Designing models using machine learning: one-body reduced density matrices and spectra

One of the biggest challenges in condensed matter physics is to calculate properties of materials taking into account the quantum many-body nature of matter. While the Coulomb interaction is universal, its effects cannot be separated from the specific material under analysis, which leads to a huge theoretical and computational effort. Recently, Machine Learning (ML) has raised new hopes as a tool that could be used to screen and predict properties of broad classes of materials. Indeed, the statistical structure of ML tools is particularly well suited to deal with quantum mechanics, thanks to the statistical information encoded in the state vector, and in recent years ML has been used for addressing various classes of problems. In quantum physics and chemistry, ML has been mainly used to predict energies and forces starting from the atomic composition. Another research line lies in the domain of Density Functional Theory (DFT): here the system is described by its electronic density, in other words, its properties are functionals of the density. These functionals are in general not known and must be approximated. So far, ML was mainly used with the aim to determine energies as density functionals. The central topic of the present thesis was the design of a density functional. However, the underlying strategy chosen differs from the usual approach in two ways: first, instead of being observable specific, we concentrated on an important building block, the one body reduced density matrix (1-RDM). The knowledge of this quantity allows one to directly access much useful information, such as the kinetic and exchange energies. We hence investigated the functional connecting the 1-RDM to the density. Second, instead of using ML merely as a clever interpolation tool, we asked a methodological question, namely "Is it possible for the model maker to learn with the machine?” This would allow one to build functionals that could then be used without creating huge data-sets beforehand. One fact is essential for modeling the 1-RDM, i.e. its diagonal is the density. Consequently, the wanted functional must be derived by exploiting the functional constraints connecting the diagonal to the off-diagonals. The fact that the object on which to build the functional is contained in the matrix to be determined assures the existence of strong correlations in between the values of the entries. This is analogous to the existing relations among the pixels of an image. Therefore, the success of ML techniques in extracting spacial information in image processing tasks was at the basis of the work. First we tried to learn from the way the machine structures the data in order to create new analytical models. Then we moved to considering how the human can inform the machine by embedding in it some pre-existing theoretical knowledge. For the discussion we used a simple two sites system, whose 1-RDM can be found exactly in some limiting cases. Many other results were derived for characterizing this system, including the functional in non-limiting cases and the observation of its applicability as an auxiliary system for the determination of the desired functional in a system with a higher number of grid-points. Another methodological question has been discussed: “How can we use ML to augment low resolution calculations and experiments?” This question could only be partially addressed, since adverse conditions did not allow us to perform the necessary experiment.