High-accuracy Nuclear Spin Dependent Parity Violating Amplitudes in $^{133}$Cs

Relativistic coupled-cluster (RCC) theory at the singles and doubles approximation has been implemented to estimate nuclear spin dependent (NSD) parity violating (PV) electric dipole (E1) transition amplitudes ($E1_{PV}^{NSD}$) among hyperfine levels of the $6s ~^2S_{1/2} \rightarrow 7s ~^2S_{1/2}$ transition in $^{133}$Cs. To validate our calculations, we reproduce the Dirac-Hartree-Fock values and results from the combined coupled-Dirac-Hartree-Fock and random phase approximation (CPDF-RPA) method reported earlier. Contributions from the double-core-polarization (DCP) effects at the CPDF-RPA method were found to be between 3-12\% among different hyperfine levels. We derived a generalized expression for $E1_{PV}^{NSD}$, which helped incorporate both the NSD PV Hamiltonian and E1 operator simultaneously in the perturbative approach to account for the DCP contributions. The RCC method subsumes the CPDF-RPA and DCP effects in addition to contributions from the Br\"uckner pair-correlations and normalization of the wave functions, and correlations among them. To improve accuracy of the $E1_{PV}^{NSD}$ amplitudes further, we replace the {\it ab initio} values of the E1 matrix elements and energies by their experimental values via a sum-over-states approach.