Krylov complexity in a natural basis for the Schr\"odinger algebra

TL;DR

This paper explores Krylov complexity in quantum systems with Schr"odinger algebra symmetry, proposing a natural basis approach that reveals a pentadiagonal structure and offers insights into non-semisimple algebras.

Contribution

It introduces a novel method to compute Krylov complexity for non-semisimple algebras using a natural basis, resulting in a pentadiagonal structure of the evolution operator.

Findings

Krylov complexity behaves as expected in Schr"odinger algebra systems.

The natural basis yields a pentadiagonal structure, contrasting with traditional methods.

This approach can be extended to other non-semisimple algebras.

Abstract

We investigate operator growth in quantum systems with two-dimensional Schr\"odinger group symmetry by studying the Krylov complexity. While feasible for semisimple Lie algebras, cases such as the Schr\"odinger algebra which is characterized by a semi-direct sum structure are complicated. We propose to compute Krylov complexity for this algebra in a natural orthonormal basis, which produces a pentadiagonal structure of the time evolution operator, contrasting the usual tridiagonal Lanczos algorithm outcome. The resulting complexity behaves as expected. We advocate that this approach can provide insights to other non-semisimple algebras.