Discretization of the nonlinear O(3) sigma model (original) (raw)

2016, arXiv: High Energy Physics - Theory

We derive a discretized version of the nonlinear O(3) sigma model through a discrete theory of complex analysis of Mercat. Adopting a simple two-dimensional rectangular lattice, we obtain a discrete energy E^{disc.} and a discrete area \cal{A}(f)^{disc.}, where a function f is related to a stereographic projection governed by a unit vector {\boldmath n} of the model. Both discrete energy and area satisfy the inequality E^{disc.} \ge |\cal{A}(f)^{disc.}|, which is saturated if and only if the function f is discrete (anti-)holomorphic. The vector {\boldmath n} parametrizing the function describes classical discrete (anti-)instanton solutions on the lattice. Two discrete instanton solutions are illustrated, which are derived from polynomials up to degree two in z. Except for a factor 2, the discrete energy and the area tend to the usual continuous energy E and the area \cal{A}(f)=4 \pi q, q \in \pi_2(S^2), respectively, as the lattice spacings tend to zero.

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