Emerging superconductivity with broken time reversal symmetry inside a superconducting $s$-wave state

In general, magnetism and superconductivity are antagonistic to each other. However, there are several families of superconductors, in which superconductivity may coexist with magnetism, and only a few examples are known, when superconductivity itself induces spontaneous magnetism. The most known compounds are Sr${}_2$RuO${}_4$ and some noncentrosymmetric superconductors. Here, we report the finding of a narrow dome of a novel $s+i{s}^{\mathrm{\prime }}$ superconducting (SC) phase with broken time-reversal symmetry (BTRS) inside the broad $s$-wave SC region of the centrosymmetric multiband superconductor Ba${}_{1-\mathrm{x}}$K$_{\rm x}$Fe$_2$As$_2$($0.7 \lesssim x \lesssim 0.85$). We observe spontaneous magnetic fields inside this dome using the muon spin relaxation ($\mu$SR) technique. Furthermore, our detailed specific heat study reveals that the BTRS dome appears very close to a change in the topology of the Fermi surface (Lifshitz transition). With this, we experimentally demonstrate the emergence of a novel quantum state due to topological changes of the electronic system.