Quantum Heisenberg Antiferromagnets versus Nonlinear σ Model Without the Large S Limit

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In this letter, I develop a new topologically invariant coherent state path integral for spin systems, and apply it to the quantum Heisenberg model on a square lattice. As a result, the quantum nonlinear σ model for arbitrary values of spin can be directly obtained. The effective coupling constant and spin wave velocity are modified by gₛ= 2 S√d+TΛ 2SJ and cₛ=2JSa √d+TΛ 2SJ, where TΛ is a natural temperature scale for the reliability of the theory. The formulation can also be extended to other generalized coherent state path integrals.