  In complex transition metal oxides (TMO) an arrested electronic phase
separation (PS) appears by tuning the system near a Lifshitz transition in
multiband Hubbard models. The PS in La2CuO4+y near insulator to metal
transition (IMT) is made of short range Charge Density Wave (CDW) order
inhomogeneity coexisting with quenched lattice disorder. While at high doping
y=0.1 percolation gives a single superconducting phase, near the IMT at y=0.06
two coexisting superconducting phases appear: the first one with a critical
temperature Tc1=16 K and the second one with Tc2=29K. It is known that the two
superconducting phases are characterized by two different space geometry
because of two different spatial distributions of both CDW order and dopants
self-organization. Here we show that these two phases show different flux
dynamic regimes using alternating current (AC) multi-harmonic susceptibility
experiments. This is a unique technique capable to investigate multi-phase
superconductors and characterize their transport properties in a percolative
scenario. Results point out that the low critical temperature phase is well
described by a bulk-like flux pinning with a 2D geometry while the phase with
higher critical temperature shows a barrier pinning mechanism providing direct
evidence of two different superconducting vortex dynamics in different complex
geometrical spaces.
