We study two versions of the XY model where the spins but also the interaction topology is allowed to change. In the free XY model, the number of links is fixed, but their positions in the network are not. We also study a more relaxed version where even the number of links is allowed to vary, we call it the freer XY model. When the interaction networks are dense enough, both models have phase transitions visible both in spin configurations and the network structure. The low-temperature phase in the free XY model is characterized by tightly connected clusters of spins pointing in the same direction and isolated spins disconnected from the rest. For the freer XY model the low-temperature phase is almost completely connected. In both models, exponents describing the magnetic ordering are mostly consistent with values of the mean-field theory of the standard XY model.

中文翻译：

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免费和免费的XY模型

我们研究了XY模型的两个版本，其中自旋以及交互拓扑都可以更改。在免费的XY模型中，链接数是固定的，但它们在网络中的位置不是固定的。我们还研究了一种更为宽松的版本，其中甚至允许链接数量发生变化，我们将其称为更自由的XY模型。当交互网络足够密集时，两个模型都具有在自旋配置和网络结构中均可见的相变。自由XY模型中的低温阶段的特征是紧密连接的自旋簇指向相同的方向，而孤立的自旋则与其余部分断开连接。对于更自由的XY模型的低温相几乎完全连接。在这两个模型中，描述磁序的指数大多与标准XY模型的平均场理论值一致。