We have precisely determined the ground state phase diagram of the quantum spin-1 bilinear-biquadratic Heisenberg model on the honeycomb lattice using the tensor renormalization group method. We find that the ferromagnetic, ferroquadrupolar, and a large part of the antiferromagnetic phases are stable against quantum fluctuations. However, around the phase where the ground state is antiferroquadrupolar ordered in the classical limit, quantum fluctuations suppress completely all magnetic orders, leading to a plaquette order phase which breaks the lattice symmetry but preserves the spin SU(2) symmetry. On the evidence of our numerical results, the quantum phase transition between the antiferromagnetic phase and the plaquette phase is found to be either a direct second order or a very weak first order transition.

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Control of nutrients as nitrogen and phosphorus after discharge into lakes is necessary since it is difficult and costly to control within wastewater plants currently in China. This paper studied the cycling of phosphorus and nitrogen with water and sediments from two lakes in China. It is found that oxygen plays a critical role in regulating phosphorus and nitrogen cycling within water and sediments. Three different oxygenation methods including aeration, calcium peroxide and hydrogen peroxide were studied to control phosphorus and nitrogen in overlying water. In anoxic conditions, the P concentration in water increased from an average 14 microg/L to 115.2 microg/L for Xili Lake, and from an average 24 microg/L to 1,000 microg/L for Jinchun Lake. The concentration of ammonia increased under anoxic conditions, while the concentration of nitrate increased under oxic conditions. In anoxic conditions, the nitrate concentration decreased probably through denitrification. Both N and P accumulation processes can be controlled under the three treatments. The phosphorus removal efficiency from the water body was in the order of CaO2 addition > aeration > H2O2 addition, while controlling effectiveness for ammonia was in the order of aeration > CaO2 addition > H2O2 addition.