量子多体哈密顿的代数动力学理论：强相互作用系统的形式化方法
Algebraic-Dynamical Theory for Quantum Many-body Hamiltonians: A Formalized Approach To Strongly Interacting Systems
摘要：
非对易代数和纠缠是多体量子系统的两个重要特征。动力学微扰方法是研究量子多体系统最常用的方法。近年来，基于量子纠缠的数值方法研究蓬勃发展，但传统的动力学微扰方法在量子纠缠的研究中并没有得到很好的应用。在这项工作中，我们制定了一个代数动力学理论（ADT）相结合的力量，量子代数和动力学方法，其中量子纠缠自然出现的组织原则。首先，我们引入了一个完整的运营商基集（COBS），一个任意的状态，无论是纯或混合，可以表示的期望值的COBS。然后，我们建立了一个完整的映射，从一个给定的状态到一个完整的动态关联函数的状态通过海森堡和Schwinger-Dyson-equations-of-motion（SDEOM）。COBS和映射的完备性保证了ADT在原则上是一个数学上完备的框架。将ADT应用于格点上的多体系统，我们发现多体COBS的期望值的累积结构可以表征量子纠缠。状态的累积量结构在相关性中形成了一个等级。更重要的是，这样的静态相关层次继承了动态的相关性和他们的SDEOM。我们建议，动力学层次也进行任何微扰计算的状态。我们证明了这样的微扰层次的有效性与一个明确的例子，在其中，我们表明，一个单粒子型微扰计算失败，而多体微扰的层次成功。我们还讨论了ADT的计算和近似方案及其对其他强耦合理论如部分子和从粒子方法的影响。
英文摘要：
Non-commutative algebras and entanglement are two of the most important hallmarks of many-body quantum systems. Dynamical perturbation methods are the most widely used approaches for quantum many-body systems. While study of entanglement-based numerical methods are booming recently, the traditional dynamical perturbation methods have not benefited from study of quantum entanglement. In this work, we formulate an algebraic-dynamical theory (ADT) by combining the power of quantum algebras and dynamical methods in which quantum entanglement naturally emerges as the organizing principle. We start by introducing a complete operator basis set (COBS), with which an arbitrary state, either pure or mixed, can be represented by the expectation values of COBS. Then we establish a complete mapping from a given state to a complete set of dynamical correlation functions of the state through the Heisenberg- and Schwinger-Dyson-equations-of-motion (SDEOM). The completeness of COBS and the mapping ensures ADT to be a mathematically complete framework in principle. Applying ADT to many-body systems on lattices, we find that the quantum entanglement is represented by the cumulant structure of expectation values of the many-body COBS. The cumulant structure of the state forms a hierarchy in correlations. More importantly, such static correlational hierarchy is inherited by the dynamical correlations and their SDEOM. We propose that the dynamical hierarchy is also carried into any perturbative calculation on that state. We demonstrate the validity of such perturbation hierarchy with an explicit example, in which we show that a single-particle-type perturbative calculation fails while a many-body perturbation following the hierarchy succeeds. We also discuss the computation and approximation schemes of ADT and its implications to other strong coupling theories like parton and slave particle methods.
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