An electric-field representation of the harmonic XY model

Michael F. Faulkner, Steven T. Bramwell, Peter C.W. Holdsworth

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The two-dimensional harmonic XY (HXY) model is a spin model in which the classical spins interact via a piecewise parabolic potential. We argue that the HXY model should be regarded as the canonical classical lattice spin model of phase fluctuations in two-dimensional condensates, as it is the simplest model that guarantees the modular symmetry of the experimental systems. Here we formulate a lattice electric-field representation of the HXY model and contrast this with an analogous representation of the Villain model and the two-dimensional Coulomb gas with a purely rotational auxiliary field. We find that the HXY model is a spin-model analogue of a lattice electric-field model of the Coulomb gas with an auxiliary field, but with a temperature-dependent vacuum (electric) permittivity that encodes the coupling of the spin vortices to their background spin-wave medium. The spin vortices map to the Coulomb charges, while the spin-wave fluctuations correspond to auxiliary-field fluctuations. The coupling explains the striking differences in the high-temperature asymptotes of the specific heats of the HXY model and the
Coulomb gas with an auxiliary field. Our results elucidate the propagation of effective long-range interactions throughout the HXY model (whose interactions are purely local) by the lattice electric fields. They also imply that global spin-twist excitations (topological-sector fluctuations) generated by local spin dynamics are ergodically excluded in the low-temperature phase. We discuss the relevance of these results to condensate physics.
Original languageEnglish
Article number085402
JournalJournal of Physics Condensed Matter
Issue number8
Early online date6 Jan 2017
Publication statusPublished - 1 Mar 2017


  • generalized lattice electrostatics
  • two-dimensional harmonic XY model
  • two-dimensional Coulomb gas
  • Berezinskii-Kosterlitz-Thouless phase transition
  • twodimensional condensates
  • ergodicity breaking


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