Eigen-microstate Signatures of Criticality in Relativistic Heavy-Ion Collisions

We develop the eigen-microstate framework as a new approach to identify criticality in relativistic heavy-ion collisions. We construct the original microstate, defined as the final-state particle fluctuations of a single event. By examining ensembles of such original microstates with and without critical signals, we demonstrate that the corresponding eigen-microstate can extract and reveal the dominant critical mode, with the largest eigenvalue serving as a robust order parameter. This framework avoids equilibrium assumptions and is insensitive to non-critical background, and the approach is directly applicable to RHIC Beam Energy Scan data, offering a powerful new tool in the search for the QCD critical point.