Entanglement and emergence in quantum states of matter

该奖项支持理论研究和教育，旨在使用最近开发的概念来描述材料中电子的拓扑状态，以深入了解电子的强相互作用系统如何在导电状态下组织自己。世界上有许多不同种类的材料，如金属、绝缘体和半导体。这些材料使电子设备，如手机、电脑、电视，甚至互联网成为可能。为了设计具有期望功能的电子器件，需要理论来描述和预测金属、绝缘体和半导体的特性。正如理论物理学家朗道所发现的那样，液体和晶体等物质的状态可以通过对给定状态进行对称变换而保持不变来组织。例如，围绕普通盐晶体的主轴旋转90度，可以将新原子旋转到原来由同一种原子占据的位置，因此晶体看起来没有变化。对称的概念也允许通过类似的考虑来组织磁性和其他状态。近年来，人们发现，拓扑学（数学的一个分支，关注物体的变形、扭曲和拉伸不会改变的几何性质）的思想，带来了对物质的新可能相的见解，称为拓扑相。拓扑绝缘体就是一个常见的例子。它们与普通绝缘体的根本区别在于，虽然它们的主体不导电，但它们的表面却导电，就好像它们属于金属一样。人们逐渐认识到，量子纠缠是材料的另一个基本原理，它导致了一类新的量子材料，也被称为拓扑材料。纠缠是一个系统的纯量子特性，在我们的日常经验中没有类似的东西。它反映了量子系统各部分属性之间的联系，即使这些部分在物理上是分开的。这种性质反映在多电子态的结构上。相应的材料理论——拓扑有序理论——预测了一类具有新的拓扑和量子特性的新型绝缘体和半导体。这些拓扑材料可能在制造量子计算机中发挥关键作用，就像硅在制造当今普遍可用的传统计算机和手机中发挥关键作用一样。该奖项支持理论研究和教育，旨在使用为物质拓扑状态开发的现代基本概念，以深入了解电子在无间隙状态下组织的强相互作用系统。过去30年的研究表明，朗道的对称破缺理论只描述了物质可能具有的一小部分相。物质的相可能比以前想象的要丰富得多。利用拓扑序和对称保护平凡序的概念来描述这些新型量子相。经过大量的研究，在1-、2-和3维空间中，对玻色子和费米子系统的所有拓扑序和SPT序的系统分类理解已经出现。现在解决下一个大问题的时机已经成熟：对强相关无间隙态的系统理解。该奖项支持PI对这一问题的研究。PI的观点是，一般来说，一个相互作用的系统想要处于间隙状态，即最稳定的状态。如果一个相互作用的系统是无间隙的，那么无间隙状态必须是非常特殊和高度有组织的，这样即使在相互作用存在的情况下，那些无间隙的激发也可以保持无间隙。这表明对无间隙量子态的普遍和系统的理解是可能的。首先，无间隙态的低能部分可以成为几个解耦的扇区，在重整化群流的红外极限下，不同扇区之间的相互作用趋于零。这似乎是普遍发生的，例如一维无间隙系统中具有不同速度的扇区在低能量时解耦。因此，在低能量极限下，经常会出现涌现对称性和更高的对称性。由于每个解耦的低能量扇区不是一个完整的系统，因此每个扇区本身通常是异常的。扇区本身可能有引力异常或更高对称性异常。众所周知，异常可以影响低能动力学，特别是在某些情况下，它可以保护低能激发，从而使它们是无间隙的。每个脱钩部门的“无间隙”可以通过其异常来理解。这可能使我们能够系统地理解强相关的无间隙态。强相关的无间隙态应该比强相关的间隙态复杂得多，丰富得多。要完全系统地理解无间隙态可能需要一些时间。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 and education that aims to use recently developed concepts for describing topological states of electrons in materials to gain insight into how strongly interacting systems of electrons organize themselves in electrically conducting states. There are many different kinds of materials in the world, such as metals, insulators, and semiconductors. These materials make electronic devices, such as cell phones, computers, TV's, and even the internet possible. To design electronic devices with desired functions, theories are needed to describe and to predict the properties of metals, insulators, and semiconductors.As the theoretical physicist Landau discovered, states of matter such as liquids and crystals can be organized through the symmetry transformations performed on a given state that leaves it unchanged. For example, a rotation of 90 degrees around a principle axis of a crystal of common salt, can rotate new atoms into positions that were originally occupied by the very same kind of atom, so the crystal appears unchanged. The concept of symmetry also allows magnetic and other states to be organized by similar considerations. In recent years, it has been discovered that ideas from topology, the branch of mathematics concerned with geometric properties that are unchanged by deformations, twisting, and stretching objects, bring insight into new possible phases of matter, called topological phases. Topological insulators are a common example. They are fundamentally different from ordinary insulators in that while the bulk does not conduct electricity, their surfaces do, as if they belonged to a metal.It is becoming understood that quantum entanglement is another underlying principle for materials, one which leads to a new class of quantum materials, also known as topological materials. Entanglement is a purely quantum property of a system that has no analog in our everyday experience. It reflects connections between the properties of a quantum system's parts, even if the parts become physically separated. The property is reflected in the structure of the many-electron state. The corresponding material theory -- the theory of topological order -- predicts a new class of insulators and semiconductors with new topological and quantum properties. These topological materials may play key roles in making quantum computers, just like silicon plays a key role in making commonly available conventional computers and cell phones of today.TECHNICAL SUMMARYThis award supports theoretical research and education that aims to use modern fundamental concepts developed for topological states of matter to gain insight into how strongly interacting systems of electrons organized in gapless states. Research over the last 30 years reveals that Landau's symmetry breaking theory only describes a small set of possible phases that matter can have. The phases of matter can be much richer than has ever imagined before. The concepts of topological order and symmetry protected trivial (SPT) order to describe those new types of quantum phases. After much research, a systematic classification understanding of all topological orders and SPT orders for both bosonic and fermionic systems, in 1-, 2-, and 3-dimensional spaces has emerged. The time is now ripe to attack the next big problem: a systematic understanding of strongly correlated gapless states. This award supports the PI's research to engage this problem.The PI takes the view that in general, an interacting system wants to be gapped, the most stable state. If an interacting system is gapless, the gapless state must be very special and highly organized, so that those gapless excitations can remain gapless even in the presence of interactions. This suggests that the general and systematic understanding of gapless quantumstates is possible. First, the low energy part of a gapless state may become several decoupled sectors, where the interactions between different sectors flow to zero in the infrared limit under renormalization group flow. This appears to happen quite generally, such as the sectors with different velocities in 1d gapless system become decoupled at low energies. Consequently, in the low energy limit, there are often emergent symmetries and higher symmetries. Since each decoupled low energy sector is not a full system, each sector byitself is often anomalous. A sector by itself may have a gravitational anomaly or higher symmetry anomaly. It is well known that an anomaly can affect low energy dynamics, in particular, it can protect the low energy excitations with the result that they are gapless in some cases. The "gaplessness" of each decoupled sector may be understood via its anomaly. This may enable a systematic understanding of strongly correlated gapless states. The strongly correlated gapless states should be much more complicated and much richer than strongly correlated gapped states. It may take some time to gain a fully systematic understanding of gapless states.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.