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We look at the general SU(2) invariant spin-1 Heisenberg model. This familyincludes the well known Heisenberg ferromagnet and antiferromagnet as well asthe interesting nematic (biquadratic) and the largely mysteriousstaggered-nematic interaction. Long range order is proved using the method ofreflection positivity and infrared bounds on a purely nematic interaction. Thisis achieved through the use of a type of matrix representation of theinteraction making clear several identities that would not otherwise benoticed. Using the reflection positivity of the antiferromagnetic interactionone can then show that the result is maintained if we also include anantiferromagnetic interaction that is sufficiently small.

Long-range order for the spin-1 Heisenberg model with a small antiferromagnetic interaction