Interacting dynamics and exotic phases of quantum lattice systems

量子晶格系统位于多体凝聚态物理学、量子信息理论和一般的数理统计物理学的交叉点。该项目的总体目标是探索这些描述微观尺度凝聚态物质的量子力学模型的分析结构。在量子霍尔效应的背景下出现的物理动机的问题，最近由量子信息理论驱动，并得到光学晶格实验的支持，应放在一个数学框架中，可以解决光谱，动力学和统计力学问题。 该项目旨在发展代数和分析方法，为严格研究以拓扑有序或多体定域为特征的新型奇异物质状态奠定基础。处理特别是由任意大量粒子之间的相互作用的非微扰性质引起的具体困难，需要扩展我们对无限量子晶格系统的统计力学的知识：动力学局域性，基态性质，本征态结构和无序效应。 该项目分为三个主要主题。第一个建立在最近的结果有关的量子晶格系统的驱动动力学，以获得一个一般的C*-代数绝热理论的无限体积极限，改进，指数误差界。第二个目的是更好地理解拓扑物质的不变量，如（分数）量子霍尔系统和拓扑绝缘体，在相互作用的设置。第三个主题是推动准绝热延拓技术超越其目前非常成功的应用领域，特别是对无间隙无序系统。预期进一步的应用程序的范围从光谱稳定性的结果，以更好地了解动态局域量子晶格系统的相关结构。 专门为该项目招募的HQP将参与所有阶段，研究问题将根据其技术能力水平和计划设定的时间范围进行调整。

Quantum lattice systems lie at the intersection of many-body condensed matter physics, quantum information theory and, generally, mathematical statistical physics. The overall objective of this project is to explore the analytic structure of these quantum mechanical models that describe condensed matter at the microscopic scale. Physically motivated questions that arose in the context of the quantum Hall effect, recently driven by quantum information theory and supported by experiments in optical lattices, shall be put in a mathematical framework where spectral, dynamical and statistical mechanical questions can be addressed. The project aims at developing algebraic and analytic methods to lay the foundations of a rigorous study of novel exotic states of matter characterized by topological order or many-body localization. Handling the specific difficulties arising in particular from the non-perturbative nature of interactions between an arbitrarily large number of particles requires an extension of our knowledge of the statistical mechanics of infinite quantum lattice systems: dynamical locality, properties of ground states, structure of eigenstates, and effects of disorder. The project is organized in three principal themes. The first builds on very recent results concerning the driven dynamics of quantum lattice systems to obtain a general C*-algebraic adiabatic theory in the infinite volume limit, with improved, exponential error bounds. The second aims at a better understanding of the invariants of topological matter, such as (fractional) quantum Hall systems and topological insulators, in an interacting setting. The third theme is to push the quasi-adiabatic continuation technique beyond the domain of its current and very successful application, in particular to gapless disordered systems. Expected further applications of the program range from spectral stability results to a better understanding of the correlation structure of dynamically localized quantum lattice systems. HQPs recruited specifically for this project will be involved in all stages, with research questions adapted to their level of technical abilities and to the time horizon set by their program.