Field theoretic and holographic approaches to strongly correlated quantum many-body systems

凝聚态物质系统，包括金属和绝缘体，是由许多十到二十三个数量级的粒子组成的。单个粒子的行为受公认的物理定律--量子力学的支配。如果粒子之间的相互作用很弱，那么将量子力学应用于凝聚态物质系统是很简单的。在弱相互作用系统中，由于粒子之间的相关性很弱，人们可以通过逐个研究单个粒子来预测整个系统的行为。然而，当粒子之间有很强的相互作用和相关性时，这种“单粒子方法”就会彻底失败。事实证明，在许多最近发现的材料中都是如此，例如高温超导体。在这些材料中，如果不了解整个系统，就无法了解物理性质，因为单个粒子的行为受到其他粒子的强烈影响。这些问题通常不可能通过第一原理计算来解决，因为有如此多的自由度通过强烈的相互作用相互纠缠在一起。另一方面，由于粒子之间的非平凡的量子力学纠缠，强关联系统经常出现新类型的意想不到的集体现象。例如，在超导状态下，即使在存在杂质的情况下，电流也没有任何电阻流动，而在量子霍尔态中，电导被量子化，而与样品的细节无关。当凝聚态物质系统表现出明显的集体行为时，这是该系统处于物质的新阶段的标志。我的研究关注的是从理论上理解强关联量子多体系统中出现的新的集体现象。我专注于一组小的集体自由度，以便以有效的方式捕捉基本现象，而不是试图理解多体系统中的所有微观细节。本研究不仅有助于提高我们对强关联量子多体系统的基本认识，而且有助于我们将各种新材料应用于信息技术等领域。

Condensed matters systems, that include metals and insulators, are made of many particles of the order of ten to the twenty three. Behaviors of individual particles are governed by the well established law of physics, quantum mechanics. It is straightforward to apply quantum mechanics to condensed matter systems if interactions between particles are weak. In weakly interacting systems, one can predict behaviors of a whole system by studying individual particles one by one because correlations between particles are weak. However, such `single-particle approaches' drastically fail when there are strong interactions and correlations between particles. This turns out to be the case in many recently discovered materials, such as high temperature superconductors. In those materials, one can not understand physical properties without understanding the system as a whole because behaviors of individual particles are strongly affected by other particles. It is generally impossible to solve those problems from first-principle calculations because there are so many degrees of freedom that are entangled with each other through strong interactions. On the other hand, strongly correlated systems often show new types of unexpected collective phenomena as a result of non-trivial quantum mechanical entanglements between particles. For example, electric current flows without any resistance even in the presence of impurities in superconducting state, and conductance is quantized independent of details of samples in quantum Hall states. When a condensed matter system shows distinct collective behaviors, it is a sign that the system is in a new phase of matter. My research is concerned about theoretically understanding novel collective phenomena that arise in strongly correlated quantum many-body systems. I focus on a small set of collective degrees of freedom in order to capture essential phenomena in an efficient way instead of trying to understand all the microscopic details in many-body systems. The present research will not only help us to improve our fundamental understanding of strongly correlated quantum many-body systems, but also to use various novel materials for applications, such as information technology.