Computer Simulations ofSU(N)⊗SU(N)Spin Systems in Two Dimensions

$\mathrm{SU}\mathrm{}\left(N\right)\mathrm{}\otimes \mathrm{SU}\mathrm{}\left(N\right)\mathrm{}$ spin systems in two dimensions are studied for $N=2, 3, \mathrm{and} 4$ on 10×10, 20×20, and 30×30 lattices with use of Monte Carlo methods. Variant forms of the action are considered. In addition to crossover phenomena, unexpected first-order transitions in multidimensional phase diagrams are discovered. These transitions are related to the presence of $Z_N$ vortices which experience short-range interactions. Contrasts with $\mathrm{SU}\mathrm{}\left(N\right)\mathrm{}$ gauge theories in four dimensions are drawn.