“…Indeed, a Heisenberg model of the zigzag ladder predicts degenerated ground states, including nematic, 3,4 topological, 5 chiral, 6 incommensurate magnetic orders, 7 the spin-Bose metal, 8 and the field-induced orbital current. 9 In the presence of the orbital degrees of freedom with the moderate electron correlation and the electron-phonon coupling, various phase transitions have been found, such as the valence bond 10,11 and ferromagnetic 12 orders in hollandites A 2 M 8 O 16 , and the antiferromagnetic metal 13 and insulator 14 in pyroferites AM 2 O 4 , where A are alkaline metals and M are transition metals. To understand these phase transitions, the electron correlation and the orbital occupancy would be the dominant parameters, which should be revealed experimentally.…”