Emergence: from nanomagnets to quantum spin liquids

具有一定规模和复杂性的实体集合可以获得新的“涌现”身份。一个经典的例子是雨滴，在室温和常压下是1020个水分子的集合。雨滴与水分子完全不同。它甚至遵循一套不同的物理定律，例如，它的形状是由表面张力决定的，这个概念在分子水平上没有意义。我们说雨滴作为一个独立的物体出现了。由于涌现是高度非线性的，它通常不能从第一性原理计算出来。例如，从水分子的知识出发，是不可能预测雨滴的形状的。然而，我的小组最近的工作表明，这种预测可以在一大类量子磁体中进行。它们遵循“涌现原理”：在低能量下，量子磁体类似于一个按照量子力学定律运动的粒子。粒子在一个抽象空间中运动，这个空间通常可以通过简单的几何参数来确定。例如，XY二聚体（具有两个自旋的简单磁铁）以沿圆周运动的粒子形式出现。磁铁的能级与粒子的能级是无法区分的。我建议以这个想法为基础，去理解构成涌现的机制，并设计出理想的涌现特性。我的第一个目标是在纳米磁铁中发现新的涌现特性。在低温下，这些小磁铁类似于简单的单粒子问题。粒子的运动可以受到量子效应的影响，如贝里相位和干涉。我希望建立纳米磁体作为研究量子动力学的平台，探索遍历性、热化和量子混沌等思想。我的第二个目标是研究本地化。我的团队最近的研究已经证明了“量子优柔寡断”，即低能粒子在十字路口冻结，无法选择方向。这在某些量子磁体的涌现物理学中可以看到，磁体有效地冻结成有序的结构。这个想法只在小型模型系统中得到了证明。我建议把它扩展到实验上可以实现的更大的系统，例如，在金属中，电子可以被认为沿着费米表面运动。我研究计划的第三个目标是研究量子自旋液体，自旋无序的磁体的涌现状态。它们是用规范理论来描述的，规范理论是一个起源于电磁学的概念。我试图解决两个广泛的开放性问题——旋转是如何进入无序状态的？是什么导致了量规结构？我建议研究由三角形和四面体基元构成的候选自旋液体。利用每个母题层面的涌现原理，构建场理论描述。这可以揭示出出现的方法，追踪磁铁能量降低时的物理运动。

A collection of entities, at a certain scale and complexity, can acquire a new 'emergent' identity. A classic example is the raindrop, a collection of 1020 water molecules at room temperature and pressure. The raindrop is completely unlike a water molecule. It even obeys a different set of physical laws, e.g., its shape is determined by surface tension, a concept that has no meaning at the molecular level. We say that the raindrop has 'emerged' as an independent object. As emergence is highly nonlinear, it cannot usually be calculated from first principles. For example, starting with knowledge of water molecules, it is impossible to predict the shape of a raindrop. However, recent work from my group shows that such predictions can be made in a large class of quantum magnets. They follow an 'emergence principle': at low energies, the quantum magnet resembles a single particle that moves according to the laws of quantum mechanics. The particle moves on an abstract space that can often be determined from simple geometric arguments. For example, an XY dimer (a simple magnet with two spins) emerges as a particle that moves on a circle. The energy levels of the magnet are indistinguishable from that of the particle. I propose to build on this idea to understand the mechanisms that underlie emergence and to engineer desirable emergent properties. My first objective is to find novel emergent properties in nanomagnets. At low temperatures, these small magnets resemble simple single-particle problems. The particle's motion can be affected by quantum effects such as Berry phase and interference. I hope to establish nanomagnets as a platform to study quantum dynamics, exploring ideas such as ergodicity, thermalization and quantum chaos. My second objective is to study localization. Recent studies from my group have demonstrated 'quantum indecision' where a low-energy particle freezes at a crossroads, unable to pick a direction. This is seen in the emergent physics of certain quantum magnets with the magnet effectively freezing into an ordered configuration. This idea has only been demonstrated in small model systems. I propose to extend it to larger systems that are experimentally realizable, e.g., in metals where electrons can be thought to move along Fermi surfaces. The third goal of my research programme is to study quantum spin liquids, emergent states of magnets where spins do not order. They have been described using gauge theory, a concept that originates from electromagnetism. I seek to address two broad open questions - how do spins settle into an unordered state? What gives rise to gauge structure? I propose to study candidate spin liquids that are built from triangular and tetrahedral motifs. Using the emergence principle at the level of each motif, I propose to construct field theory descriptions. This can reveal the approach to emergence, tracking the physics of the magnet as its energy is lowered.