Existence of N\'eel order in the S=1 bilinear-biquadratic Heisenberg model via random loops

We consider the general spin-1 SU(2) invariant Heisenberg model with a two-body interaction. A random loop model is introduced and relations to quantum spin systems is proved. Using this relation it is shown that for dimensions 3 and above N\'eel order occurs for a large range of values of the relative strength of the bilinear ($-J_1$) and biquadratic ($-J_2$) interaction terms. The proof uses the method of reflection positivity and infrared bounds. Links between spin correlations and loop correlations are proved.Comment: 24 pages, 5 figure

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

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.Comment: 15 pages, 1 figur

Phase transition for loop representations of Quantum spin systems on trees

We consider a model of random loops on Galton-Watson trees with an offspring distribution with high expectation. We give the configurations a weighting of $\theta^{\#\text{loops}}$. For many $\theta>1$ these models are equivalent to certain quantum spin systems for various choices of the system parameters. We find conditions on the offspring distribution that guarantee the occurrence of a phase transition from finite to infinite loops for the Galton-Watson tree.Comment: 16 pages, 1 figur

Correlation inequalities for the quantum XY model

We show the positivity or negativity of truncated correlation functions in the quantum XY model with spin 1/2 (at any temperature) and spin 1 (in the ground state). These Griffiths-Ginibre inequalities of the second kind generalise an earlier result of Gallavotti.Comment: 9 page

Henri Temianka Correspondence; (lees)

https://digitalcommons.chapman.edu/temianka_correspondence/2294/thumbnail.jp

Henri Temianka Correspondence; (lees)

https://digitalcommons.chapman.edu/temianka_correspondence/2285/thumbnail.jp

Henri Temianka Correspondence; (lees)

https://digitalcommons.chapman.edu/temianka_correspondence/2289/thumbnail.jp

Henri Temianka Correspondence; (lees)

https://digitalcommons.chapman.edu/temianka_correspondence/2293/thumbnail.jp

Henri Temianka Correspondence; (lees)

https://digitalcommons.chapman.edu/temianka_correspondence/2297/thumbnail.jp

Griffiths inequalities for the O(N)O(N)-spin model

We prove Griffiths inequalities for the $O(N)$-spin model with inhomogeneous coupling constants and external magnetic field for any $N\geq 2$. This is achieved by using a representation of $O(N)$-spins in terms of random paths that reduces to the random current representation of the Ising model for $N=1$ and an identity that is analogous to the switching lemma for random currents.Comment: 20 pages, 1 figure. Improvements to Lemma 3.1 and fixing of some typos. Comments welcome