Geometry and Topology in Complex Quantum Systems

在过去的十年里，量子信息已经从一个全新的话题发展成为物理学中最活跃和最富有成果的领域之一。这种惊人的转变不仅归功于不断发现的创新科学的财富，也归功于它作为信息处理的一种全新理论的应用。纠缠是一种描述非局域关联的现象，是量子信息技术的主要资源之一。它的脆弱性质要求采用新的适当技术来产生、保存和操纵它。在理论方面，人们已经考虑了许多复杂的模型，并发现了许多有趣的性质，这些性质在量子信息处理中具有潜在的应用。事实上，在过去的几年里，量子信息通过引入各种纠缠措施、复杂的数值技术和新颖的分析方法，为理解许多粒子效应做出了重大贡献。特别令人感兴趣的是可以在实验室中实际实现的模型，例如通过光学晶格、离子陷阱或约瑟夫森结。除了对这些旨在理解临界现象、拓扑效应和高温超导电性的系统有直接兴趣外，它们还为实现无误差量子计算提供了令人兴奋的可能性。这项提议有三个不同的目标。首先，利用Berry相，即由Michael Berry首先提出的几何演化，来研究临界现象及其纠缠性质。临界现象涉及相互作用的多粒子系统行为的突然变化，是凝聚态和理论物理领域的中心兴趣。除了智力上的趣味性，几何相也可以用来理解复杂的系统。一个例子是在化学中，Berry相被广泛用于探测复杂分子的势面(总能量)的结构，并检测它们的锥形交点。当几何相被用来探索临界现象时，例如相互作用的自旋链，可以进行等价的研究。这是一个新建立的领域，具有在关键现象领域提供有价值的见解的令人兴奋的可能性。其次，开发能够支持拓扑量子信息处理的模型，同时可以用现在或不久的将来的技术来实现。量子计算机相对于经典计算机的显著加速受到由于不完善的控制程序或与环境的不希望的相互作用而引入的错误的阻碍。通过使用弗兰克·威尔切克命名为任意子的特殊奇异粒子，实现了对量子信息最直接的无误差处理。有了这些任意子系统，信息被编码在许多粒子量子系统的全局拓扑属性中，因此可以防止任何类型的局部错误发生。原则上可以实现这些性质的主要物理系统有两种，一种是分数量子霍尔态的二维电子气，另一种是呈现拓扑相的特定自旋晶格系统。大多数提议的系统要么难以操纵，要么需要要求高得令人望而却步的交互。确定能够支持拓扑阶段的替代设置是至关重要的，也是提案这一部分的中心主题。最终目的是将围绕临界点进行的几何演化推广到简并基态的情况，后者是拓扑模型的主要特征。提案第一部分的这一令人兴奋的应用将使拓扑系统的明确检测成为可能。

Within the last decade, quantum information has progressed from being a completely novel topic to one of the most active and fruitful areas within physics. This amazing transition is not only due to the wealth of innovative science that is being continuously discovered, but also to its applications as a radical new theory for information processing. Entanglement, a phenomenon that portrays non-local correlations, is one of the main resources for quantum information technology. Its fragile nature requires the introduction of new and appropriate technologies for its generation, preservation and manipulation. From the theoretical front many complex models have been considered and a variety of intriguing properties have been discovered that have potential applications to quantum information processing. Indeed, in the last few years quantum information has made a significant contribution to the understanding of many particle effects by introducing a variety of entanglement measures, sophisticated numerical techniques and novel analytical methods. Of particular interest are models that can actually be realized in the laboratory for example by optical lattices, ion traps, or Josephson Junctions. Apart from the direct interest in these systems aiming at the understanding of critical phenomena, topological effects and high-Tc superconductivity they offer the exciting possibility of implementing error-free quantum computation. There are three distinct aims of this proposal. Firstly, to employ Berry phases, i.e. geometrical evolutions first introduced by Michael Berry, to the study of critical phenomena and their entanglement properties. Critical phenomena are concerned with the abrupt changes in the behavior of interacting many-particle systems and are of central interest in the fields of condensed matter and theoretical physics. Apart from being intellectually interesting, geometrical phases can also be used for the understanding of complex systems. An example is in chemistry where Berry phases are widely used to probe the structure of the potential surfaces (total energy) of complex molecules and to detect their conical intersections. An equivalent study can be performed when geometrical phases are used to probe critical phenomena, e.g. of interacting spin chains. This is a newly established field with the exciting possibility of providing valuable insights in the area of critical phenomena. Secondly, to develop models that can support topological quantum information processing while at the same time being realizable with present or near future technology. The significant speedup of quantum computers with respect to their classical counterparts is hindered by the introduction of errors due to imperfect control procedures or undesired interactions with the environment. The most straightforward error-free processing of quantum information is achieved by employing particular exotic particles named anyons by Frank Wilczek. With these anyonic systems information is encoded in global topological properties of many particle quantum systems and thus it is protected from any type of errors that can occur locally. There are two main physical systems that can in principle realize these properties, two-dimensional electron gases in the fractional quantum Hall state or specific spin lattice systems that exhibit topological phases. Most of the proposed systems are either difficult to manipulate or require interactions that are prohibitively demanding. Identification of alternative setups that can support topological phases is of vital importance and is the central theme of this part of the proposal. The final aim is to generalize the geometrical evolutions that are performed around critical points to the case of degenerate ground states, the latter being the main characteristic of topological models. This exciting application of the first part of the proposal will enable the unambiguous detection of topological systems.