The flow equation approach to many particle systems

Overthepastdecade, the? owequationmethodhasdevelopedintoanewv- satile theoretical approach to quantum many-body physics. Its basic concept was conceived independently by Wegner [1] and by G lazek and Wilson [2, 3]: the derivation of a unitary ? ow that makes a many-particle Hamiltonian - creasingly energy-diagonal. This concept can be seen as a generalization of theconventionalscalingapproachesinmany-bodyphysics, wheresomeult- violet energy scale is lowered down to the experimentally relevant low-energy scale [4]. The main di? erence between the conventional scaling approach and the ? ow equation approach can then be traced back to the fact that the ? ow equation approach retains all degrees of freedom, i. e. the full Hilbert space, while the conventional scaling approach focusses on some low-energy subspace. One useful feature of the ? ow equation approach is therefore that it allows the calculation of dynamical quantities on all energy scales in one uni? ed framework. Since its introduction, a substantial body of work using the ? ow eq- tion approach has accumulated. It was used to study a number of very d- ferent quantum many-body problems from dissipative quantum systems to correlated electron physics. Recently, it also became apparent that the ? ow equation approach is very suitable for studying quantum many-body n- equilibrium problems, which form one of the current frontiers of modern theoretical physics. Therefore the time seems ready to compile the research literature on ? ow equations in a consistent and accessible way, which was my goal in writing this book.