Ising Model of the commensurate charge-density waves in under-hole-doped monolayer, bilayer, and trilayer cuprate superconductors

By generalizing the one-dimensional Hubbard model for electrons hopping between near-neighbor sites and repelling each other when two of them with opposite spins are on the same site to a square lattice in two dimensions appropriate for the cuprate superconductor Bi2201, and further to the two- and three-layer square lattices appropriate for Bi2212 and Bi2223, we found that the difference in the mean charge density on each site corresponds to an Ising spin on that site. The effective Ising interactions between electrons on neighboring sites are J and J’ for intralayer and interlayer neighboring sites respectively. When the magnitude of the interlayer effective coupling J’/k_BT approaches infinity, the effective intralayer near-neighbor interactions J respectively approach 2J and 3J for Bi2212 and Bi2223, strongly stabilizing the commensurate charge-density waves in the under-hole-doped regions of those high-temperature superconductors. These charge density waves exhibit d_x2-y2-wave orbital symmetry, which many workers have misinterpreted to be the symmetry of the superconducting state.