双射 BKT 相变
Bijecting the BKT transition
AI总结：
本文通过概率方法证明二维 XY 模型的相关长度恰为对偶高度函数相关长度的两倍，从而严格验证了 BKT 相变与高度函数退局域化相变相重合的启发式预测。
摘要：
我们考虑二维正方格图上的经典 XY 模型（或经典转子模型）及其对偶模型，后者是一个高度函数模型。XY 模型具有一个称为 Berezinskii-Kosterlitz-Thouless 相变的相变。有一个启发式预测认为，该相变应与高度函数的局部化-退局域化相变一致：原始模型与对偶模型共享相同的配分函数，并且当用逆温度表示时，任一模型的相变都应与该配分函数唯一的非解析点重合。我们使用概率论证证明了 XY 模型的相关长度（质量的倒数）恰好是高度函数相关长度的两倍，这特别意味着该启发式预测是正确的：即 XY 模型的 BKT 相恰好与对偶高度函数的退局域化相一致。
英文摘要：
We consider the classical XY model (or classical rotor model) on the two-dimensional square lattice graph as well as its dual model, which is a model of height functions. The XY model has a phase transition called the Berezinskii-Kosterlitz-Thouless transition. There is a heuristic which predicts that this phase transition should coincide with the localisation-delocalisation transition for height functions: the primal and dual model share the same partition function, and the phase transition of either model should coincide with the unique non-analytic point of the partition function when expressed in terms of the inverse temperature. We use probabilistic arguments to prove that the correlation length (the reciprocal of the mass) of the XY model is exactly twice the correlation length of the height function, which implies in particular that the prediction of this heuristic is correct: namely, that the BKT phase for the XY model coincides exactly with the delocalised phase of the dual height function.
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