Non-analyticity of Krylov-complexity derivatives at spectral changes (Conjecture)

Establish the conjecture that derivatives of Krylov complexity (including Krylov spread complexity) generically exhibit non-analytic behavior—either discontinuities or divergences—whenever the spectrum of the underlying quantum system undergoes certain changes, such as a transition from gapful to gapless or other spectral non-analyticities, thereby enabling the detection of quantum phase transitions via Krylov-based measures.

Background

The paper discusses prior evidence from a monitored 1D Ising chain where second derivatives of the Krylov density diverge as the imaginary part of the spectrum changes from gapless to gapped, suggesting a broader phenomenon linking spectral changes to non-analytic behavior in Krylov measures.

Motivated by this, the authors propose a general conjecture that such non-analyticities in derivatives of Krylov complexity occur at spectral changes. In the present work, they provide supporting examples in a non-Hermitian SSH model but emphasize the need to establish the conjecture more generally.