Fig. 5: The cubic fixed point.

Stochastic data collapse with both the vison order parameter ∣ϕ∣ and its Binder cumulant. a, b plot the vison order parameter and its Binder cumulant U₂. All data points are connected with polynomial fits to the third order and χ²/d.o.f (χ² per degree of freedom) is close to 1. The location of the crossing in b shifts due to finite-size effects. c, d are the distributions of β and ν with V_c obtained from the bootstrap sampling process. Each data point is determined from a fitted curve of the data collapse for ∣ϕ∣ and B₂, and the the χ²/d.o.f of all the curves is close to 1. We obtain V_c = 0.59(2), β = 0.33(5), and ν = 0.75(8). The values of ν and β are consistent with the standard O(3) value, ν = 0.7112(5) and β = 0.3689(3). e, f show the data collapse of the vison order parameter and its Binder ratio B₂. Here, the values of V_c, β, and ν are determined independently from our fitting procedures. The error bars here represent the standard error of the mean, which is calculated as \(\sigma /\sqrt{N}\), where σ is one standard deviation and N is the total number of independent samples.