CAREER: Nonlinear Algebra for Quantum Materials

Quantum materials are central to modern materials research due to their impact on energy-efficient electronics, sustainable catalysis, precision sensing, and quantum information technologies. This project develops new mathematical foundations and computational methods that make simulations of such materials more reliable and systematically improvable. Computer-based predictions offer one of the fastest and most cost-effective ways to advance this area. Reliable simulations reveal how electrons organize and interact in quantum materials, enabling researchers to focus costly and painstaking experiments only on the most promising directions. Today’s most widely used computational tools are efficient, but they rely on approximations that are known to become unreliable for materials whose electrons interact strongly, including systems related to superconductivity and other emergent quantum phases. This project addresses this limitation by developing high-accuracy wavefunction-based methods, which have long been regarded as the most reliable approaches for molecular prediction, practical for extended crystalline materials. Beyond the scientific advances, the project will train students in a truly interdisciplinary setting that brings together applied mathematics, physics, computer science, and materials science, reflecting that progress in quantum materials research depends on the close interaction of these fields. This project will develop a rigorous and scalable framework for high-accuracy electronic-structure simulation of quantum materials using periodic coupled-cluster theory. The central challenge is to overcome the slow convergence and high computational cost that currently limit wavefunction-based methods for strongly correlated periodic systems. The project is organized around three integrated directions: (1) Periodic coupled-cluster theory will be recast as a nonlinear Petrov--Galerkin method for a pseudo-differential operator on a high-dimensional torus, creating a mathematical framework for deriving sharp error estimates and exploiting multiscale and low-rank compression techniques to reduce computational complexity. (2) The coupled-cluster equations will be studied as low-degree polynomial systems using algebraic geometry. This will provide new tools for characterizing their solution structure and identifying the physically relevant states, especially in regimes where multiple phases or competing solutions may coexist. (3) Machine learning and randomized numerical linear algebra will be used to accelerate and stabilize the resulting solvers by learning high-quality initial guesses, low-dimensional solution structures, and efficient tensor-compression strategies. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.