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Time-reversal symmetry breaking superconductivity in the coexistence phase with magnetism in Fe-pnictides

We argue that superconductivity in the coexistence region with spin-density-wave (SDW) order in weakly doped Fe-pnictides differs qualitatively from the ordinary $s^{+-}$ state outside the coexistence region, as it develops an additional gap component which is a mixture of intra-pocket singlet ($s^{++}$) and inter-pocket spin-triplet pairings (the $t-$state). The coupling constant for the $t-$channel is proportional to the SDW order and involves interactions that do not contribute to superconductivity outside of the SDW region. We argue that the $s^{+-}$ and $t-$type superconducting orders coexist at low temperatures, and the relative phase between the two is in general different than $0$ or $π$, manifesting explicitly the breaking of the time-reversal symmetry promoted by long-range SDW order. We show that this exotic state emerges already in the simplest model of Fe-pnictides, with one hole pocketand two symmetry-related electron pockets. We argue that in some parameter range time-reversal gets broken even before long-range superconducting order develops.