An electric-field representation of the harmonic XY model

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

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Abstract

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
Volume29
Issue number8
Early online date6 Jan 2017
DOIs
Publication statusPublished - 1 Mar 2017

Keywords

  • 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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