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The mean-field quantum Heisenberg ferromagnet via representation theory

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arxiv 1811.10530 v1 pith:MIZLJZZH submitted 2018-11-26 math.PR

Gil Alon , Gady Kozma This is my paper · ORCID
classification math.PR
keywords alphaferromagnetformulaheisenbergmagnetisationmean-fieldquantumrepresentation
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abstract

We use representation theory to write a formula for the magnetisation of the quantum Heisenberg ferromagnet. The core new result is a spectral decomposition of the function $\alpha_k 2^{\alpha_1+\dotsb+\alpha_n}$ where $\alpha_k$ is the number of cycles of length k of a permutation. In the mean-field case, we simplify the formula further, arriving at a closed-form expression for the magnetisation, which allows to analyse the phase transition.

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    Infinite loops in random loop models on bounded-degree graphs occur strictly later than infinite clusters in the associated Bernoulli percolation.

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