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 $$\mathrm{H=}\frac{1}{2}\hspace{0.17em}{\displaystyle \sum _{\mathrm{ijn}}}\hspace{0.17em}{\mathcal{I}}_{\mathrm{ij}}^{\mathrm{(n)}}({\mathbf{S}}_i·{\mathbf{S}}_j{)}^n$$ , where the exchange interactions ${\mathcal{I}}_{\mathrm{ij}}^{\mathrm{(n)}}$ are of finite range, cannot order in the sense that $${\mathrm{〈O}}_i\text{〉=0}$$ 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.