Holographic subregion complexity in unbalanced Stückelberg holographic superconductors

Abstract

Within the subregion complexity-volume conjecture, we numerically compare holographic subregion complexity (HSC) and holographic entanglement entropy (HEE) for a strip in unbalanced Stückelberg holographic superconductors. Varying the Stückelberg parameter $γ$ yields both second- and first-order transitions. Both observables signal these transitions, but with markedly different robustness. The qualitative HEE signatures persist across strip widths, and the finite part of HEE remains smaller in the superconducting phase than in the normal phase. The HSC is instead strongly width dependent: its temperature trend is opposite to that of HEE at small $\ell$ and agrees with it at large $\ell$. Consequently, the superconducting and normal HSC branches reverse their relative ordering, creating a crossover region where they nearly coincide. There, HSC alone cannot reliably determine the occurrence or order of the transition, and the physical branch must be selected from the grand potential. Thus, HEE provides a more robust diagnostic, whereas HSC is a scale-dependent probe whose interpretation depends explicitly on the subsystem size.