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

Abstract

We look at the general SU(2) invariant spin-1 Heisenberg model. This family includes the well known Heisenberg ferromagnet and antiferromagnet as well as the interesting nematic (biquadratic) and the largely mysterious staggered-nematic interaction. Long range order is proved using the method of reflection positivity and infrared bounds on a purely nematic interaction. This is achieved through the use of a type of matrix representation of the interaction making clear several identities that would not otherwise be noticed. Using the reflection positivity of the antiferromagnetic interaction one can then show that the result is maintained if we also include an antiferromagnetic interaction that is sufficiently small. * b.lees@warwick.ac.uk

Existence of Néel Order in the S=1 Bilinear-Biquadratic Heisenberg Model via Random Loops

“…The dark blue region 0 ≤ J 1 ≤ 1 2 J 2 has nematic order at low temperatures [23], with Néel order on the line J 2 > 0, J 1 = 0. The adjacent dark yellow region has been proved to exhibit nematic order in high enough dimension [16]. Antiferromagnet order is expected here but is not yet proved.…”

“…Reflection positivity for this quadrant is already known; for J 1 < 0 = J 2 it was shown in [8] and for J 1 = 0 < J 2 one can see, for example, [16] lemma 3.4 for an explicit proof. It was proved in [8] that Néel order occurs for J 1 < 0 = J 2 , it is clear the result extends to a neighbourhood of the axis J 1 < 0 < J 2 with J 2 sufficiently small.…”

“…This representation can be used to show reflection positivity of the loop model, however in light of the famous paper of Dyson, Lieb and Simon [8] showing reflection positivity for J 1 < 0 = J 2 and the recent paper [16] showing reflection positivity for J 1 = 0 < J 2 we have reflection positivity for J 1 < 0 < J 2 more directly, as we shall see. where α, β, γ ∈ {1, 2, 3} and E αβγ is the Levi-Civita symbol.…”

“…The result can be proved directly using the loop model. However in light of the proof of reflection positivity for the J 2 interaction [16], we can complete the proof more directly, appealing to Proposition 5.3 when required. To begin we define a Hamiltoniañ…”

Ferromagnetism, antiferromagnetism, and the curious nematic phase ofS=1quantum spin systems

“…The system has actually Néel order in the direction J 1 = 0, J 2 > 0 [11]. Occurrence of quadrupolar order has recently been proved for J 2 > 0 and small negative J 1 in d 6 [12]; the system should have the stronger Néel order, though.…”