Closed-form analytic expression for the time-averaged Krylov spread in the non-Hermitian SSH model

Derive a closed-form analytic expression for the time-averaged Krylov spread density \bar{\mathcal{C}}(h,\gamma) of the non-Hermitian Su-Schrieffer-Heeger model, particularly in parameter regions where the spectrum is complex and the imaginary part exhibits a continuum of gapless modes.

Background

The authors define the time-averaged Krylov spread density \bar{\mathcal{C}} and compare it to the spread \mathcal{C}\Omega obtained via a unitary evolution to the non-Hermitian vacuum. They observe that \bar{\mathcal{C}} equals \mathcal{C}\Omega only in regions with purely imaginary spectra and at most finitely many gapless modes.

In regions with complex spectra, the time-averaged spread involves a complicated dependence on momentum k, and the authors state that they were unable to obtain a closed-form expression, highlighting an analytic gap that, if resolved, could clarify the relationship between the two spreads and aid in probing phase transitions.