Dimerization in Quantum Spin Chains with O(n) Symmetry

Björnberg, Jakob E.; Mühlbacher, Peter; Nachtergaele, Bruno; Ueltschi, Daniel

doi:10.1007/s00220-021-04148-1

Cited by 2 publications

(4 citation statements)

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“…We now turn to the case of the complete bipartite graph, given by a = b = 0. By our comments below (10), the wb-model with a = b = 0, c = 1, has Hamiltonian unitarily equivalent to…”

Section: Introduction and Resultsmentioning

confidence: 98%

“…We next consider another two-block model but where the interaction "between" the blocks uses the operator Q defined in (1). We let (10)…”

Section: Introduction and Resultsmentioning

confidence: 99%

“…(See also [31], but beware that the predictions using Gell-Mann matrices there are most likely wrong. The corresponding one-dimensional spin chain has a different phase-diagram, exhibiting dimerization, see [1,10].) The biquadratic model (J 1 = 0, J 2 = 1) lies on the boundary of the nematic phase of that diagram, but actually belongs to a Néel-ordered (or antiferromagnetic) phase for bipartite graphs.…”

Section: Introduction and Resultsmentioning

confidence: 99%

“…Walled Brauer algebra: proof of Theorem 1.2. As noted above, our analysis of the model in (10) uses the walled Brauer algebra. We will now define this algebra, and collect some facts which allow us to approach a proof in a similar way to that of Theorem 1.1.…”

Section: 1mentioning

confidence: 99%

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Heisenberg models and Schur--Weyl duality

Björnberg¹,

Rosengren²,

Ryan³

2022

Preprint

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We present a detailed analysis of certain quantum spin systems with inhomogeneous (non-random) mean-field interactions. Examples include, but are not limited to, the interchange-and spin singlet projection interactions on complete bipartite graphs. Using two instances of the representation theoretic framework of Schur-Weyl duality, we can explicitly compute the free energy and other thermodynamic limits in the models we consider. This allows us to describe the phase-transition, the ground-state phase diagram, and the expected structure of extremal states. Contents 1. Introduction and results 1.1. Free energy 1.2. Phase transition and critical temperature 1.3. Correlations and magnetisation 1.4. Ground-state phase diagrams 1.5. Heuristics for extremal Gibbs states 1.6. Acknowledgements 2. Free energy and correlations 2.1. Interchange model: proof of Theorem 1.1 2.2. Walled Brauer algebra: proof of Theorem 1.2 2.3. Correlation functions: proof of Theorem 1.7 2.4. Magnetisation term: proof of Theorem 1.8 3. The phase-transition 3.1. Existence of a phase transition: proof of Proposition 1.3 3.2. Formulas for β c : proofs of Propositions 1.4 and 1.5 3.3. Form of the maximiser of F for c > 0 4. The ground-state phase diagram 4.1. Diagram for c > 0 4.2. Diagram for c < 0 5. Multi-block models Appendix A. The trace-inequality (75) Appendix B. Equivalence of Q i,j and P i,j in the wb-model References

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“…We now turn to the case of the complete bipartite graph, given by a = b = 0. By our comments below (10), the wb-model with a = b = 0, c = 1, has Hamiltonian unitarily equivalent to…”

Section: Introduction and Resultsmentioning

confidence: 98%

“…We next consider another two-block model but where the interaction "between" the blocks uses the operator Q defined in (1). We let (10)…”

Section: Introduction and Resultsmentioning

confidence: 99%