplitude, while small lattices can fall outside this asymptotic form. The crossing value itself also depends on shape and boundary conventions.

A simulation at nearby couplings may use reweighting to interpolate the moments, but the two moment curves are correlated because they use the same configurations. Numerically solving their Binder crossing adds another nonlinear operation. Use matched block resamples to reconstruct moments, cumulants, and the crossing inside each resample. Treating many reweighted coupling points as independent measurements would make the apparent precision grow without adding information. Independent sizes may still share analysis or tuning uncertainties that need explicit handling.

Report the crossing pair, coupling convention, interpolation or reweighting range, and assumed drift model. Stability under larger minimum sizes and independent runs is more persuasive than many digits from one intersection. Compare another dimensionless observable such as a correlation-length ratio when available, since different correction amplitudes can expose hidden drift. The quadratic lab below evaluates the squared denominator ingredient only; it neither generates magnetization moments nor locates a thermodynamic transition.