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arXiv:1507.04942 · Communications in Mathematical Physics

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

Benjamin Lees

Published May 24, 2016• 6 citations
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Abstract

We consider the general spin-1 SU(2) invariant Heisenberg model with a two-body interaction. A random loop model is introduced and relation to quantum spin systems is proved. Using this relation it is shown that for dimensions 3 and above Néel 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.

DOI
10.1007/s00220-016-2656-1

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We consider the general spin-1 SU(2) invariant Heisenberg model with a two-body interaction. A random loop model is introduced and relation to quantum spin systems is proved. Using this relation it is shown that for dimensions 3 and above Néel 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.
Abstract. We describe random loop models and their relations to a family of quantum spin systems on finite graphs. The family includes spin 1 2 Heisenberg models with possibly anisotropic spin interactions and certain spin 1 models with SU(2)-invariance. Quantum spin correlations are given by loop correlations. Decay of correlations is proved in 2D-like graphs, and occurrence of macroscopic loops is proved in the cubic lattice in dimensions 3 and higher. As a consequence, a magnetic long-range order is rigorously established for the spin 1 model, thus confirming the presence of a nematic phase. Dedicated to Elliott Lieb on the occasion of his eightieth birthday Contents
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.
We investigate the existence of N\'eel long-range order in a spatially anisotropic antiferromagnetic Heisenberg model. As the anisotropical coupling ${J}_{\ensuremath{\perp}}$ increases from 000 to 1,1,1, the model interpolates between an infinite number of decoupled one-dimensional gapped spin chains and a two-dimensional gapless isotropic Heisenberg antiferromagnet. By applying a method based on reflection positivity [F. J. Dyson, E. H. Lieb, and B. Simon, J. Stat. Phys. 18,18,18, 335 (1978)], we determine the upper bound of the critical values of ${J}_{\ensuremath{\perp}}^{c}$ such that when ${J}_{\ensuremath{\perp}}>~{J}_{\ensuremath{\perp}}^{c},$ the ground state of this model has an antiferromagnetic long-range order. For the case of spin S=1,S=1,S=1, our result confirms and strengthens previous results derived by numerical calculations on finite-size samples, while for other values of S, our results are new.
We explore quantum and classical correlations along with coherence in the ground states of spin-1 Heisenberg chains, namely the one-dimensional XXZ model and the one-dimensional bilinear biquadratic model, with the techniques of density matrix renormalization group theory. Exploiting the tools of quantum information theory, that is, by studying quantum discord, quantum mutual information, and three recently introduced coherence measures in the reduced density matrix of two nearest neighbor spins in the bulk, we investigate the quantum phase transitions and special symmetry points in these models. We point out the relative strengths and weaknesses of correlation and coherence measures as figures of merit to witness the quantum phase transitions and symmetry points in the considered spin-1 Heisenberg chains. In particular, we demonstrate that, as none of the studied measures can detect the infinite-order Kosterlitz-Thouless transition in the XXZ model, they appear to be able to signal the existence of the same type of transition in the biliear biquadratic model. However, we argue that what is actually detected by the measures here is the SU(3) symmetry point of the model rather than the infinite-order quantum phase transition. Moreover, we show in the XXZ model that examining even single site coherence can be sufficient to spotlight the second-order phase transition and the SU(2) symmetry point.
By using the concept of negativity, we study entanglement in spin-one Heisenberg chains. Both the bilinear chain and the bilinear-biquadratic chain are considered. Due to the SU(2) symmetry, the negativity can be determined by two correlators, which greatly facilitate the study of entanglement properties. Analytical results of negativity are obtained in the bilinear model up to four spins and the two-spin bilinear-biquadratic model, and numerical results of negativity are presented. We determine the threshold temperature before which the thermal state is doomed to be entangled.