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 {\em long-range} correlated random scalar and vector potential. It is shown that $ \kappa_{xx}/T = F_{1}(a \frac{T}{T_{c}} \sqrt{\frac{H_{c2}}{H}}) $ acquires vertex correction. At high enough magnetic field $ H \gg H_{c2} (\frac{T}{T_{c}})^{2} $, $ \kappa_{xx}/T $ approaches a {\em different} value than its zero field counterpart. However the thermal Hall conductivity is much smaller $ <\kappa_{xy}(H, T)/T>= \frac{T_{c}}{\epsilon_{F}}\sqrt{\frac{H}{H_{c2}}} F_{2}(b \frac{T}{T_{c}} \sqrt{\frac{H_{c2}}{H}}) $. At $ H \gg H_{c2} (\frac{T}{T_{c}})^{2} $, it {\em increases} as $ \sqrt{H} $. Due to the previously neglected random magnetic fields, the properly defined thermal Hall {\em conductance} fluctuation in a mesoscopic sample satisfies {\em similar} scaling as $ \kappa_{xx}/T $: $ \sqrt{< (\tilde{\kappa}_{xy}/T)^{2} >} = F_{3}(c \frac{T}{T_{c}} \sqrt{\frac{H_{c2}}{H}}) $. Therefore it is much larger than its average $ < \kappa_{xy}/T > $ and should also approach a constant at high enough magnetic field. The implications for experiments are discussed.