Entanglement and topology of time-reversal symmetric fractional topological insulators

传统上，物理学是通过一种简化的方法来发展的，用越来越小的组成部分来解释我们所看到的世界，直到达到我们认为是基本的粒子水平。然而，即使我们知道物质组成部分的确切性质以及它们如何相互作用，仍然有很多惊喜的空间。事实证明，当许多粒子相互作用时，它们经常表现出真正新的、不同的行为。从这个意义上说，那句老话“整体不仅仅是各部分的总和”证明是正确的！在这个提议中，我们研究了在一个特别有利于观察强量子效应的环境中相互作用的多粒子系统的后果。增强量子效应的一个常用途径是应用强磁场：在这种情况下，粒子的动能变得不重要，所有的物理都是由相互作用驱动的。这是分数量子霍尔效应的基础。分数量子霍尔效应是一种宏观的量子现象，当电子被限制在一层薄薄的材料中，被一个非常强的磁场刺穿。令人惊讶的是，磁场的影响可以被最近发现的某些类型的材料自发地模拟，包括张力石墨烯，或者可能包含重元素（如铱）的材料薄膜。此外，这种材料产生的磁场可以比在实验室中实验产生的磁场高10 - 100倍。然而，这个故事有一个转折：在这些材料中产生的合成磁场的方向取决于自旋的方向，即单个电子旋转的轴。强磁场的结合，加上相反自旋的相反场方向的复杂性，导致了我们项目将要研究的新领域。天真地说，这是一项非常困难的任务，因为通常用于此目的的数值方法在包括额外的自旋自由度时不堪重负。除了材料物理学的新机遇之外，我们还将利用另一个及时的发展，并通过从量子信息的角度描述各自的波函数来采用一种新的方式来看待多体量子系统。而不是给出整个系统的全局波函数，我们将使用一种有效的方法来编码个体自由度如何依赖于它们近邻的自由度。在更专业的术语中，我们根据其纠缠特性来描述系统，其中纠缠量化了给定位置的粒子的量子状态在多大程度上取决于系统中其他粒子的量子状态。通过建立一种有效的方法来模拟我们目标系统的复杂多体物理，我们实现了对所产生的集体行为的预测能力。通常，我们期望这种行为对系统参数非常敏感。因此，在数值上有效地模拟行为的能力为寻找物理上有趣的状态提供了巨大的优势。我们将利用这一能力，以确定所期望的量子效应最强大的材料类型。我们也将超越理解在一个固定状态下会发生什么，当系统松弛到它的绝对基态。通过将我们的系统暴露在参数的快速变化中，并研究之后它是如何演变的，我们可以更多地了解组成粒子的复杂相互作用。想象一下，拿着一把锤子，听铃铛在不同的地方被敲击时发出的声音。显然，这比只看钟的自生自弃要有用得多。

Traditionally, physics has advanced via a reductionist approach, explaining the world we see in terms of its smaller and smaller components - until reaching the level of particles that we believe to be elementary. However, even as we know the precise nature of components of matter and how they interact with each other, there is still a lot of room for surprises. It turns out that when many particles interact with each other, they often display genuinely new and different behaviour. In this sense, the old saying that the whole is more than just the sum of its parts turns out to be true!In this proposal, we investigate the consequences of interacting many-particle systems in a setting that is particularly conducive to observing strong quantum effects. One frequently exploited route for enhancing quantum effects is the application of strong magnetic fields: in this setting, the kinetic energy of particles becomes unimportant, and all of the physics is driven by the interactions. This is the foundation of the fractional quantum Hall effect, which is a macroscopic quantum phenomenon observed when electrons are confined in a thin layer of material pierced by a very strong magnetic field.Surprisingly, it turns out that the effect of the magnetic field can be emulated spontaneously by certain types of materials that have recently been discovered, including strained graphene, or possibly thin films of materials involving heavy elements like iridium. Furthermore, the fields generated by such materials can be 10 - 100 times higher than magnetic fields which can otherwise be experimentally generated in the laboratory. Still, there is a twist to this story: the direction of the synthetic magnetic fields generated in these materials depends on the orientation of the spin, i.e., the axis around which individual electrons turn.The combination of strong magnetic fields, along with the added complexity of the opposite field directions for opposite spins leads to the new area that will be investigated in our project. Naively, this is a very difficult task, however, as the numerical methods that are usually used for this purpose are overwhelmed when the extra spin degree of freedom is included. In addition to new opportunities from materials physics, we will thus exploit another timely development and adopt a new way of looking at many-body quantum systems by characterising the respective wave functions from a quantum information point of view. Rather than giving the wave function globally for the entire system, we will use an efficient approach to encode how individual degrees of freedom depend on those of their immediate neighbours. In more technical terms, we describe the system in terms of its entanglement properties, where entanglement quantifies to which extent the quantum state of a particle at a given location is conditional on the quantum states of other particles in the system.By establishing an effective means of simulating the complex many-body physics of our target systems, we achieve predictive power over the resulting collective behaviour. Typically, we expect that this behaviour is very sensitive to the system parameters. Hence, the ability to effectively simulate the behaviour numerically provides a tremendous advantage in finding physically interesting regimes. We will use this ability in order to identify types of materials in which the desired quantum effects are most robust. We will also go beyond understanding what happens in a stationary regime, when the system is left to relax to its absolute ground state. We can learn even more about the intricate interplay of the constituent particles by exposing our system to a rapid change in parameters, and studying how it evolves afterwards. Think of taking a hammer and listening to the resulting sound of a bell when it is being struck in different places. Clearly, this is a lot more informative than just looking at the bell left to its own devices.