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Origins of spontaneous magnetic fields in Sr$_2$RuO$_4$

The nature of the broken time reversal symmetry (BTRS) state in Sr$_2$RuO$_4$ remains elusive, and its relation to superconductivity remains controversial. There are various universal predictions for the BTRS state when it is associated with a multicomponent superconducting order parameter. In particular, in the BTRS superconducting state, spontaneous fields appear around crystalline defects, impurities, superconducting domain walls and sample surfaces. However, this phenomenon has not yet been experimentally demonstrated for any BTRS superconductor. Here, we aimed to verify these predictions for Sr$_2$RuO$_4$ by performing muon spin relaxation ($μ$SR) measurements on Sr$_{2-y}$La$_{y}$RuO$_4$ single crystals at ambient pressure and stoichiometric Sr$_2$RuO$_4$ under hydrostatic pressure. The study allowed us to conclude that spontaneous fields in the BTRS superconducting state of Sr$_2$RuO$_4$ appear around non-magnetic inhomogeneities and, at the same time, decrease with the suppression of $T_{\rm c}$. The observed behaviour is consistent with the prediction for multicomponent BTRS superconductivity in Sr$_2$RuO$_4$. The results of the work are relevant to understanding BTRS superconductivity in general, as they demonstrate, for the first time, the relationship among the superconducting order parameter, the BTRS transition, and crystal-structure inhomogeneities.