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Here we have performed detailed first principles calculations for theelectronic structure and magnetic properties of Ba$_6$Cr$_2$S$_{10}$ to studythe origin of the anti-ferromagnetic exchange interaction between spins on Crions for the spin-chain compound Ba$_6$Cr$_2$S$_{10}$ synthesised recently.Most importantly, we have found the co-existence of a charge density wave phasealong one line and an anti-ferromagnetic spin chain along another. Thedimerization of sulfur atoms loosely bonded with Ba atoms drives the systeminto an insulating state owing to the formation of charge density wave.Meanwhile, the small size of the effective Hubbard $U$ parameter ($\sim 0.5$eV) due to electrostatic screening mainly accounts for the anti-ferromagneticground state. This co-existence equips us with a platform to tune the chargeand spin degrees of freedom independently. Moreover, there exists anext-nearest-neighbouring anti-ferromagnetic interaction along the chain, whichcould bring forward spin frustration and hence quantum spin liquid.

Co-existence of charge density wave and anti-ferromagnetic coupling in the spin-chain compound Ba$_6$Cr$_2$S$_{10}$