Holographic Schwinger-Keldysh effective field theories including a non-hydrodynamic mode

We derive the Schwinger-Keldysh effective field theories for diffusion including the lowest non-hydrodynamic degree of freedom from holographic Gubser-Rocha systems. At low temperature the dynamical non-hydrodynamic mode could be either an IR mode or a slow mode, which is related to IR quantum critical excitations or encodes the information of all energy scales. This additional dynamical vector mode could be viewed as an ultraviolet sector of the diffusive hydrodynamic theory. We construct two different effective actions for each case and discuss their physical properties. In particular we show that the Kubo-Martin-Schwinger symmetry is preserved.