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Physical Review Letters

Absence of Ferromagnetism or Antiferromagnetism in One- or Two-Dimensional Isotropic Heisenberg Models

N. David Mermin, H. Wagner

Published Dec 26, 1966• 441 citations
Publisher
DOI
10.1103/physrevlett.17.1307

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It is rigorously proved that at any nonzero temperature, a one- or two-dimensional isotropic spin-SSS Heisenberg model with finite-range exchange interaction can be neither ferromagnetic nor antiferromagnetic. The method of proof is capable of excluding a variety of types of ordering in one and two dimensions.
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=12 ∑ ijn Iij(n) (Si·Sj)n,where the exchange interactionsIij(n) are of finite range, cannot order in the sense that 〈Oi〉=0for all traceless operators Oi defined at a single site i. Mermin and Wagner have proved the above for the case n = 1 with Oi = Si, i.e., for the Heisenberg Hamiltonian. The proof allows us to rule out the possibility that a small isotropic biquadratic exchange (Si·Sj)2 could induce ferromagnetism or antiferromagnetism in a two-dimensional Heisenberg system.