Random magnetic field and quasi-particle transports in high $T_c$ cuprates

By a singular gauge transformation, the quasi-particle transport in the mixed state of high $T_c$ cuprates is mapped into charge-neutral composite Dirac fermion moving in short-range correlated random scalar and long-range correlated vector potential. A fully quantum mechanical approach to longitudinal and transverse thermal conductivities and their fluctuations in mesoscopic sample is presented. The quasi-particle scattering from the random magnetic field which was completely missed in all the previous semi-classical approaches is stressed. The implications for experiments are discussed.