Time-reversal symmetry breaking in frustrated superconductor Re$_2$Hf

Geometrical frustration leads to novel quantum phenomena such as the spin-liquid phase in triangular and Kagom\'e lattices. Intra-band and inter-band Fermi surface (FS) nesting can drive unique superconducting (SC) ground states with $d$-wave and $s^{\pm}$ pairing symmetries, respectively, according to the criterion that the SC gap changes sign across the nesting wavevector. For an odd number of FSs, when multiple inter-band nesting is of comparable strength, the sign-reversal criterion between different FS sheets can leads to frustration, which promotes novel SC order parameters. Here we report the experimental observation of a time-reversal symmetry breaking pairing state in Re$_2$Hf resulting from FS nesting frustration. Furthermore, our electronic specific heat and transverse-field $\mu$SR experiments suggest a fully gaped pairing symmetry. The first-principle electronic structure calculation reveals multiple Fermi surface sheets with comparable inter-band nesting strength. Implementing the {\it ab-initio} band structure, we compute spin-fluctuation mediated SC pairing symmetry which reveals a $s+is'$-pairing state - consistent with experimental observations. Our investigation demonstrates a novel SC state which provides a putative setting for both applied and fundamental study.