Effective dynamics from a multi-timescale and symmetry-preserving exact renormalization group approach and its applications

We provide a useful method of deriving the slow, effective dynamics of an autonomous system that preserves some desired symmetry properties of the original system. By an appropriate infinitesimal transformation from the original system into a simpler, effective system by means of an exponential map, the group symmetries are preserved independent of the truncation order. The formalism relies on a simple and intuitive approach to address fast and slow timescales common within experiments that allows for higher-order corrections from the fast, high-frequency terms. The formalism was previously introduced in the context of quantum information science as quantum averaging theory (QAT) but can be formally generalized to any autonomous differential equation allowing for a wider range of applications. We will demonstrate the exciting capabilities of the QAT method with examples from quantum information science and discuss how this technique can be used to develop group-preserving effective models of multi-scale classical dynamics.