Dynamics and Ground States in Quantum Statistical Mechanics

该奖项支持对平衡和非平衡量子统计力学问题的数学研究，主要关注量子晶格模型的动力学，包括量子自旋系统和量子振荡器晶格系统。在凝聚态物理中，这些类型的模型通常用于描述三种类型的自由度（单独或组合）：晶格中的原子振荡，自旋磁矩和流动电子。在图上定义的同一类型的系统，目前在量子信息和计算理论中得到了广泛的研究。在这种情况下，自由度是量子位（或者更一般地说，qudits），它是携带量子信息的基本实体，量子算法在其上实现为一系列幺正变换。该项目侧重于研究这些系统的动力学（哈密顿或不可逆），并将导致诸如新类型系统的Lieb-Robinson型传播估计等基本结果。在这个项目的过程中，研究者和他的研究生还将分析量子晶格模型的低洼谱，并研究诸如基态以上的谱隙行为和海森堡模型中低洼激发的顺序等重要性质。数学结果将进一步应用于分析在其他情况下出现的类似转移矩阵的对象，例如目前在生物应用中使用的网络分析技术。这是对涉及大量相互作用组件的系统动力学的一系列数学模型的基础研究。在这个项目中，主要关注的是这些组成部分是微观的，并由量子力学描述的系统。研究者将与研究生一起工作，在他们的研究过程中，他们将获得量子点阵系统的分析和代数方法的最新状态的工作知识。随着基础研究和技术的进步，本项目开发的成果和技术预计将对量子信息处理、量子密码学、量子计算以及利用新材料量子特性的电子或自旋电子设备的建模等技术应用产生影响。为这个项目开发的一些数学技术也将在结构化网络的分析中有用，例如，在系统生物学中。

This award supports mathematical research on problems in equilibrium and non-equilibrium quantum statistical mechanics with the primary focus on the dynamics of quantum lattice models, which include quantum spin systems and quantum oscillator lattice systems. In condensed matter physics these types of models are commonly used to describe three types of degrees of freedom (separately or in combination): atomic oscillations in a crystal lattice, spin magnetic moments, and itinerant electrons. The same type of system, defined on a graph, is currently studied extensively in the theory of quantum information and computation. In that context, the degrees of freedom are qubits (or, more generally, qudits), which are the basic entities carrying quantum information, on which quantum algorithms are implemented as a sequence of unitary transformations. The project focuses on studying the dynamics (Hamiltonian or irreversible) of these systems and will lead to fundamental results such as propagation estimates of Lieb-Robinson type for new classes of systems. In the course of this project the investigator and his graduate students will also analyze the low-lying spectrum of quantum lattice models and investigate important properties such as the behavior of the spectral gap above the ground state and the ordering of the low-lying excitations in Heisenberg-type models. The mathematical results will furthermore be applied to the analysis of transfer-matrix-like objects that appear in other contexts, such as techniques of network analysis currently employed in biological applications.This is fundamental research on a range of mathematical models for the dynamics of systems that involve a large number of interacting components. In this project, the main focus is on systems in which these components are microscopic and described by quantum mechanics. The investigator will work with graduate students, who in the process of their research will acquire a working knowledge of the state of the art in analytic and algebraic methods for quantum lattice systems. As fundamental research and technology advances, the results and techniques developed in this project are expected to have an impact in technological applications such as quantum information processing, quantum cryptography, quantum computing, and the modeling of electronic or spintronic devices that exploit the quantum properties of novel materials. Some of the mathematical techniques developed for this project will also be useful in the analysis of structured networks such as arise, for example, in systems biology.