Coupled Cluster Degree of the Grassmannian

TL;DR

This paper calculates the number of solutions to a nonlinear eigenvalue problem on the Grassmannian, relevant to quantum chemistry, using algebraic geometry and toric degenerations.

Contribution

It proves a conjectured formula for the Grassmannian of lines and introduces a squarefree Gr"obner basis related to its birational parametrization.

Findings

Derived the solution count formula for the Grassmannian eigenproblem

Established a squarefree Gr"obner basis for the graph of the Grassmannian

Connected the problem to toric degenerations in representation theory

Abstract

We determine the number of complex solutions to a nonlinear eigenvalue problem on the Grassmannian in its Pl\"ucker embedding. This is motivated by quantum chemistry, where it represents the truncation to single electrons in coupled cluster theory. We prove the formula for the Grassmannian of lines which was conjectured in earlier work with Fabian Faulstich. This rests on the geometry of the graph of a birational parametrization of the Grassmannian. We present a squarefree Gr\"obner basis for this graph, and we develop connections to toric degenerations from representation theory.