Charge-dependent nucleon-nucleon interaction at N$^3$LO in nuclear lattice effective field theory

The nuclear lattice effective field theory (NLEFT) is an efficient tool for solving nuclear many-body problems, which takes high-fidelity lattice chiral interactions as input and computes nuclear low-energy observables via quantum Monte Carlo techniques. In this work, we present the first next-to-next-to-next-to-leading order (N$^3$LO) chiral forces on the lattice with the isospin-breaking effects fully taken into account. We focus on both the charge-independence breaking (CIB) and charge-symmetry breaking (CSB) effects. Specifically, we include the isospin-breaking effect from the mass difference between the charged and neutral pions in the one-pion-exchange potential (OPEP), the Coulomb force for the $pp$ interaction and the contribution of two additional charge-dependent contact operators. We also explicitly incorporate the two-pion-exchange potentials which was mostly neglected in previous NLEFT calculations. With these improvements, we are able to accurately reproduce the $np$ and $pp$ scattering phase shifts up to relative momentum $p \sim 200$ MeV as well as the deuteron properties. The construction of these charge-dependent lattice nuclear forces establishes a solid foundation for future high-precision nuclear ab initio calculations within the NLEFT framework.