his work is attempting to answer a few big questions–how do we model the electronic behavior of materials where interactions are important, and how do we find a unifying principle for the emergent behavior of these materials? Electrons are quantum particles, so computing the behavior of many of them is challenging computationally.  There is no known general way to solve the Schrödinger equation which scales polynomially in the number of electrons.

Quantum Monte Carlo (QMC) methods are particularly effective to study collections of particles because they explicitly simulate the correlations between electrons, which many other techniques try to smooth over at the cost of sometimes severe approximation. While these QMC techniques are often accurate, they still require some approximation to be stable at large numbers of particles, and they are computationally demanding.

Fortunately, they scale extremely well in parallel, so they can take advantage of massively parallel resources such as Blue Waters. Recently, through better computational resources like Blue Waters and better implementations, QMC algorithms can be applied to very realistic models of strongly correlated materials and much higher accuracy and new insights have arisen from these simulations. However, these simulations are typically limited to roughly 1000 quantum particles (about 1 nm³ in many materials), which is large enough to compute some useful properties, but many physical effects occur over much larger length scales.

To address emergent behavior at larger length scales, it is necessary to use multiscale models; in physics and other fields, these are often most easily expressed through effective Hamiltonians for quantum coarse- grained models. There are several techniques to create quantum coarse-grained models, but they are inefficient (scaling exponentially in the system size) or involve approximations that are difficult to systematically improve. Our group has recently developed a new technique to compute accurate coarse-grained Hamiltonian from first principles quantum calculations. This technique, which we call density matrix downfolding (DMD) recasts the coarse-graining problem as a data analysis and fitting problem, which allows for the application of machine learning techniques.

In this project, we use QMC methods to derive effective models using density matrix downfolding. The project consists of several interrelated subprojects, which range from development of a new Python package to perform QMC calculations for model generation, to calculations closely coupled to experiments performed at Illinoiis.