Research

My research is lies at the intersection between applied math and condensed matter physics with a particular focus on electronic properties of materials. Most recently, I’ve been interested in understanding electron-electron interactions in moiré materials (in particular twisted bilayer graphene). Previously, I worked on sparse basis construction for insulating materials. A list of my publications is below.

Moiré Materials

When two periodic patterns are twisted relative to each other, a large scale pattern, called a moiré pattern, is formed. Recently, it was discovered that by placing two thin sheets of a material on top of each other with a relative twist, the interplay between the small scale atomic configuration and the large scale moiré pattern leads to remarkable behaviors such as correlated insulators and superconductivity. These behaviors are due in part to the fact that the interactions between the two scales (atomic and moiré) can suppress the kinetic energy of the electrons making electron-electron repulsion dominant. In my work, I have studied the role of electron-electron repulsion in twisted bilayer graphene and its implications for the ground states.

Sparse Basis Construction

Given a subspace of a vector space, a standard step in most algorithms is to construct an orthogonal basis for this subspace. Not all orthogonal bases are equally well suited for computation however. For studying systems with local interactions, bases which decay quickly in space lead to sparse equations which drastically speed up calculation. In electron structure theory, orthogonal bases which decay exponentially quickly in space are called exponentially localized (generalized) Wannier functions (ELWFs). In my work, I proposed an algorithm for constructing ELWFs and proved its correctness for a large class of subspaces (Left: A low energy eigenfunction of -Δ + V and an ELWF which is contained in the low energy eigenspace).

Publications