Finite-size behavior of higher-order cumulant ratios near criticality in two-dimensional Potts models

Abstract

Theoretical considerations predict a specific hierarchy among ratios of net-baryon number cumulants ($\chi_n$, where $n$ is the order of cumulant) in the vicinity of the transition from the low-temperature hadronic phase to the high temperature quark-gluon plasma phase at small baryon chemical potential, $\mu_\mathrm{B}$, in the QCD phase diagram. This hierarchy, $\frac{\chi_6}{\chi_2} < \frac{\chi_5}{\chi_1} < \frac{\chi_4}{\chi_2} < \frac{\chi_3}{\chi_1}$, has been observed by the STAR experiment in net-proton number (a proxy of net-baryon number) cumulant ratios over a broad range of collision energies. Motivated by these findings, we investigate whether similar ordering emerges generically in finite statistical systems undergoing second-order phase transitions. We employ two different spin models: the two-state and three-state Potts models in two dimensions, both exhibiting a transition from an ordered phase to a disordered phase at their respective critical temperatures. Monte Carlo simulations are performed on square lattices of varying sizes using the Wolff cluster algorithm. Cumulants of the total magnetization are calculated up to sixth order in both of these models in a temperature range near their corresponding critical temperatures. Higher-order cumulants exhibit extrema (peaks/troughs) whose magnitudes grow with both cumulant order and lattice size, reflecting enhanced critical fluctuations. Except within a narrow temperature window above the critical temperature, neither the complete hierarchy nor its exact reverse is realized over the studied temperature range in either model.