  The field dependence of low-temperature thermal conductivity $\kappa(H)$
observed on cuprates is explained by calculating $\kappa(H)$ microscopically.
The heat current carried by low-lying quasiparticles around a vortex core
decreases with $H$ due to its depletion by the $H$-induced moment centered at a
core for an underdoped case. This leads to a common and consistent picture on
an ``anomalous core''. $\kappa(H,T)$ is found to be a useful tool for probing
quasiparticle structures in the mixed state in general.
