具有竞争作用的非局部自由能泛函的泡沫极小元
Froth-like minimizers of a non local free energy functional with competing interactions
摘要：
本文研究了在平均场水平上描述同时具有铁磁和反铁磁相互作用的自旋系统的一维非局域自由能泛函的基态和低能态。特别地，反铁磁相互作用被假定为具有比铁磁相互作用大得多的范围。这两种效应之间的竞争，预计将导致自发出现的一个定期交替的长时间间隔上的自旋剖面被磁化或向上或向下，与一个振荡规模之间的范围的铁磁和反铁磁相互作用。在这个意义上，最佳或准最佳的配置文件是“泡沫状”：如果在反铁磁势的尺度上看，他们看起来中性，但如果在显微镜下看，他们实际上是由两个不同的阶段相互交替的大气泡。在本文中，我们证明了这一图片的有效性，我们计算的准最佳配置文件的振荡规模，我们量化他们的距离在范数从一个参考周期性的配置文件。证明包括两个主要步骤：我们首先粗粒的范围之间的铁磁势和预期的最佳振荡规模的范围内的中间规模的系统，以这种方式，我们减少了原来的功能，一个有效的“尖锐的接口”。接下来，我们研究后者的反射积极性的方法，这需要作为一个关键成分的准确的局部性的短期。我们的证明具有将粗粒化与反射正性方法相结合的概念性兴趣，这一想法可能在比这里研究的更一般的上下文中有用。
英文摘要：
We investigate the ground and low energy states of a one dimensional non local free energy functional describing at a mean field level a spin system with both ferromagnetic and antiferromagnetic interactions. In particular, the antiferromagnetic interaction is assumed to have a range much larger than the ferromagnetic one. The competition between these two effects is expected to lead to the spontaneous emergence of a regular alternation of long intervals on which the spin profile is magnetized either up or down, with an oscillation scale intermediate between the range of the ferromagnetic and that of the antiferromagnetic interaction. In this sense, the optimal or quasi-optimal profiles are "froth-like": if seen on the scale of the antiferromagnetic potential they look neutral, but if seen at the microscope they actually consist of big bubbles of two different phases alternating among each other. In this paper we prove the validity of this picture, we compute the oscillation scale of the quasi-optimal profiles and we quantify their distance in norm from a reference periodic profile. The proof consists of two main steps: we first coarse grain the system on a scale intermediate between the range of the ferromagnetic potential and the expected optimal oscillation scale; in this way we reduce the original functional to an effective "sharp interface" one. Next, we study the latter by reflection positivity methods, which require as a key ingredient the exact locality of the short range term. Our proof has the conceptual interest of combining coarse graining with reflection positivity methods, an idea that is presumably useful in much more general contexts than the one studied here.
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