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Recent experiments strongly indicate deep connections between transports ofstrange metal and high $T_c$ superconductors. For example, the dependence ofthe zero-temperature phase stiffness on the critical superconductingtemperature is generally linear, which is incompatible with the standardBardeen-Cooper-Schrieffer description. We explore the scaling relations amongsuperconducting critical temperature, superfluid density, and momentumdissipation (disorder) strength for the Gubser-Rocha model with extensions inthe probe limit. The critical temperature is evaluated by using both theSturm-Liouville eigenvalue method and numerical calculations. In the normalphase, we show that the critical temperature is proportional to the momentumdissipation (disorder) strength in a certain parameter range. In thesuperconducting phase, studying the AC conductivity analytically andnumerically, we find linear dependence of zero-temperature superfluid density(phase stiffness) on the critical superconducting temperature, which isconsistent with recent experiments of high $T_c$ superconductors. These resultsfurther underpin the deep connections between strange metal and high $T_c$superconductors.

Dependence of the critical temperature and disorder in holographic superconductors on superfluid density