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Classical lattice spin models involving singular interactions isotropic in spin space

We address here a few classical lattice--spin models, involving $n-$component unit vectors ($n=2,3$), associated with a $D-$dimensional lattice $\mathbb{Z}^D,\,D=1,2$, and interacting via a pair potential restricted to nearest neighbours and being isotropic in spin space, i.e. defined by a function of the scalar product between the interacting spins. When the potential involves a continuous function of the scalar product, the Mermin--Wagner theorem and its generalizations exclude orientational order at all finite temperatures in the thermodynamic limit, and exclude phase transitions at finite temperatures when $D=1$; on the other hand, we have considered here some comparatively simple functions of the scalar product which are bounded from below, diverge to $+\infty$ for certain mutual orientations, and are continuous almost everywhere with integrable singularities. Exact solutions are presented for $D=1$, showing absence of phase transitions and absence of orientational order at all finite temperatures in the thermodynamic limit; for $D=2$, and in the absence of more stringent mathematical results, extensive simulations carried out on some of them point to the absence of orientational order at all finite temperatures, and suggest the existence of a Berezinski\vı-Kosterlitz-Thouless transition.