  We explore the rich nature of correlations in the ground state of ultracold
atoms trapped in state-dependent optical lattices. In particular, we consider
interacting fermionic ytterbium or strontium atoms, realizing a two-orbital
Hubbard model with two spin components. We analyze the model in one-dimensional
setting with the experimentally relevant hierarchy of tunneling and interaction
amplitudes by means of exact diagonalization and matrix product states
approaches, and study the correlation functions in density, spin, and orbital
sectors as functions of variable densities of atoms in the ground and
metastable excited states. We show that in certain ranges of densities these
atomic systems demonstrate strong density-wave, ferro- and antiferromagnetic,
as well as antiferroorbital correlations.
