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Monte Carlo simulations are performed on the three-dimensional (3D) Isingmodel with the 2-1-4 layered perovskite structure as a minimal model forcheckerboard charge ordering phenomena in layered perovskite oxides. Due to theinterlayer frustration, only 2D long-range order emerges with a finitecorrelation length along the c axis. Critical exponents of the transitionchange continuously as a function of the interlayer coupling constant. Theinterlayer long-range Coulomb interaction decays exponentially and isnegligible even between the second-neighbor layers. Instead, monoclinicdistortion of a tetragonal unit cell lifts the macroscopic degeneracy to inducea 3D charge ordering. The dimensionality of the charge order inLa$_{0.5}$Sr$_{1.5}$MnO$_4$ is discussed from this viewpoint.Comment: 5 pages including 6 figures, with major changes including discussion  on charge ordering phenomena in layered perovskite oxide

Two-Dimensional Charge Order in Layered 2-1-4 Perovskite Oxides