  Using the path-integral formulation we derive microscopically a
Ginzburg-Landau free energy with a Zeeman coupling between the magnetic field
and the orbital angular momentum of the Cooper pairs in a superconductor with
singlet pairing in the $d_{x^{2}-y^{2}}$- and the sub-dominant $d_{xy}$-
channels. The Zeeman coupling induces a time-reversal-symmetry-breaking pairing
state. Based on careful examinations of the energy gain due to the Zeeman
coupling, the energy lost due to the kinetic energy of the excess superfluid,
and the Doppler energy shift for quasi-particle excitations, we present a
coherent interpretation for the puzzling and conflicting thermal conductivites
observed at above 5K (K. Krishana, {\it et al}, Science {\bf 277}, 83(1997))
and at sub-Kelvins (H. Aubin, {\it et al}, Phys. Rev. Lett. {\bf 82}, 624
(1999)).
