Topological phases by momentum space braiding

量子力学理论的发展是物理学乃至整个科学领域最具影响力的成就之一。站在凝聚态领域的基础上，它早期的成功之一是阐明了为什么有些材料表现为绝缘体，而另一些材料表现为金属性质。在利用粒子的波动解释时，很容易发现周期性势中的电子产生能带；光谱显示由间隙分隔的连续能级带。因此，填满“费米海”，这样一个整数数量的能带被填满，确保有一个激发间隙，从而绝缘行为达到那个能量尺度，而少量填满能带会产生金属相。虽然带理论已经非常成功，但由于与拓扑学数学领域的意想不到的联系，它在过去的几年里重新焕发了活力。拓扑学本质上表征物体在平滑变形下保留的特性。这通常以咖啡杯和甜甜圈的拓扑等价为例。在不撕裂或戳洞的情况下，一个可以变形成另一个，它们的一般类别可以用所谓的不变量来量化，即计算洞数的整数。值得注意的是，这一原理已经被发现在电子物质的相中具有关键的重要性，其中波函数可以结合不同的集体结，在拓扑上区分不同类别的绝缘体和金属。这些拓扑绝缘体和金属不仅从纯理论的角度来看很有吸引力，而且还表现出非凡的物理现象，例如可以塑造下一代电子产品的受保护金属边缘状态，或者可以存储量子信息的激发，使拓扑材料成为量子计算平台的潜在关键组件。虽然拓扑材料的影响已经被广泛的研究兴趣和该领域的快速发展所支持，但在过去的一年里，人们发现了一类全新的拓扑金属。这些系统的特点是，除了带成对接触的特殊点外，带在任何地方都是间隙的。由此产生的带节点进一步携带可以以高度非平凡的方式改变的奇异类型的拓扑电荷。也就是说，当不同带之间的节点在动量空间中相互编织时，它们的电荷会发生转换，从而在集体波函数中产生无法解开的特定相位因子。因此，出现了一种新的拓扑结构，可以通过一种新的不变量来量化，称为欧拉类。然而，有明确的迹象表明，这些结果只是冰山一角，并且存在一类全新的此类欧拉金属，特别是当其他晶体对称性存在时，对拓扑分类施加了新的条件。该计划旨在利用这些及时的迹象，并研究这些令人兴奋的新物质形式。这阐明了三个主要支柱，旨在(i)推进对这些欧拉相的理论理解，（ii）揭示它们的物理性质，（iii）设计具体途径，将它们带到实验领域。对于后一个目标，这包括明确整合实验和ab-initio项目合作伙伴，我们打算与他们建立长期联盟，从而在拓扑材料的突出领域创建一个强大的计划。鉴于强有力的迹象表明，这些新的欧拉相具有奇异的物理特性，除了它们巨大的科学潜力外，还可能最终影响未来的技术，我们预计该计划将产生深远的影响，从而进一步巩固英国强大的研究地位。

The development of the theory of quantum mechanics entails one of the most influential achievements in Physics and, arguably, science as a whole. Standing at the basis of the field of condensed matter, one of its earlier triumphs was to shed light on the question why some materials behave as insulators, while others exhibit metallic properties. Upon utilising the wave interpretation of particles, it was readily found that electrons in a periodic potential give rise to energy bands; the spectrum shows bands of continuous energy levels separated by gaps. Hence, filling up the 'Fermi Sea' such that an integer amount of bands are filled ensures that there is an excitation gap, and thus insulating behavior up to that energy scale, whereas filling a band fractionally gives rise to a metallic phase. Although band theory has been extraordinarily successful, it has been reinvigorated the past years due unexpected connections with the mathematical domain of topology. Topology in essence characterises properties of objects that are preserved under smooth deformations. This is usually exemplified by the topological equivalence of a coffee cup and doughnut. Without tearing or poking holes one can be deformed into the other and their general class may be quantified in terms of a so-called invariant, being the integer that counts the number of holes. Rather remarkably, this principle has been found to be of pivotal importance in phases of electronic matter, where the wavefunctions can tie distinctive collective knots, topologically distinguishing different classes of insulators and metals. These topological insulators and metals are not only appealing from a purely theoretical point of view, but also exhibit remarkable physical phenomena such as protected metallic edge states that could shape next-generation electronics, or excitations that can store quantum information, making topological materials a potentially key component of quantum computing platforms.While the impact of topological materials has been underpinned by a vast research interest and a rapid advancement of the field, it was discovered the past year that a whole new class of topological metals exists. These systems feature bands that are gapped everywhere except for special points at which bands pairwise touch. The resulting band nodes furthermore carry exotic kinds of topological charges that can be altered in a highly non-trivial manner. Namely, when such nodes between different sets of bands are braided along each other in momentum space, their charges are converted, inducing specific phase factors in the collective wave function that cannot be untangled. As a result, a new topological structure emerges that can be quantified by a novel type of invariant, known as Euler class. There are however clear indications that these results comprise the tip of the iceberg and that a whole new class of such Euler metals exists, especially when other crystalline symmetries are present that enforce new conditions on the topological classification. This programme aims to exploit these timely indications and investigate these new exciting forms of matter. This articulates around three main pillars that aim to (i) advance the theoretical understanding of these Euler phases, (ii) uncover their physical properties and (iii) design concrete pathways to bring them to the experimental domain. For the latter objective this includes an explicit integration of experimental and ab-initio project partners, with whom we intend to foster long-term alliances, thereby creating a strong programme in the prominent field of topological materials.Given the strong indications that these new Euler phases host exotic physical properties that, apart from their immense scientific potential, could culminate impact future technologies, we anticipate that this programme will generate profound impact, thereby further underpinning the strong research position of the UK.