The Mermin-Wagner theorem states that spontaneous continuous symmetry breaking is prohibited in systems with short-range interactions at spatial dimension D ≤ 2. For long-range interactions with a power-law form (1/r^α), the theorem further forbids ferromagnetic or antiferromagnetic order at finite temperature when α ≥ 2D. However, the situation for α ∈ (2, 4) at D = 2 is not covered by the theorem. To address this, we conduct large-scale quantum Monte Carlo simulations and field theoretical analysis. Our findings show spontaneous breaking of SU(2) symmetry in the ferromagnetic Heisenberg model with 1/r^α-form long-range interactions at D = 2. We determine critical exponents through finite-size analysis for α < 3 (above the upper critical dimension with Gaussian fixed point) and 3 ≤ α < 4 (below the upper critical dimension with non-Gaussian fixed point). These results reveal new critical behaviors in 2D long-range Heisenberg models, encouraging further experimental studies of quantum materials with long-range interactions beyond the Mermin-Wagner theorem’s scope.

Mermin-Wagner 定理指出，在空间维度D ≤ 2的短程相互作用的系统中，自发连续对称破缺是被禁止的。 对于具有幂律形式 (1/ r ^α ) 的长程相互作用，该定理进一步禁止铁磁或有限温度下的反铁磁序，当α  ≥​​ 2 D时。 然而，该定理并未涵盖D = 2 时α  ε  (2, 4)的情况。为了解决这个问题，我们进行了大规模量子蒙特卡罗模拟和现场理论分析。我们的研究结果表明，在D = 2 时，具有 1/ r ^α形式长程相互作用的铁磁海森堡模型中S U (2) 对称性自发破缺。我们通过α < 3 的有限尺寸分析 （高于上限）确定临界指数。高斯不动点的临界尺寸）和 3 ≤  α  < 4（低于非高斯不动点的上临界尺寸）。这些结果揭示了二维长程海森堡模型中新的关键行为，鼓励对具有超出默明-瓦格纳定理范围的长程相互作用的量子材料进行进一步的实验研究。