Re-order parameter of interacting thermodynamic magnets.

Phase diagrams of materials are typically based on a static order parameter, but it faces challenges when distinguishing subtle phase changes, such as re-ordering. Here, we report a dynamic nonequilibrium order parameter termed re-order parameter to determine subtle phases and their transitions in interacting magnets. The dynamical precession of magnetization, so-called magnon, premises as a reliable re-order parameter of strong spin-orbit coupled magnets. We employ orthoferrites YFeO and its Mn-doped variations, where diverse magnetic phases, including canted antiferromagnetic (Γ) and collinear antiferromagnetic (Γ) states, have been well-established. Low-energy magnon uncovers the spin-orbit coupling-induced subtle magnetic structures, resulting in distinct terahertz emissions. The temporal and spectral parameters of magnon emission exhibit characteristics akin to BCS-type order parameters, constructing the magnetic phase diagram of Mn-doped YFeO. This approach further reveals a concealed ferrimagnetic phase within the Γ state, underscoring its potential to search for hidden phases of materials, completing their phase diagrams.