  Quasi-one-dimensional correlated electronic multi-orbital systems with either
ladder or chain geometries continue attracting considerable interest due to
their complex electronic phases arising from the interplay of the hopping
matrix, the crystal-fields splitting, the electronic correlations, and strong
quantum fluctuations. Recently, the intriguing cobalt zigzag chain system
BaCoTe$_2$O$_7$, with electronic density $n = 7$, was prepared experimentally.
Here, we systematically study the electronic and magnetic properties of this
quasi-one-dimensional compound from the theory perspective. Based on
first-principles density functional theory calculations, strongly anisotropic
one-dimensional electronic Co $3d$ bands were found near the Fermi level. By
evaluating the relevant hopping amplitudes, we provide the magnitude and origin
of the nearest-neighbor (NN) and next nearest-neighbor (NNN) hopping matrices
in BaCoTe$_2$O$_7$. With this information, we constructed a three-orbital
electronic Hubbard model for this zigzag chain system, and studied two cases:
with only a NN hopping matrix, and with NN plus NNN hopping matrices.
Introducing the Hubbard and Hund couplings and studying the model via the
density matrix renormalization group method, we constructed the ground-state
phase diagram. A robust staggered antiferromagnetic (AFM) region was found when
only the NN hopping matrix in the chain direction was employed. However, for
the realistic case where the NNN hopping matrix is also included, the dominant
state becomes instead a block AFM order, in agreement with experiments. The
system displays Mott insulator characteristics with three half-filled orbitals,
when the block AFM order is stable. Our results for BaCoTe$_2$O$_7$ provide
guidance to experimentalists and theorists working on this zigzag
one-dimensional chain and related materials.
