Hysteresis loop signatures of phase transitions in a mean-field model of disordered Ising magnet

In accordance with recent experiments the mean-field type theories predict the presence of numerous metastable minima (states) in the rugged free-energy landscape of frustrated disordered magnets. This multiplicity of long-lived states with lifetimes greater than $10^5 s$ makes the task to experimentally determine which of them has the lowest free energy (and thus what thermodynamic phase the sample is in) seem rather hopeless the more so as we do not know a protocol (such as field-cooling or zero-field-cooling) leading to the equilibrium state(s). Nevertheless here we show in the framework of Landau-type phenomenological model that signatures of the mean-field equilibrium phase transitions in such highly nonequilibrium systems may be found in the evolution of the hysteresis loop form. Thus the sequence of transitions from spin-glass to mixed phase and to ferromagnetic one results in the changes from inclined hysteresis loop to that with the developing vertical sides and to one with the perfectly vertical sides. Such relation between loop form and the location of global minimum may hold beyond the mean-field approximation and can be useful in the real experiments and Monte-Carlo simulations of the problems involving rugged potential landscape. Also the very existence of the quasi-static loops in spin glass and mixed phases implies that the known disorder-smoothing of the first-order transition can be always accompanied by the emergence of multiple metastable states.