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Variational Analysis of the Asymptotics of the XY Model

Archive for Rational Mechanics and Analysis volume 192, pages 501–536 (2009)Cite this article

Abstract

In this paper we consider the XY (N-dimensional possibly anisotropic) spin-type model and, by comparison with a Ginzburg–Landau-type functional, we perform a variational analysis in the limit when the number of particles diverges. In particular we show how the appearance of vortex-like singularities can be described by properly scaling the energy of the system through a Γ-convergence procedure. We also address the problem in the case of long-range interactions and solve it in two dimensions.

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Authors and Affiliations

  1. DAEIMI, Università di Cassino, via Di Biasio 43, 03043, Cassino (FR), Italy

    R. Alicandro

  2. Dipartimento di Matematica e Applicazioni ‘R. Caccioppoli’, Università di Napoli, via Cintia, 80126, Napoli, Italy

    M. Cicalese

Authors
  1. R. Alicandro
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  2. M. Cicalese
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Correspondence to M. Cicalese.

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Communicated by G. Dal Maso

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Alicandro, R., Cicalese, M. Variational Analysis of the Asymptotics of the XY Model. Arch Rational Mech Anal 192, 501–536 (2009). https://doi.org/10.1007/s00205-008-0146-0

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  • DOI: https://doi.org/10.1007/s00205-008-0146-0

Keywords

  • Vortex
  • Variational Analysis
  • Continuum Limit
  • Integral Boundary
  • Landau Energy