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A Statistical Model of Current Loops and Magnetic Monopoles

We formulate a natural model of current loops and magnetic monopoles for arbitrary planar graphs, which we call the monopole-dimer model, and express the partition function of this model as a determinant. We then extend the method of Kasteleyn and Temperley-Fisher to calculate the partition function exactly in the case of rectangular grids. This partition function turns out to be a square of the partition function of an emergent monomer-dimer model when the grid sizes are even. We use this formula to calculate the local monopole density, free energy and entropy exactly. Our technique is a novel determinantal formula for the partition function of a model of vertices and loops for arbitrary graphs.

preprint2015arXivOpen access
Arvind Ayyer
math.COcond-mat.stat-mechmath-phmath.PRmath.MP
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Arvind Ayyer

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