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

Abstract

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 $g_s= {2\over S}\sqrt{d+{T_Λ\over 2SJ}}$ and $c_s=2JSa \sqrt{d+{T_Λ\over 2SJ}}$, where $T_Λ$ is a natural temperature scale for the reliability of the theory. The formulation can also be extended to other generalized coherent state path integrals.