Research Areas

Chan and Umrigar will collaborate to advance renormalization group, tensor network, and quantum Monte Carlo methods towards the goal of realistic strongly correlated materials simulations. Work by Chan and Umrigar in the last decade helped establish the ab-initio density matrix renormalization group and determinant-based quantum Monte Carlo methods at the heart of first principles strongly correlated electron simulations. In our current work, we are

a) Carrying out the fundamental, long-needed generalization of the ab-initio density matrix renormalization group (DMRG) to arbitrary two-dimensional and three-dimensional tensor networks. This will lower the computational cost to treat (non-pathological) strongly correlated electronic materials from exponential to linear in the volume, the theoretical optimal scaling. This will enable the study of hitherto inaccessible strongly correlated systems, including key correlated electron materials. The mathematical techniques we employ lift numerical analytical tools in lower dimensions to the high-dimensional setting of quantum many-body physics.

b) Accelerating determinant-based quantum Monte Carlo and related modern selected configuration interaction algorithms by multiple orders of magnitude, and extend their capabilities to treat complex material properties. These efforts will produce a method for strongly correlated systems with the smallest computational prefactor, which becomes the method of choice for medium scale simulations.

c) Developing infinite basis and infinite system formalisms for both ab-initio tensor networks and quantum Monte Carlo. The accurate description of strongly correlated phenomena requires taking two limits: the continuum limit in spatial resolution (i.e. the complete basis set limit) as well as the thermodynamic limit (i.e. infinite system size). Reaching both these limits in practice presents a critical computational bottleneck. We will implement techniques to accelerate convergence to the basis set and infinite size limits, and explore new mathematical formulations, to provide the unprecedented ability to simulate directly at the basis set and thermodynamic limit.

d) Implementing our algorithms not only for ground-state bulk properties, but also for electronic spectral functions, so that our advances in correlated methods can immediately be combined with the Green's function embedding advances of Thrust 2 to study complex bulk and non-periodic materials.

e) Using our ability to efficiently solve the high-dimensional quantum problem to inform approximate reduced dimensional techniques, for example, density functionals in the strictly correlated limit, or density matrix methods.

Lin, Garcia-Cervera, and Chan will invent and extend new Green's function embedding techniques, both to study complex non-periodic materials, as well as to study bulk properties, through a self-consistent formulation. While Green's function embedding has a long history, the recent mathematical work of Lin has opened up the exciting possibility of combining embedding with fast numerical Green's function techniques for density functional theory, especially the Pole Expansion and Selected Inversion (PEXSI) method. The PEXSI method can already treat insulating and metallic systems with tens of thousands of atoms, and has been integrated into multiple electronic structure software packages.

Recent developments in correlated quantum embedding, such as the dynamical mean-field theory (DMFT) of Georges and Kotliar, or the density matrix embedding theory (DMET) of Knizia and Chan, further show how to extend quantum embedding to correlated dopants and bulk correlated materials. We are implementing efficient Green's function embedding algorithms to be used with PEXSI type methods.

This will radically change the type of environment that can be embedded with the Green's function, from the current practical limit of a few hundred atoms, to tens of thousands of atoms.

a) We are finding low rank, sparse approximations of the Dirichlet-to-Neumann map (i.e. self-energy operator) in PEXSI and implementing these techniques efficiently in parallel. These approximations is expected to further lower the prefactor of Green's function embedding methods by 1-2 orders of magnitude.

b) We are combining these advances with descriptions of strong correlations through the methods of Thrust 1, both for isolated correlated impurities (e.g. magnetic atoms in the Kondo problem), as well as for bulk correlated materials (such as high-temperature superconductors) through the dynamical mean-field and density matrix embedding theories.