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Quantum Heisenberg antiferromagnets on pyrochlore and checkerboard lattices in a strong external magnetic field are mapped onto hard-core lattice gases with an extended exclusion region. The effective models are studied by the exchange Monte Carlo simulations and by the transfer-matrix method. The transition point and the critical exponents are obtained numerically for a square-lattice gas of particles with the second-neighbor exclusion, which describes a checkerboard antiferromagnet. The exact structure of the magnon crystal state is determined for a pyrochlore antiferromagnet.
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Zhitomirsky et al. (2007) studied this question.
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