Time-reversal symmetry breaking and anomalous Hall effect in heavy fermion metals

Motivated by recent experimental evidence for a possible chiral spin liquid phase in the metallic pyrochlore heavy fermion iridates (Pr$_2$ Ir$_2$ O$_7$) [Phys.Rev.Lett, {\bf 96}, 087204 (2006), Phys.Rev.Lett {\bf 98}, 057203 (2007), Nature {\bf 463}, 210 (2010), Phys.Rev.Lett, {\bf 106}, 217204 (2011)], we study the effect of Kondo coupling on a time-reversal symmetry breaking state of the $J_1-J_2$ model on square lattices. We use a slave fermion representation for the $f$-moments which are coupled to conduction electrons, and study the mean field solution in the large-N limit. We calculate the ground state energies of various feasible states, and map out the mean field phase diagram by energetic consideration. As the probe for time-reversal symmetry breaking, we calculate the anomalous Hall response for the chiral phase. Finally we discuss the implications of our results on the pyrochlore heavy-fermion iridates.