Wandering range of robust quantum symmetries

TL;DR

This paper defines and analyzes the wandering range of robust symmetries in Hamiltonian systems, providing conditions and bounds for how these symmetries deviate under perturbations.

Contribution

It introduces the wandering range concept for robust symmetries and derives nonperturbative bounds and conditions for linear scaling under perturbations.

Findings

Wandering range measures deviation of symmetries under perturbation.

Linear scaling of wandering range occurs under specific conditions.

Explicit nonperturbative bounds are established.

Abstract

This paper introduces the concept of the wandering range of a robust symmetry $S$ of a Hamiltonian $H$. This quantity measures how the perturbed time evolution $\mathrm{e}^{\mathrm{i}t(H+\varepsilon V)} S \mathrm{e}^{-\mathrm{i} t(H+\varepsilon V)}$ deviates from its unperturbed counterpart $\mathrm{e}^{\mathrm{i} tH} S\mathrm{e}^{-\mathrm{i} tH} = S$. Although the wandering range does not necessarily scale linearly with the perturbation strength $\varepsilon$, we identify conditions under which this linear behavior is recovered and we obtain explicit nonperturbative bounds.