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 fluctuations, leading to a wide variety of phases and types of phase transitions. Here, we identify that the true long-range ordered phase encompasses distinct scaling regimes, which we term enhanced long-range ordered (EnLRO) and reduced long-range ordered (ReLRO) regimes. In the former regime, the spin-spin correlation function decays exponentially to a finite value, whereas in the latter regime it decays algebraically to a finite value. In the one-dimensional XY model, the crossover from EnLRO to ReLRO regimes occurs around σ≈1.575, while in two dimensions, the crossover happens near σ≈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 one- and two-dimensional long-range Heisenberg models, where we find the EnLRO-ReLRO crossover at σ≈1.575 and σ≈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. The persistence of EnLRO regime can be attributed to the interplay between the short-range spin wave and the long-range order.