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是由于不断发现的创新科学的财富，而且也是由于它作为一种激进的信息处理新理论的应用。纠缠是描述非局域相关的一种现象，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和适当的技术来产生、保存和操纵它。从理论方面考虑了许多复杂的模型，并发现了各种有趣的性质，这些性质在量子信息处理中具有潜在的应用前景。事实上，在过去的几年里，量子信息通过引入各种纠缠测量、复杂的数值技术和新的分析方法，对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、离子阱或约瑟夫森结。除了对这些系统的直接兴趣，旨在理解临界现象，拓扑效应和高tc超导性，它们提供了实现无错误量子计算的令人兴奋的可能性。这项提议有三个不同的目的。首先，采用Berry相，即由Michael Berry首先引入的几何演化，来研究临界现象及其纠缠特性。临界现象与相互作用的多粒子系统行为的突变有关，是凝聚态物质和理论物理领域的核心兴趣。除了在智力上有趣之外，几何相位还可以用于理解复杂系统。一个例子是在化学中，Berry相被广泛用于探测复杂分子的潜在表面（总能量）的结构和探测它们的锥形相交。当几何相用于探测关键现象时，例如相互作用的自旋链，可以进行等效的研究。这是一个新建立的领域，具有令人兴奋的可能性，在关键现象领域提供有价值的见解。其次，开发既能支持拓扑量子信息处理，同时又能用当前或不久的将来的技术实现的模型。量子计算机相对于经典计算机的显著加速受到了由于不完善的控制程序或与环境的不期望相互作用而引入的错误的阻碍。最直接的无错误的量子信息处理是通过使用Frank Wilczek命名为任意子的特殊奇异粒子来实现的。在这些任意子系统中，信息被编码在许多粒子量子系统的全局拓扑特性中，因此它可以免受任何可能发生的局部错误的影响。原则上有两种主要的物理系统可以实现这些特性，分数量子霍尔态的二维电子气体或表现出拓扑相的特定自旋晶格系统。大多数建议的系统要么难以操作，要么需要非常苛刻的交互。确定可以支持拓扑阶段的替代设置是至关重要的，也是该部分提案的中心主题。最后的目的是将在临界点附近进行的几何演化推广到退化基态的情况下，后者是拓扑模型的主要特征。这一令人兴奋的应用方案的第一部分将使拓扑系统的明确检测。

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.