Non-Gaussian statistics of the order parameter across a phase transition

Second-order phase transitions are characterized by critical scaling and universality¹. The singular behaviour of thermodynamic quantities at the transition, in particular, is determined by critical exponents of the universality class of the transition. However, critical properties are also characterized by the probability distribution of the order parameter across the transition^2,3, in which non-Gaussian statistics are expected^4-6, but remain largely unexplored⁷. Here, making use of single-atom-resolved detection in momentum space⁸, we measure the full probability distribution of the order-parameter amplitude across a continuous phase transition in an interacting lattice Bose gas⁹. We find that fluctuations are captured by an effective potential-reconstructed from the measured probability distribution by analogy with Landau theory¹⁰-displaying a non-trivial minimum in the superfluid (ordered) phase, which vanishes at the transition point. Moreover, we observe non-Gaussian statistics of the order parameter near the transition, distinguished by non-zero high-order cumulants undergoing abrupt sign changes. We show numerically that these sign changes of the cumulants obey critical scaling in homogeneous systems, and that their experimental behaviour is not reproduced by classical models, whereas it is captured by a low-temperature quantum model. Our results underscore the crucial role of order parameter statistics in probing critical phenomena and universality.