The flow equation approach to many particle systems

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Over the past decade, the flow equation method has emerged as a versatile theoretical approach to quantum many-body physics. Its core concept, developed independently by Wegner and Głazek and Wilson, involves deriving a unitary flow that progressively diagonalizes a many-particle Hamiltonian. This method generalizes conventional scaling approaches in many-body physics, which typically lower an ultraviolet energy scale to a relevant low-energy scale. The key distinction lies in the flow equation approach retaining all degrees of freedom within the full Hilbert space, while traditional scaling focuses on a low-energy subspace. This feature enables the calculation of dynamical quantities across all energy scales within a unified framework. Since its inception, a significant body of work has utilized the flow equation approach to address various quantum many-body problems, ranging from dissipative quantum systems to correlated electron physics. Recently, it has also proven effective for exploring quantum many-body non-equilibrium problems, a current frontier in theoretical physics. Consequently, this book aims to compile the research literature on flow equations in a coherent and accessible manner, reflecting the method's growing importance and versatility in the field.