Thermoelectric signatures of time-reversal symmetry breaking states in multiband superconductors

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 mechanism it has a singular behavior near the time-reversal symmetry breaking phase transition. A local hot spot in such superconductors is surrounded by a multipolar magnetic field, sensitive to the presence of domain walls and crystalline anisotropy of superconducting states. A non-stationary heating process produces an electric field and a charge imbalance in different bands. These effect can be measured and used to distinguish s$+$is and s$+$id superconducting states in the candidate materials such as Ba$_{\mathrm{1-x}}$K$_{\mathrm{x}}$Fe$_{\mathrm{2}}$As$_{\mathrm{2}}$.