Topological phases by momentum space braiding

量子力学理论的发展带来了物理学以及可以说是整个科学领域最有影响力的成就之一。站在凝聚态领域的基础上，它的早期成就之一是阐明了为什么有些材料表现为绝缘体，而另一些材料表现出金属性质。利用对粒子的波的解释，很容易发现周期势中的电子产生能带；光谱显示由间隙隔开的连续能级的带。因此，填充“费米海”以填充整数个带可确保存在激发间隙，从而使绝缘行为达到该能级，而填充一个带则会产生金属相。虽然带理论非常成功，但在过去的几年里，由于与拓扑学的数学领域的意外联系，它重新焕发了活力。拓扑本质上表征了在平滑变形下保留的对象的属性。这通常以咖啡杯和甜甜圈的拓扑等价为例。在没有撕开或戳洞的情况下，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，不变量是计算孔数的整数。值得注意的是，这一原理已被发现在电子物质的各个阶段具有关键的重要性，在电子物质的各个阶段，波函数可以连接不同的集体结，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具有吸引力，而且还表现出非凡的物理现象，如可以塑造下一代电子的受保护的金属边缘状态，或者可以存储量子信息的激发，使拓扑材料成为潜在的量子计算平台的关键组件。尽管拓扑材料的影响得到了广泛的研究兴趣和该领域的快速发展，但在过去的一年里，人们发现了一类全新的拓扑金属的存在。这些系统的特点是，除了带子成对接触的特殊点外，其他地方都有带子的间隙。由此产生的频带节点还携带着奇特的拓扑电荷，这些电荷可以以非常不平凡的方式改变。也就是说，当不同带组之间的这些节点在动量空间中彼此编织时，它们的电荷被转换，在集体波函数中产生无法解缠的特定相位因子。结果，一种新的拓扑结构出现了，它可以用一种新型的不变量来量化，称为欧拉类。然而，有明显的迹象表明，这些结果只是冰山一角，而且存在一种全新的此类欧拉金属，特别是当存在其他晶体对称时，这些对称对拓扑分类施加了新的条件。该方案旨在利用这些及时的迹象并研究这些新的令人兴奋的物质形式。它围绕三个主要支柱展开，旨在(I)促进对这些欧拉阶段的理论理解，(Ii)揭示它们的物理性质，以及(Iii)设计将它们带入实验域的具体途径。对于后一个目标，这包括实验和从头计划合作伙伴的明确整合，我们打算与他们建立长期联盟，从而在突出的拓扑材料领域创建一个强大的计划。鉴于强有力的迹象表明，这些新的欧拉阶段具有奇异的物理性质，除了它们巨大的科学潜力，可能最终影响未来的技术，我们预计这个计划将产生深远的影响，从而进一步巩固英国的强大研究地位。

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.