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arXiv:2507.10018 · Physical review. E

Crossover in the ordered phase in the non-Mermin-Wagner-Hohenberg regime of spin models with long-range coupling

Jiewei Ding, Jiahao Su, Ho-Kin Tang, Wing Chi Yu

Published Jun 10, 2026• 0 citations
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Abstract

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.

DOI
10.1103/sctc-q3cq

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The d-dimensional long-range Ising model, defined by spin-spin interactions decaying with the distance as the power 1/r^{d+s}, admits a second-order phase transition with continuously varying critical exponents. At s=s_{*}, the phase transition crosses over to the usual short-range universality class. The standard field-theoretic description of this family of models is strongly coupled at the crossover. We find a new description, which is instead weakly coupled near the crossover, and use it to compute critical exponents. The existence of two complementary UV descriptions of the same long-range fixed point provides a novel example of infrared duality.
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The phase of spins in the quasi-two-dimensional (q2D) XY model has emerged as a topic of significant interest across multiple subfields of physics. Conventional wisdom, rooted in the Mermin-Wagner theorem and supported by existing paradigms, asserts that true long-range (LR) order is prohibited in q2D systems with continuous symmetries and short-range (SR) interactions. In this Letter, we propose a strictly SR q2D XY model defined on a plane perpendicularly intersected by a group of parallel planes, where each plane consists of XY spins coupled via nearest-neighbor interactions. Through large-scale Monte Carlo simulations complemented by finite-size scaling analysis, we establish the complete phase diagram of the setup. A LR ordered phase emerges in the q2D model when the spins on the parallel planes develop a Berezinskii-Kosterlitz-Thouless critical phase. The LR ordered phase is anisotropic: true LR correlations develop exclusively along the direction of the intersection lines, while the perpendicular direction exhibits quasi-long-range order. Furthermore, the LR order exhibits Goldstone-mode physics. Our findings reveal a mechanism for stabilizing LR order in low-dimensional systems with continuous symmetries, thereby establishing a new platform for studying exotic superfluidity.
Critical exponents for a one-dimensional general continuous Ising model with long-range ferromagnetic interactions decaying as 1/r1+σ are calculated using a histogram Monte Carlo technique. A continuous Ising model means that a spin can take any value between −1 and 1. The critical point behavior is investigated. It is found that the system exhibits a second-order phase transition with nonstandard critical exponents; the singularities in the specific heat and susceptibility depend on σ. For σ=1, there are extremely weak finite size effects: the first-order and the second-order cumulants of the order parameter yield ν=2.42(1). The susceptibility exponent γ=2.259(9). Results for σ=0.7 will be shown and discussed.