Oscillating hysteresis in theq-neighbor Ising model

We modify the kinetic Ising model with Metropolis dynamics, allowing each spin to interact only with $q$ spins randomly chosen from the whole system, which corresponds to the topology of a complete graph. We show that the model with $q\ge 3$ exhibits a phase transition between ferromagnetic and paramagnetic phases at temperature $T^{*}$, which linearly increases with $q$. Moreover, we show that for $q=3$ the phase transition is continuous and that it is discontinuous for larger values of $q$. For $q>3$, the hysteresis exhibits oscillatory behavior—expanding for even values of $q$ and shrinking for odd values of $q$. Due to the mean-field-like nature of the model, we are able to derive the analytical form of transition probabilities and, therefore, calculate not only the probability density function of the order parameter but also precisely determine the hysteresis and the effective potential showing stable, unstable, and metastable steady states. Our results show that a seemingly small modification of the kinetic Ising model leads not only to the switch from a continuous to a discontinuous phase transition, but also to an unexpected oscillating behavior of the hysteresis and a puzzling phenomenon for $q=5$, which might be taken as evidence for the so-called mixed-order phase transition.