Absence of ordering in certain isotropic systems

The method of Mermin and Wagner [Phys. Rev. Lett. 17, 1133 (1966)] is used to show that one- and two-dimensional spin systems interacting with a general isotropic interaction H=1/2 Σ _ijn ℐ_ij ⁽ⁿ⁾ (S_i·S_j)ⁿ, where the exchange interactionsI_ij⁽ⁿ⁾ are of finite range, cannot order in the sense that 〈O_i〉=0 for all traceless operators O_i defined at a single site i. Mermin and Wagner have proved the above for the case n = 1 with O_i = S_i, i.e., for the Heisenberg Hamiltonian. The proof allows us to rule out the possibility that a small isotropic biquadratic exchange (S_i·S_j) ² could induce ferromagnetism or antiferromagnetism in a two-dimensional Heisenberg system.