Lattice formulation of the CPn−1 non-linear σ models

We set up a general lattice version of non-linear σ models defined on homogeneous spaces. We then apply this to the CP^n?1 models which are the correct extension of the SU(2) σ model to SU(N). We exhibit their “confinement” property: the elementary multiplets Z_α used to describe the system do not appear as physical particles but only as bound states $\text{Z}{}_{\mathrm{\alpha }}\text{λ}{}_{\mathrm{\alpha \beta }}{}^{\mathrm{a}}\text{Z}{}_{\mathrm{\beta }}$. The method enable us to examine the “θ vacua” in the strong coupling limit by using a “dilute loop” approximation. We discuss the effect of the low activation energy for instantons which means that on a lattice, topological number is not conserved.