Abstract
We propose and analyze a generalized two dimensional XY model, whose interaction potential has n weighted wells, describing corresponding symmetries of the system. As the lattice spacing vanishes, we derive by \({\Gamma}\)-convergence the discrete-to-continuum limit of this model. In the energy regime we deal with, the asymptotic ground states exhibit fractional vortices, connected by string defects. The \({\Gamma}\)-limit takes into account both contributions, through a renormalized energy, depending on the configuration of fractional vortices, and a surface energy, proportional to the length of the strings. Our model describes in a simple way several topological singularities arising in Physics and Materials Science. Among them, disclinations and string defects in liquid crystals, fractional vortices and domain walls in micromagnetics, partial dislocations and stacking faults in crystal plasticity.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Alberti G., Baldo S., Orlandi G.: Variational convergence of functionals of Ginzburg–Landau type. Indiana Univ. Math. J. 54(5), 1411–1472 (2005)
Alicandro R., Braides A., Cicalese M.: Phase and anti-phase boundaries in binary discrete systems: a variational viewpoint. Netw. Heterog. Media 1, 85–107 (2006)
Alicandro R., Cicalese M.: Variational analysis of the asymptotics of the XY model. Arch. Ration. Mech. Anal. 192(3), 501–536 (2009)
Alicandro R., Cicalese M., Ponsiglione M.: Variational equivalence between Ginzburg–Landau, XY spin systems and screw dislocations energies. Indiana Univ. Math. J. 60(1), 171–208 (2011)
Alicandro R., De Luca L., Garroni A., Ponsiglione M.: Metastability and dynamics of discrete topological singularities in two dimensions: a \({\Gamma}\)-convergence approach. Arch. Ration. Mech. Anal. 214(1), 269–330 (2014)
Alicandro R., De Luca L., Garroni A., Ponsiglione M.: Dynamics of discrete screw dislocations along glide directions. J. Mech. Phys. Solids 92, 87–104 (2016)
Alicandro R., Ponsiglione M.: Ginzburg–Landau functionals and renormalized energy: a revised \({\Gamma}\)-convergence approach. J. Funct. Anal. 266(8), 4890–4907 (2014)
Ambrosio L.: Existence theory for a new class of variational problems. Arch. Ration. Mech. Anal. 111(4), 291–322 (1990)
Ambrosio L., Braides A.: Functionals defined on partitions of sets of finite perimeter, II: semicontinuity, relaxation and homogenization. J. Math. Pures Appl. 69, 307–333 (1990)
Ambrosio L., Fusco N., Pallara D.: Functions of Bounded Variation and Free Discontinuity Problems. Clarendon Press, Oxford (2000)
Ball J.M., Bedford S.: Discontinuous order parameters in liquid crystal theories. Mol. Cryst. Liq. Cryst. 612(1), 1–23 (2015)
Ball J.M., Zarnescu A.: Orientability and energy minimization in liquid crystal models. Arch. Ration. Mech. Anal. 202(2), 493–535 (2011)
Bedford S.: Function spaces for liquid crystals. Arch. Ration. Mech. Anal. 219(2), 937–984 (2016)
Bethuel, F., Brezis, H., Hèlein, F.: Ginzburg–Landau Vortices, Progress in Nonlinear Differential Equations and Their Applications, vol. 13. Birkhäuser Boston, Boston (1994)
Bonnes L., Wessel S.: Half-vortex unbinding and Ising transition in constrained superfluids. Phys. Rev. B 85(9), 094513 (2012)
Braides A., Cicalese M.: Interfaces, modulated phases and textures in lattice systems. Arch. Ration. Mech. Anal. 223(2), 977–1017 (2017)
Braides A., Cicalese M., Solombrino F.: Q-tensor continuum energies as limits of head-to-tail symmetric spin systems. SIAM J. Math. Anal. 47(4), 2832–2867 (2015)
Braides A., Conti S., Garroni A.: Density of polyhedral partitions. Calc. Var. PDE 56, 28 (2017)
Brezis, H.: Degree theory: old and new. Topological nonlinear analysis, II (Frascati, 1995). In Matzeu, M., Vignoli, A. (eds.) Progress in Nonlinear Differential Equations and Their Applications, vol. 27, pp. 87–108. Birkhäuser Boston, Boston (1997)
Brezis H., Nirenberg L.: Degree theory and BMO: Part I: compact manifolds without boundaries. Sel. Math. (N.S.) 1(2), 197–263 (1995)
Caffarelli L.A., de la Llave R.: Interfaces of ground states in Ising models with periodic coefficients. J. Stat. Phys. 118, 687–719 (2005)
Carpenter D.B., Chalker J.T.: The phase diagram of a generalised XY model. J. Phys. Condens. Matter. 1, 4907–4912 (1989)
Ciarlet P.G.: The Finite Element Method for Elliptic Problems. North Holland, Amsterdam (1978)
De Luca L.: \({\Gamma}\)-convergence analysis for discrete topological singularities: the anisotropic triangular lattice and the long range interaction energy. Asymptot. Anal. 96(3–4), 185–221 (2016)
Dávila J., Ignat R.: Lifting of BV functions with values in S 1. C. R. Math. Acad. Sci. Paris 337(3), 159–164 (2003)
Federer, H.: Geometric Measure Theory, GrundlehrenMath. Wiss. 153, Springer, New York (1969)
Giusti, E.: Minimal Surfaces and Functions of Bounded Variation. Monographs in Mathematics, vol. 80. Birkhäuser Verlag, Basel (1984)
Goldman, M., Merlet, B., Millot, V.: A Ginzburg–Landau model with topologically induced free discontinuities (forthcoming paper)
Hull D., Bacon D.J.: Introduction to Dislocations. Butterworth-Heinemann, Oxford (2011)
Jerrard R.J., Soner H.M.: The Jacobian and the Ginzburg–Landau energy. Calc. Var. 14(2), 151–191 (2002)
Korshunov S.E.: Phase diagram of the modified XY model. J. Phys. C Solid State Phys. 19(23), 4427–4441 (1986)
Kosterlitz J.M., Thouless D.J.: Ordering, metastability and phase transitions in two-dimensional systems. J. Phys. C 6, 1181–1203 (1973)
Lebwohl P.A., Lasher G.: Nematic-liquid-crystal order—a Monte Carlo calculation. Phys. Rev. A 6(1), 426–429 (1972)
Lee D.H., Grinstein G.: Strings in two-dimensional classical XY models. Phys. Rev. Lett. 55(5), 541–544 (1985)
Lin F.H.: Some dynamical properties of Ginzburg–Landau vortices. Commun. Pure Appl. Math. 49(4), 323–359 (1996)
Longa L., Trebin H.-R.: Structure of the elastic free chiral nematic liquid crystals. Phys. Rev. A 39(4), 2160–2168 (1989)
Pang J., Muzny C.D., Clark N.A.: String defects in freely suspended liquid-crystal films. Phys. Rev. Lett. 69(19), 2783–2787 (1992)
Ponsiglione M.: Elastic energy stored in a crystal induced by screw dislocations: from discrete to continuous. SIAM J. Math. Anal. 39(2), 449–469 (2007)
Sandier, E., Serfaty, S.: Vortices in the Magnetic Ginzburg–Landau Model, Progress in Nonlinear Differential Equations and Their Applications, vol. 70. Birkhäuser Boston, Boston (2007)
Sandier E., Serfaty S.: Gamma-convergence of gradient flows with applications to Ginzburg–Landau. Commun. Pure Appl. Math. 57(12), 1627–1672 (2004)
Schön R., Uhlenbeck K.: Boundary regularity and the Dirichlet problem for harmonic maps. J. Differ. Geom. 18(2), 253–268 (1983)
Tchernyshyov O., Chern G.-W.: Fractional vortices and composite domain walls in flat nanomagnets. Phys. Rev. Lett. 95(19), 197204 (2005)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by L. Caffarelli
Rights and permissions
About this article
Cite this article
Badal, R., Cicalese, M., De Luca, L. et al. \({\Gamma}\)-Convergence Analysis of a Generalized XY Model: Fractional Vortices and String Defects. Commun. Math. Phys. 358, 705–739 (2018). https://doi.org/10.1007/s00220-017-3026-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-017-3026-3