Neutrinoless double-$\beta$ decay of $^{124}$Sn, $^{130}$Te, and $^{136}$Xe in the Hamiltonian-based generator-coordinate method

We present a generator-coordinate method for realistic shell-model Hamiltonians that closely approximates the full shell model calculations of the matrix elements for the neutrinoless double-beta decay of $^{124}$Sn, $^{130}$Te, and $^{136}$Xe. We treat axial quadrupole deformations and also triaxial quadrupole deformations, including the proton-neutron pairing amplitudes as generator coordinates. We validate this method by calculating and comparing spectroscopic quantities with the exact shell model results. A detailed analysis of the $0\nu\beta\beta$ decay nuclear matrix elements for $^{124}$Sn, $^{130}$Te, and $^{136}$Xe is presented. Our Hamiltonian-based generator-coordinate method produces $0\nu\beta\beta$ matrix elements much closer to the shell model ones, when compared to the existing energy density functional-based approaches. The remaining overestimation of $0\nu\beta\beta$ nuclear matrix element suggests that additional correlations may be needed to be taken into account for $^{124}$Sn, $^{130}$Te, and $^{136}$Xe when calculating with the Hamiltonian-based generator-coordinate method.