A Statistical Model of Current Loops and Magnetic Monopoles

Abstract

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.

17 pages, 5 figures, significant stylistic revisions. In particular, rewritten with a mathematical audience in mind. Numerous errors fixed. This is the final published version. Maple program file can be downloaded from the link on the right of this page

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• Boundary monomers in the dimer model
• The signed loop approach to the Ising model: foundations and critical point
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Cited by in corpus (2)

• A Pfaffian formula for monomer-dimer partition functions
• Pfaffian Correlation Functions of Planar Dimer Covers