Unconventional thermoelectric effect in superconductors that break time-reversal symmetry

We demonstrate that superconductors which break time-reversal symmetry can exhibit thermoelectric properties, which are entirely different from the Ginzburg mechanism. As an example, we show that in the $s+is$ superconducting state there is a reversible contribution to thermally induced supercurrent, whose direction is not invariant under time-reversal operation. Moreover in contrast to Ginzburg's mechanism it has a singular behavior near the time-reversal symmetry breaking phase transition. The effect can be used to confirm or rule out the $s+is$ state, which is widely expected to be realized in pnictide compounds Ba$_{1-x}$K$_{x}$Fe$_2$As$_2$ and stoichiometric LiFeAs.