Solid State Quantum Simulation: From Hyperbolic Space to Emergent Spin-3/2 Fermions

关于我们的世界，我们所知道的基本真理是什么？首先，在很长一段时间里，人类都相信太空是平的。一个很好的近似值是：如果你画一个三角形，内角的和是180度。后来我们发现我们的地球实际上是弯曲的。如果你要在地球表面画一个非常大的三角形，它的内角之和将大于180度。数学家称这为正曲率。他们还意识到，还有一种更耐人寻味的几何学，这种几何被称为负曲线，其中三角形的内角之和可以小于180度，甚至是零度！这种空间被称为双曲空间，它有许多逃避我们直觉的特征，但数学家们喜欢它，因为它很美，物理学家在过去几十年里意识到，我们物理世界的许多属性可能秘密地与双曲空间有关。另一个这样的事实，尽管你可能不经常想到它，是电子的存在-那些亚原子带电粒子，负责固体中的电流，并填充在原子的电子壳层中。电子有一种本征性质，称为自旋量子数，它有两个值，通常称为“向上”或“向下”，或“+1/2”和“-1/2”。(由于后者，我们说电子自旋为1/2。)电流和我们周围所有化学元素的性质是由于电子可以以这两种状态出现的事实造成的。长期以来，物理学家一直在寻找自旋为3/2的类电子粒子，这种粒子有四种状态，分别称为“+3/2”、“+1/2”、“-1/2”和“-3/2”，但没有发现具有这种性质的基本粒子。这个研究项目将研究如果世界是负弯曲的，或者如果电子自旋为3/2会发生什么。当然，我们这样做是为了满足我们的好奇心，但这样的分析的结果也将影响我们对物理基本定律的理解，并可能有技术应用。在过去的几年里，有两项试验性的突破推动了这个项目。一种是用超导谐振器实现双曲晶格，这是一种只存在于负弯曲空间的七角形瓦片。在这样的晶格上运动的粒子认为它们是在双曲空间中。另一种是拓扑半金属材料的合成，在这种材料中，带电粒子的行为就像自旋为3/2的电子。这两种设置都被称为“量子模拟”，因为它们创造了一个人造世界，以便从实验上研究它的性质。具体地说，我们将分析双曲空间如何改变许多粒子的物理，并从理论上帮助创建另外两个实验平台，用于他们的研究：电路和超冷量子气体。我们将研究自旋3/2粒子之间的相互作用如何改变固体的性质，以及如何在实验中测量这一点。

What are the basic truths we know about our world? For one, humankind believed for a long time that space is flat. And to a good approximation, it is: If you draw a triangle, the sum of the interior angles is 180 degrees. Later we discovered that our Earth is actually curved. If you were to draw a very large triangle on Earth's surface, its sum of interior angles would be larger than 180 degrees. Mathematicians call this a positive curvature. They also realized that a much more intriguing geometry exists, one that is called negatively curved, where the sum of interior angles in a triangle can be less than 180 degrees, even zero! This space, called hyperbolic space, has many features that evade our intuition, but Mathematicians love it because of its beauty, and Physicists realized over the last decades that many properties of our physical world may secretly be related to hyperbolic space. Another such truth, although you might not think about it often, is the existence of electrons --- those subatomic charged particles that are responsible for electronic current in solids and that populate the electronic shell of atoms. Electrons have an intrinsic property, called their spin quantum number, which takes two values, commonly called "up" or "down", or "+1/2" and "-1/2". (Because of the latter, we say that the electron has spin 1/2.) The properties of electrical currents and all chemical elements around us result from the fact that the electron can appear in these two states. Physicists have long searched for electron-like particles with spin 3/2, which have four states, called "+3/2", "+1/2", "-1/2", and "-3/2", but no fundamental particle with this property was found. This research project will investigate what would happen if the world would be negatively curved or if electrons had spin 3/2. We do this to satisfy our curiosity, of course, but the outcome of such an analysis will also affect our understanding of the fundamental Laws of Physics and could have technological applications. The last few years have seen two experimental breakthroughs that motivate this project. One is the realization of a hyperbolic lattice with superconducting resonators, which is a heptagonal tiling that only exists in negatively curved space. Particles moving on such a lattice think they are in hyperbolic space. The other is the synthesization of topological semimetal materials, where the charge carrying particles act like electrons with spin 3/2. Both settings are called "quantum simulations", because they create an artifical world in order to experimentally study its properties. Specifically, we will analyse how the physics of many particles is altered by hyperbolic space and we will theoretically help to create two more experimental platforms for their investigation: electrical circuits and ultracold quantum gases. We will investigate how the interactions between spin-3/2 particles alter the properties of solids and how this can be measured in experiment.