Analytic continuation by combining sparse modeling with the Pade approximation

Numerical methods based on the imaginary-time path-integral such as the path-integral Monte Carlo method are powerful tools to investigate a quantum many-body system both at absolute zero and finite temperatures. For example, these can calculate the imaginary-time Green's function directly, and other important quantities such as spectrum function can be transformed from this by the analytic continuation (AC) or by solving the Lehmann representation as an integral equation. In practice, however, this transformation is unstable against noise of the imaginary-time Green's function.

Several methods have been developed to solve this problem so far. The noise reduction by the sparse modeling (SpM) is one of them. In this method, we transform basis by the singular matrices of the integral kernel and truncate noisy components in the new basis by the sparse modeling. However, this truncation introduces an unphysical oscillation to the obtained spectrum as a systematic error.

In this study, we have improved SpM method by combining it with AC by the Pade approximation. AC by the Pade approximation gives a stable and smooth spectrum in low frequency region, which can be used to make the SpM spectrum smooth and high accuracy.