Bayesian inference and the analytic continuation of imaginary-time quantum Monte Carlo data

Abstract

We present a way to use Bayesian statistical inference and the principle of maximum entropy to analytically continue imaginary-time quantum Monte Carlo data. We supply the details that are lacking in the seminal literature but are important for the motivated reader to understand the assumptions and approximations embodied in these methods. First, we summarize the general relations between quantum correlation functions and spectral densities. We then review the basic principles, formalism, and philosophy of Bayesian inference and discuss the application of this approach in the context of the analytic continuation problem. Next, we present a detailed case study for the symmetric, infinite-dimension Anderson Hamiltonian. We chose this Hamiltonian because the qualitative features of its spectral density are well established and because a particularly convenient algorithm exists to produce the imaginary-time Green's function data. Shown are all the intermediate steps of data and solution qualification. The importance of careful data preparation and error propagation in the analytic continuation is discussed in the context of this example. Then, we review the different physical systems and physical quantities to which these, or related, procedures have been applied. Finally, we describe other features concerning the application of our methods, their possible improvement, and areas for additional study.