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One approach to probe the still controversial superconductivity inSr$_2$RuO$_4$ is to apply external perturbations that break the underlyingtetragonal crystalline symmetry. Chiral $p_x+ip_y$ and $d_{xz}+id_{yz}$ statesrespond to such perturbations in ways that may help to distinguish them fromother superconducting pairings. However, past experimental efforts along thisline, using uniaxial strains and magnetic fields parallel to the RuO$_2$ plane,have not been able to reach an unambiguous conclusion. In this study, wepropose to further examine the possibility of chiral superconducting order inSr$_2$RuO$_4$ using an alternative tetragonal-symmetry-breaking perturbation --in-plane supercurrent. We study the superconducting phase diagram as a functionof both temperature and the applied supercurrent. Supercurrent genericallysplits the transition of the two chiral order parameter components, and we showthat the splitting can give rise to visible specific heat anomalies.Furthermore, supercurrent parallel and anti-parallel to the unidirectionalpropagation of the chiral edge modes impact the edge states in differentmanner. This difference manifests in the tunneling spectrum, thereby providingan additional means to probe the chirality even when the related spontaneousedge current is vanishingly small. Finally, we discuss the distinction ofsupercurrent responses in non-chiral time-reversal-symmetry-breakingsuperconducting states. Our proposal can be applied to other candidate chiralsuperconductors.

Examining the possibility of chiral superconductivity in Sr$_2$RuO$_4$ and other compounds via applied supercurrent