Analytic continuation of the imaginary-frequency Green's function via causal smoothing spline approach

Many of practical many-body calculations are conducted in imaginary time and frequency. To obtain dynamic physical quantities such as electronic spectral functions, we need to obtain real-frequency Green's functions from imaginary-frequency Green's functions. This procedure is known as the analytic continuation, and it is known to be ill-posed. In this work, we suggest a causal smoothing spline approach for the analytic continuation. Our causal smoothing spline approach discretizes the problem by using the cubic spline on nonuniformly generated real-frequency grids and regularizes the ill-posedness by using the second derivatives of the spectral function. We determine the regularization parameter by balancing the regularization with data fit. As a result, our approach conducts the analytic continuation stably with few control parameters, and it can be applied straightforwardly to matrix-valued Green's functions as well. By applying it to systems of known spectral functions, we demonstrate that our approach is robust and precise. Moreover, we show the use of our causal smoothing spline approach in dynamical mean-field theory calculations.