We investigate the relationship between the N-clock model (also known as planar Potts model or -model) and the XY model (at zero temperature) through a Γ-convergence analysis of a suitable rescaling of the energy as both the number of particles and N diverge. We prove the existence of rates of divergence of N for which the continuum limits of the two models differ. With the aid of Cartesian currents we show that the asymptotics of the N-clock model in this regime features an energy that may concentrate on geometric objects of various dimensions. This energy prevails over the usual vortex-vortex interaction energy. © 2022 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.

中文翻译：

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N时钟模型变分分析中集中效应的出现

我们通过Γ收敛分析对能量的适当重新标度作为粒子的数量和N发散。我们证明了两个模型的连续极限不同的N的发散率的存在。在笛卡尔电流的帮助下，我们证明了N -clock 模型在该状态下的渐近线具有可能集中在各种维度的几何对象上的能量。这种能量胜过通常的涡旋-涡旋相互作用能。© 2022 作者。纯数学与应用数学通讯由威利期刊有限责任公司出版。