Understanding the robustness of topological phases of matter in the presence of interactions poses a difficult challenge in modern condensed matter, showing interesting connections to high energy physics. In this work, we leverage these connections to present a complete analysis of the continuum long-wavelength description of a generic class of correlated topological insulators: Wilson-Hubbard topological matter. We show that a Wilsonian renormalization group (RG) approach, combined with the so-called topological Hamiltonian, provide a quantitative route to understand interaction-induced topological phase transitions that occur in Wilson-Hubbard matter. We benchmark two-loop RG predictions for a quasi-1D Wilson-Hubbard model by means of exhaustive numerical simulations based on matrix product states (MPS). The agreement of the RG predictions with MPS simulations motivates the extension of the RG calculations to higher-dimensional topological insulators.
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