Abstract
Solving the fundamental microscopic equations of interacting quantum particles is a goal of many-body physicists. Statistical methods reduce the complexity of the problem by sampling phase space selectively using random-walks and real states. Many interesting physical phenomena (e.g., electrons in external magnetic fields) involve systems whose state functions are inherently complex-valued. The Fixed-Phase method is a stochastic approach to deal with such problems. Its key ingredient is a trial phase that plays the role of gauge function in the transformation that maps the original fermion (or boson) problem to a boson problem for the modulus of the state function. The Released-Phase method relaxes that constraint and allows us to obtain, in principle, the "exact" properties, although it is subjected to the infamous "phase problem." In our tour of the (complex) Quantum World, we will show how these methods have been successfully applied to a wide variety of physical phenomena ranging from quantum Hall topological fluids and Wigner crystals to the study of the core structure of vortices in superfluid 4 He and atomic systems in superstrong magnetic fields found in astrophysical settings.
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