A Statistical Model of Current Loops and Magnetic Monopoles

Ayyer, Arvind (2015) A Statistical Model of Current Loops and Magnetic Monopoles. In: MATHEMATICAL PHYSICS ANALYSIS AND GEOMETRY, 18 (1).

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Abstract

We formulate a natural model of loops and isolated vertices for arbitrary planar graphs, which we call the monopole-dimer model. We show that the partition function of this model can be expressed 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 a polynomial with positive integer coefficients when the grid lengths are even. Finally, we analyse this formula in the infinite volume limit and show that the local monopole density, free energy and entropy can be expressed in terms of well-known elliptic functions. Our technique is a novel determinantal formula for the partition function of a model of isolated vertices and loops for arbitrary graphs.

Item Type:	Journal Article
Publication:	MATHEMATICAL PHYSICS ANALYSIS AND GEOMETRY
Publisher:	SPRINGER
Additional Information:	Copy right for this article belongs to the SPRINGER, VAN GODEWIJCKSTRAAT 30, 3311 GZ DORDRECHT, NETHERLANDS
Keywords:	Monopole-dimer model; Dimer model; Determinantal formula; Kasteleyn orientation; Partition function; Free energy; Entropy
Department/Centre:	Division of Physical & Mathematical Sciences > Mathematics
Date Deposited:	19 Jul 2015 09:42
Last Modified:	19 Jul 2015 09:42
URI:	http://eprints.iisc.ac.in/id/eprint/51896

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