Correlation Effects in Topological Insulators

控制单个电子和原子行为的定律简单、优雅，而且寥寥无几。但是，我们如何解释我们观察到的物质所呈现的无穷无尽的各种形式呢？凝聚态物理学试图解释这些观测结果，并发现物质的新状态。物质的不同状态可以被认为是一组电子或原子可以自我组织成的不同模式，就像一组舞者可以表演不同的舞蹈一样。这个领域的一个令人着迷的问题是，有多少这样的模式符合量子力学定律，并导致物质的量子态。最近发现的拓扑绝缘体(TIS)是物质的一种新的量子状态，引发了凝聚态物理学的一次名副其实的爆炸。顾名思义，这种绝缘子在拓扑结构上有别于传统绝缘子。拓扑学是关于几何形状的全局性质的数学，它指出，如果两个几何形状不能平滑地相互变形，那么它们在拓扑上是不同的。所有绝缘子都可以用抽象的几何形状进行数学描述。描述传统绝缘子的抽象几何形状不能平滑地变形为描述TI的几何形状，就像橡皮筋不经过切割就不能转换为Möbius带一样。这种数学上的区别体现在TIS的物理性质上。与钻石或硅等普通电绝缘体一样，TIS不允许电流在其内部流动。与普通的电绝缘体不同，它确实在其表面导电，而且几乎没有耗散。如果这种惊人的物理特性能够被利用到微处理器中，那么TIS就可以设计出更小、更高效的移动计算设备。TIS的表面也可能提供量子计算机的基本成分，量子计算机是一种基于量子力学定律的计算设备，其运行速度将以指数级快于传统计算机。最初几年的理论TI研究，包括我自己的工作，专注于忽略电子之间静电斥力的TIS的理想化模型。最近的实验和理论发展，以及设计可行的微电子应用的目标，现在正将该领域引向一个新的方向。有迹象表明，电子之间的相互作用可能导致新的和意想不到的现象，甚至增强TIS的应用潜力。通过将我们的研究集中在基本上未被探索的区域，其中TI内部的电子相互作用强烈，我们将有助于我们对物质的量子态的基本理解，并希望进一步将它们用于现实世界的应用。我们对TIS领域关键问题的调查将有助于保持加拿大在量子材料研究方面的领先地位。与其他资金来源相结合，这笔NSERC发现补助金将支持8名人员--本科生、研究生和博士后研究员--在现代凝聚态理论交易工具方面的高水平专业知识的培训。这包括在学术界内外广泛适用的各种高级分析和数值技能，并将为加拿大知识经济的发展做出贡献。

The laws governing the behavior of individual electrons and atoms are simple, elegant, and few. But then how can we account for the endless variety of forms which we observe matter to assume? Condensed matter physics seeks to explain these observations and to discover new states of matter. Distinct states of matter can be thought of as distinct patterns into which a group of electrons or atoms can organize themselves, just as a group of dancers can perform different choreographies. A fascinating question in this field is how many such patterns are consistent with the laws of quantum mechanics, and lead to quantum states of matter. The recent discovery of topological insulators (TIs), a new quantum state of matter, triggered a veritable explosion in condensed matter physics. As their name suggests, TIs are topologically distinct from conventional insulators. Topology, the mathematics of the global properties of geometrical shapes, states that two geometrical shapes are topologically distinct if they cannot be smoothly deformed into one another. All insulators can be described mathematically by abstract geometrical shapes. The abstract geometrical shape describing a conventional insulator cannot be smoothly deformed into one describing a TI, in the same way that a rubber band cannot be transformed into a Möbius strip without first cutting it. This mathematical distinction reveals itself in the physical properties of TIs. TIs, like ordinary electrical insulators such as diamond or silicon, do not allow electrical currents to flow in their interior. Unlike ordinary electrical insulators, TIs do conduct electricity on their surface, and do so almost without dissipation. If this striking physical property can be harnessed in microprocessors, TIs could enable the design of smaller and more efficient mobile computing devices. The surfaces of TIs might also provide the essential ingredient for a quantum computer, a computational device based on the laws of quantum mechanics that is poised to operate exponentially faster than a conventional computer. The first few years of theoretical TI research, including my own work, focused on idealized models of TIs that ignored the electrostatic repulsive force between electrons. Recent experimental and theoretical developments, as well as the goal of designing viable microelectronics applications, are now pointing the field in a new direction. There are indications that interactions between electrons may lead to novel and unexpected phenomena and even enhance the potential of TIs for applications. By focusing our research on largely unexplored regimes where electrons inside a TI interact strongly with one another, we will contribute to our basic understanding of quantum states of matter and hope to further their use in real-world applications. Our investigations into key issues in the field of TIs will help maintain Canada’s leadership in quantum materials research. Combined with other sources of funding, this NSERC Discovery Grant would support the training of eight personnel – undergraduate and graduate students, and postdoctoral fellows – to high levels of expertise in the tools of the trade of modern condensed matter theory. This includes a variety of advanced analytical and numerical skills that are broadly applicable inside and outside of academia, and will contribute to growing Canada’s knowledge-based economy.