A 1D$_s$$\times$1D$_c$ Heisenberg-Kondo Lattice compound Nb$_{12}$O$_{29}$

Local moments embedded in conducting systems form a rich platform for unusual phases, with phenomena including Kondo, heavy fermion, and non-Fermi liquid physics. Using first principles based methods and the refined crystal structure based on columns of 3$\times$4 planar units of NbO$_6$ octahedra, we determine that mixed valent Nb$_{12}$O$_{29}$ displays tightly bound local moments forming spin chains along one direction criss-crossed by conducting ``nanowires'' in the perpendicular direction. Just how local moments -- very rare for Nb -- emerge and coexist with itinerant electrons, an enigma for decades in this system, is elucidated based on the local structure of the NbO$_6$ octahedra and orbital+spin ordering. The resulting 1D$_s$$\times$1D$_c$ Heisenberg-Kondo lattice ($s$=spin, $c$=charge) picture will be discussed.