Electronic Fractals in Strongly Correlated Quantum Materials

该奖项支持精炼和扩展技术的理论研究，旨在最大限度地从强相关电子材料的实验中提取信息。冰晶和树木的脆弱性和复杂性在强相关电子材料中找到了对应。在传统的金属和半导体内部，电子通常是均匀分布的，就像液体填充容器一样。但在相关量子材料中，电子彼此之间以及与材料的原子核之间的相互作用很强，电子的行为更像是一种奇异的秋葵汤。这些材料表面的纳米级图像显示，至少在表面上，电子聚集成复杂的形状。了解这些模式的形成对于我们理解电子特性以及我们最终对这些材料的技术控制至关重要。PI为解释和理解在这些材料表面观察到的纳米级电子结构定义了新的范例。这些新的分析方法可以根据仅在材料外部进行的观察得出关于材料内部的结论。关键的观点是，分形的几何结构揭示了它们所处的维度：当分形只在材料表面形成时（如窗户上的霜），当分形延伸到材料深处时（如树根深入地下），分形具有不同的形状。这个新的分析领域揭示了几个强相关电子材料家族的普遍行为。PI计划揭示为什么这些模式在相关材料表面如此普遍。该奖项还支持PI的教育和外展活动。PI继续是科学的普及者。她已经创建了YouTube频道www.youtube.com/profcarlson（浏览量超过65000次），该频道免费提供了她关于电磁学入门的讲座，以及与the Great Courses合作的流行视频系列“理解量子世界”。PI将为普渡大学量子科学与工程学院开设一个YouTube频道，并与The Great Courses一起制作第二个关于量子材料的视频系列。该奖项支持理论研究，旨在改进和扩展由PI在强相关电子系统领域开创的几何聚类分析技术，以便最大限度地利用PI的新方法从实验中提取信息，并促进这些技术在各种材料和图像探针中的广泛应用。为了做到这一点，PI将通过数值模拟，在干净和随机的伊辛模型中推进几何临界理论。在这些计算的成功完成后，几种传统的（和广泛可用的）实验技术将拥有新的数据采集和分析模式，以及能够检测和表征物质新相的新方法。这些研究是变革性的，因为PI将分形数学和无序统计力学的概念和技术引入相关量子材料领域，以便更好地理解分形电子图案的形成及其对这些材料的影响。这些想法的成功实施预计将继续对几种相关量子材料产生潜在的重要影响，包括铜和铁镍超导体、锰、镍酸盐、钴酸盐和钒氧化物。该奖项还支持PI的教育和外展活动。PI继续是科学的普及者。她已经创建了YouTube频道www.youtube.com/profcarlson（浏览量超过65000次），该频道免费提供了她关于电磁学入门的讲座，以及与the Great Courses合作的流行视频系列“理解量子世界”。PI将为普渡大学量子科学与工程学院开设一个YouTube频道，并与The Great Courses一起制作第二个关于量子材料的视频系列。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

NONTECHNICAL SUMMARYThis award supports theoretical research on refining and extending techniques aimed at maximizing the information that can be extracted from experiments performed on strongly correlated electronic materials. The fragility and complexity of ice crystals and trees have found their counterparts in strongly correlated electronic materials. Inside conventional metals and semiconductors, electrons are typically evenly spread out, like liquid filling a container. But in correlated quantum materials, where electrons interact strongly with each other and with the atomic nuclei of the material, electrons act more like an exotic gumbo. Nanoscale images of the surfaces of these materials show that the electrons clump into complicated shapes, at least at the surface. Understanding the formation of these patterns is vital to our understanding of the electronic properties and to our eventual technological control of these materials. The PI has defined new paradigms for interpreting and understanding nanoscale electronic textures observed at the surface of these materials. These new analysis methods allow conclusions to be drawn about the inside of a material, based on observations that are made only on the outside of the material. The key insight is that the geometric structure of fractals reveals the dimension in which they reside: Fractals have different shapes when they form only on the surface of a material (like frost on a window), from when the fractals extend deep inside the material (like a tree whose roots reach deep underground). This new field of analysis has revealed universal behavior across several families of strongly correlated electronic materials. The PI proposes to uncover why these patterns are so ubiquitous at the surface of correlated materials. This award also supports the PI's educational and outreach activities. The PI continues to be a popularizer of science. She has already produced the YouTube channel www.youtube.com/profcarlson (with over 65,000 views) which has made her lectures on introductory electricity and magnetism freely available, as well as the popular video series, “Understanding the Quantum World,” with The Great Courses. The PI will start a YouTube channel for the Purdue Quantum Science and Engineering Instutite, and also produce a second video series with The Great Courses, on Quantum Materials. TECHNICAL SUMMARYThis award supports theoretical research that is aimed at refining and extending the geometric cluster analysis technique pioneered by the PI in the field of strongly correlated electronic systems, in order to maximize the information that can be extracted from experiments using the PI's new methods, and to facilitate the broad application of these techniques to various materials and image probes. In order to do this, the PI will advance the theory of geometric criticality in clean and random Ising models, via numerical simulations. At the successful completion of these calculations, several conventional (and widely available) experimental techniques will have at their disposal new modes of datataking and analysis and new methods enabling the detection and characterization of novel phases of matter.These studies are transformational in that the PI is importing concepts and techniques from fractal mathematics and disordered statistical mechanics into the field of correlated quantum materials in order to better understand fractal electronic pattern formation and its impact on these materials. The successful implementation of these ideas is expected to continue to have potentially important impact in several correlated quantum materials, including cuprate and iron pnictide superconductors, manganites, nickelates, cobaltates, and vanadium oxides.This award also supports the PI's educational and outreach activities. The PI continues to be a popularizer of science. She has already produced the YouTube channel www.youtube.com/profcarlson (with over 65,000 views) which has made her lectures on introductory electricity and magnetism freely available, as well as the popular video series, “Understanding the Quantum World,” with The Great Courses. The PI will start a YouTube channel for the Purdue Quantum Science and Engineering Instutite, and also produce a second video series with The Great Courses, on Quantum Materials.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.