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Low Energy Effective Field Theories of Sp(4) Spin systems

We study the classical and quantum phase transitions of Sp(4) spin systems on three dimensional stacked square and triangular lattices. We present general Ginzburg-Landau field theories for various types of Sp(4) spin orders with different ground state manifolds such as CP(3), S^7/Z_2, Grassmann manifold G_{2,5}, G_{2,6} and so on, based on which the nature of the classical phase transitions are studied, and a global phase diagram is presented. The classical phase transitions close to quantum phase transitions toward spin liquid states are also discussed based on renormalization group (RG) flow. Our results can be directly applied to the simplest Sp(4) and SU(4) Heisenberg models which can be realized using spin-3/2 atoms and Alkaline earth atoms trapped in optical lattice.