The mean-field quantum Heisenberg ferromagnet via representation theory

Alon, Gil; Kozma, Gady

doi:10.1214/20-aihp1067

Cited by 5 publications

(15 citation statements)

References 30 publications

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“…A key step in our working involves restricting irreducible characters of the Brauer algebra to the symmetric group, and then decomposing them into irreducibles of the symmetric group. Many other authors have approached similar problems with representation theory, for example, [2], [3], [5], [6], [15].…”

Section: Introductionmentioning

confidence: 99%

On a class of orthogonal-invariant quantum spin systems on the complete graph

Ryan

2020

Preprint

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We study a two-parameter family of quantum spin systems in spin 1 2 and 1 on the complete graph, which is equivalent in spin 1 2 to the XXZ model. We give an explicit formula for the free energy. The key tool is expressing the partition function in terms of the irreducible characters of the symmetric group and the Brauer algebra. The parameters considered include, and go beyond, those for which the systems have probabilistic representations as interchange processes.

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Section: Introductionmentioning

confidence: 99%

On a class of orthogonal-invariant quantum spin systems on the complete graph

Ryan

2020

Preprint

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“…Let us note that the present work follows a line of papers analysing the interchange process and Heisenberg model with algebraic methods (including the aforementioned [8], [9], [26]). Alon and Kozma [4] analysed the interchange process on a general graph, and estimated the number of k-cycles at a given time; Berestycki and Kozma [7] gave an exact formula for the same on the complete graph; Alon and Kozma [5] gave an exact formula for the magnetisation of the mean-field spin-1 2 Heisenberg model. In this work we carry the methods described above further, to inhomogeneous models on the complete graph where the coupling constants between different vertices take finitely many different values.…”

Section: Introduction and Resultsmentioning

confidence: 99%

“…Using the results mentioned above, in Section 1.4 we completely describe the gound-state phase diagram of the models; and in Section 1. 5 we give heuristic descriptions of the extremal Gibbs states and phase driagrams at finite temperature. At the end of the paper, in Section 5, we give the free energy for what we call multi-block models, where coupling constants can take any finite number of values, and where we allow certain many-body interactions.…”

Section: Introduction and Resultsmentioning

confidence: 99%

“…, n}. The form of the Hamiltonian (5) means that spins at two vertices within A interact with coupling constant a, spins at two vertices within B interact with coupling constant b, and the spin at a vertex in A interacts with the spin at a vertex in B with coupling constant c. In the homogeneous case a = b = c we obtain the interchange model on the complete graph (2), while if a = b = 0 and c = 0 we obtain a model on the complete bipartite graph K m,n−m .…”

Section: Introduction and Resultsmentioning

confidence: 99%

“…Let . Note that we have the scaling factor n in front of (191) rather than 1 n as in (5). This is because the sizes of the conjugacy classes C γ A depend on n, for example for transpositions we have |C…”

Section: Multi-block Modelsmentioning

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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Quantum Spins and Random Loops on the Complete Graph

Björnberg

Fröhlich

Ueltschi

2019

Commun. Math. Phys.

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We present a systematic analysis of quantum Heisenberg-, xy-and interchange models on the complete graph. These models exhibit phase transitions accompanied by spontaneous symmetry breaking, which we study by calculating the generating function of expectations of powers of the averaged spin density. Various critical exponents are determined. Certain objects of the associated loop models are shown to have properties of Poisson-Dirichlet distributions.

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The mean-field quantum Heisenberg ferromagnet via representation theory

Cited by 5 publications

References 30 publications

On a class of orthogonal-invariant quantum spin systems on the complete graph

On a class of orthogonal-invariant quantum spin systems on the complete graph

Heisenberg models and Schur--Weyl duality

Quantum Spins and Random Loops on the Complete Graph

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