Non-Equilibrium Critical Phenomena From Probe Brane Holography in Schr\"odinger Spacetime

We study the non-equilibrium steady-state phase transition from probe brane holography in $z=2$ Schr\"odinger spacetime. Concerning differential conductivity, a phase transition could occur in the conductor state. Considering constant current operator as the external field and the conductivity as an order parameter, we derive scaling behavior of order parameter near the critical point. We explore the critical exponents of the non-equilibrium phase transition in two different Schr\"odinger spacetimes, which originated $1)$ from supergravity, and $2)$ from AdS blackhole in the light-cone coordinates. Interestingly, we will see that even at the zero charge density, in our first geometry, the dynamical critical exponent of $z=2$ has a major effect on the critical exponents.