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We report the first detection of the dark matter distribution around Lyman break galaxies (LBGs) at high redshift through the cosmic microwave background (CMB) lensing measurements with the public Planck PR3 κ map. The LBG sample consists of 1 473 106 objects with the median redshift of z∼4 that are identified in a total area of 305 deg^{2} observed by the Hyper Suprime-Cam Strategic Survey Program survey. After careful investigations of systematic uncertainties, such as contamination from foreground galaxies and cosmic infrared background, we obtain the significant detection of the CMB lensing signal at 5.1σ that is dominated by 2-halo term signals of the LBGs. Fitting a simple model consisting of the Navarro-Frenk-White profile and the linear-bias model, we obtain the typical halo mass of M_{h}=2.9_{-2.5}^{+9.5}×10^{11} h^{-1} M_{⊙}. Combining the CMB lensing and galaxy-galaxy clustering signals on the large scales, we demonstrate the first cosmological analysis at z∼4 that constrains (Ω_{m0},σ_{8}). We find that our constraint on σ_{8} is roughly consistent with the Planck cosmology, while this σ_{8} constraint is lower than the Planck cosmology over the 1σ level. This study opens up a new window for constraining cosmological parameters at high redshift by the combination of CMB and high-z galaxies, as well as studying the interplay between galaxy evolution and large-scale structure at such high redshift, by upcoming CMB and optical and near-infrared imaging surveys.

A novel nonlinear $\ensuremath{\sigma}$ model method is proposed for the two-dimensional ${J}_{1}\mathrm{\text{\ensuremath{-}}}{J}_{2}$ model, which is extended to include plaquette-type distortion. The nonlinear $\ensuremath{\sigma}$ model is properly derived without spoiling the original spin degrees of freedom. The method shows that a single disordered phase continuously extends from a frustrated uniform regime to an unfrustrated distorted regime. By the continuity and Oshikawa's commensurability condition, the disordered ground states for the uniform ${J}_{1}\mathrm{\text{\ensuremath{-}}}{J}_{2}$ model are plaquette states with fourfold degeneracy.

A novel nonlinear sigma model method is proposed for the two-dimensional J1-J2 model, which is extended to include plaquette-type distortion. The nonlinear sigma model is properly derived without spoiling the original spin degrees of freedom. The method shows that a single disordered phase continuously extends from a frustrated uniform regime to an unfrustrated distorted regime. By the continuity and Oshikawa's commensurability condition, the disordered ground states for the uniform J1-J2 model are plaquette states with fourfold degeneracy.