Optimal broadening of finite energy spectra in the numerical renormalization group: application to dissipative dynamics in two-level systems

Numerical renormalization group (NRG) calculations of quantum impurity models, based on a logarithmic discretization in energy of electronic or bosonic Hamiltonians, provide a powerful tool to describe physics involving widely separated energy scales, as typically encountered in nanostructures and strongly correlated materials. This main advantage of the NRG was however considered a drawback for resolving sharp spectral features at finite energy, such as dissipative atomic peaks. Surprisingly, we find a bunching of many-body levels in NRG spectra near dissipative resonances, and exploit this by combining the widely-used Oliveira's $z$-trick, using an averaging over {\it few} discrete NRG spectra, with an optimized {\it frequency-dependent} broadening parameter $b(\w)$. This strategy offers a tremendous gain in computational power and extracts all the needed information from the raw NRG data without {\it a priori} knowledge of the various energy scales at play. As an application we investigate with high precision the crossover from coherent to incoherent dynamics in the spin boson model.