Recently machine-learning approaches are developing rapidly as tools to analyze accumulated data across various research domains. Machine learning also has potential for innovating ways of exposing physical observables, which is invisible in the direct scientific measurements. Solving underdetermined inverse problem to extract fundamental physical quantities entangled each other in existing experimental data is a crucial step towards the machine learning innovation of the scientific measurements. A combination of properly chosen prior knowledge and machine learning holds the key to find a solution of the inverse problem.

An enigmatic inverse problem in condensed matter physics is found in open issues of high temperature superconductors, which has long been a major challenge in physics. How electrons are mutually interacting is the key to identify the origin of superconductivity. We are, however, able to observe motion of electrons only after projection onto experimentally accessible degrees of freedom.

 In this study, we utilized the Boltzmann machine [1,2], combined with physically sound prior knowledge, to solve the inverse problem and extract physical quantities hidden in experimental data. The method is applied to the angle-resolved photoemission spectroscopy data [3] for spectra of copper oxide superconductors. From the spectra of the cuprates [4], we extracted metallic (normal) and superconducting (anomalous) components of the self-energy separately, in which mutual interactions among electrons are encoded. We found prominent peak structures emerging both in the normal and anomalous self-energies, Σ^nor and W, respectively, which are canceled in the total self-energy Σ and hence invisible in experiments, as the origin of high-temperature superconductivity [5]. The peak structures also explain doping dependence of T_c and strange-metal behaviors observed in the cuprate superconductors. The present achievement may open avenues for innovative machine-learning spectroscopy.

Keywords: High-temperature superconductors, Spectroscopy