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Algebraic-Dynamical Theory for Quantum Many-body Hamiltonians: A Formalized Approach To Strongly Interacting Systems

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.