Crossover in the Ordered Phase in the Non-Mermin-Wagner-Hohenberg Regime of Spin Models with Long-Range Coupling

Continuous spin models with long-range interactions of the form $r^{-σ}$, where $r$ is the distance between two spins and $σ$ controls the decay of the interaction, exhibit enhanced order that competes with thermal disturbances, leading to a richer variety of phases and types of phase transitions. In-depth research in this area not only aids in comprehending the complex behaviors in theoretical models but also provides valuable insights into the diverse phase transitions observed in real materials. Here, we identify that the true long-range ordered phase encompasses distinct scaling regimes, which we term Enhance Long-Range Ordered (EnLRO) and Reduce Long-Range Ordered (ReLRO) regimes. In the one-dimensional XY model, the crossover from EnLRO to ReLRO regimes occurs around $σ\approx 1.575$, while in two dimensions, the crossover happens near $σ\approx 3.2$. Applying finite-size scaling analysis, we extract the critical exponents that characterize the order-to-disorder phase transitions in the EnLRO and ReLRO regimes, constructing comprehensive phase diagrams. The analysis is further extended to the 1D and 2D long-range Heisenberg models, where we find the EnLRO-ReLRO crossover at $σ\approx 1.575$ and $σ\approx 3.22$, respectively. The similar crossover points suggest that the distinction between EnLRO and ReLRO regimes is a generic feature in continuous spin models with long-range interactions.