Low-rank eigenvalue solvers for block-sparse matrix product states

TL;DR

This paper introduces a low-rank eigenvalue solver tailored for fermionic Schr"odinger equations, leveraging matrix product states and rank truncation to efficiently approximate multiple eigenstates.

Contribution

It develops a novel iterative eigensolver combining preconditioned inverse iteration with rank truncation, and extends it to approximate multiple eigenspaces simultaneously.

Findings

The solver effectively approximates eigenfunctions with controlled accuracy.

Numerical tests demonstrate the method's efficiency on model problems.

Abstract

We consider an iterative eigensolver for Schr\"odinger equations that constructs low-rank approximations of eigenfunctions with accuracy-adapted ranks, with particular focus on fermionic Schr\"odinger equations in second-quantized form and on matrix product state approximations enforcing particle number conservation. We provide a complete analysis of a solver based on preconditioned inverse iteration combined with rank truncation and propose a generalization to subspace iteration for the joint approximation of several eigenspaces. The practical performance of the method is illustrated by numerical tests for several model problems.