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We show that the observed time-reversal symmetry breaking (TRSB) of thesuperconducting state in $\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$ can beunderstood as originating from inhomogeneous strain fields near edgedislocations of the crystal. Specifically, we argue that, without straininhomogeneities, $\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$ is asingle-component, time-reversal symmetric superconductor, likely with$d_{x^{2}-y^{2}}$ symmetry. However, due to the strong strain inhomogeneitiesgenerated by dislocations, a slowly-decaying sub-leading pairing statecontributes to the condensate in significant portions of the sample. As itphase winds around the dislocation, time-reversal symmetry is locally broken.Global phase locking and TRSB occur at a sharp Ising transition that is notaccompanied by a change of the single-particle gap and yields a very small heatcapacity anomaly. Our model thus explains the puzzling absence of a measurableheat capacity anomaly at the TRSB transition in strained samples, and thedilute nature of the time-reversal symmetry broken state probed by muon spinrotation experiments. We propose that plastic deformations of the material maybe used to manipulate the onset of broken time-reversal symmetry.

Inhomogeneous time-reversal symmetry breaking in $\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$