Interacting dynamics and exotic phases of quantum lattice systems

量子点阵系统位于多体凝聚态物理、量子信息理论和数学统计物理的交叉点。该项目的总体目标是探索在微观尺度上描述凝聚态物质的这些量子力学模型的解析结构。在量子霍尔效应背景下产生的物理动机问题，最近由量子信息理论驱动并由光学晶格实验支持，应该放在一个数学框架中，其中可以解决光谱，动力学和统计力学问题。******该项目旨在发展代数和分析方法，为以拓扑秩序或多体定位为特征的新型奇异物质状态的严格研究奠定基础。处理特定的困难，特别是由任意大量粒子之间相互作用的非摄动性质引起的困难，需要扩展我们对无限量子点阵系统的统计力学知识：动态局部性，基态的性质，特征态的结构和无序的影响。******该项目分为三个主要主题。第一个是建立在最近关于量子点阵系统驱动动力学的结果上，获得了无限体积极限下的一般C*-代数绝热理论，具有改进的指数误差界限。第二个目标是更好地理解拓扑物质的不变量，例如（分数）量子霍尔系统和拓扑绝缘体，在相互作用的环境中。第三个主题是推动拟绝热延拓技术超越其目前非常成功的应用领域，特别是在无间隙无序系统中。预计该程序的进一步应用范围从光谱稳定性结果到更好地理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，研究问题将根据他们的技术能力水平和项目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.